algebraic geometry and geometric modeling barcelona ... · stefka gueorguieva, pascal desbarats,...
TRANSCRIPT
Algebraic Geometry and Geometric Modeling
Barcelona, September 4-7 2006
Forewords
Algebraic Geometry and Geometric Modeling both deal with curves and surfaces generated by polynomialequations. AG investigates the theoretical properties of polynomial curves and surfaces; GM uses polynomial,piecewise polynomial, and rational curves and surfaces to build computer models of mechanical componentsand assemblies for industrial design and manufacture.
The aim of this workshop is to present new results, algorithms, developments or applications of effectivealgebraic geometry in Geometric Modeling. On the one hand, Algebraic Geometry has developed an impressivetheory targeting the understanding of geometric objects defined algebraically. On the other hand, GeometricModeling is using every day, in practical and difficult problems, virtual shapes based on algebraic models.Could these two domains benefit from each other? Recent and interesting developments in this direction areabout to convince us to answer yes. The workshop will try to reinforce the natural bridge which exists betweenthese two areas, expecting as a result, a better analysis of the key problems and of the related approaches.
Previous AGGM’s were held in Vilnius 2002 and Nice 2004. This edition is a Satellite Conference of theInternational Conference of Mathematicians (Madrid 2006).
Organization
Local organisation : Carlos D’Andrea, Martın SombraCo-organisators: : Laurent Buse, Ron Goldman, Laureano Gonzalez-Vega, Bernard Mourrain
Supporting Institutions
• IMUB (Institut de Matematica, Universitat de Barcelona)• Departament d’Algebra i Geometria, Universitat de Barcelona• Universidad de Cantabria• AIM@SHAPE, NoE IST 506766 (partner: INRIA, GALAAD)• “La Caixa” bank
Participants
Name Affiliation Country e-mailHesham AbdelmoezMohamed
Helwan University Egypt [email protected]
Abdallah Al-Amrani Universite Louis Pasteur France [email protected]
Abdolali Basiri Damghan University of Basic Sciences Iran [email protected]
Laurent Buse GALAAD, INRIA Sophia Antipolis France [email protected]
Fernando Carreras Universidad de Cantabria Spain [email protected]
Falai ChenUniversity of Scienceand Technology of China
China [email protected]
Laura Costa Universitat de Barcelona Spain [email protected]
Carlos D’Andrea Universitat de Barcelona Spain [email protected]
Ferran Espuny Universitat de Barcelona Spain [email protected]
Rida Farouki University of California at Davis USA [email protected]
Ignacio Fernandez Rua Universidad de Cantabria Spain [email protected]
Andre Galligo Universite de Nice Sophia Antipolis France [email protected]
Vıctor Gonzalez-Aguilera Universidad Santa Marıa Chile [email protected]
Laureano Gonzalez-Vega Universidad de Cantabria Spain [email protected]
Stefka Gueorguieva Universite de Bordeaux I France [email protected]
Miklos Hoffman Eszterhazy Karoly College Hungary [email protected]
Robert Joan-Arinyo Universitat Politecnica de Catalunya Spain [email protected]
Imre Juhasz University of Miskolc Hungary [email protected]
Bert Juttler Johannes Kepler University Austria [email protected]
Rimvydas Krasauskas Vilnius University Lithuania [email protected]
Oliver Labs Universitat des Saarlandes Germany [email protected]
Miroslav Lavicka University of West Bohemia Czech Republic [email protected]
Pedro Macias Marques Universidade de Evora Portugal [email protected]
Esmeralda Mainar Maza Universidad de Cantabria Spain [email protected]
Ana Marco Universidad de Alcala Spain [email protected]
Rosa Marıa Miro-Roig Universitat de Barcelona Spain [email protected]
Nicolas Montes Universidad Politecnica de Valencia Spain [email protected]
Marc Moreno Maza University of Western Ontario Canada [email protected]
Bernard Mourrain GALAAD, INRIA Sophia Antipolis France [email protected]
Boris Odehnal Technische Universitat Wien Austria [email protected]
Martin Peternell Technische Universitat Wien Austria [email protected]
Fernando San Segundo Universidad de Alcala Spain [email protected]
Josef SchichoJohann Radon Institute forComput. and Appl. Mathematics
Austria [email protected]
Juan Rafael Sendra Pons Universidad de Alcala Spain [email protected]
Zbynek Sır Johannes Kepler University Austria [email protected]
Martın Sombra Universitat de Barcelona Spain [email protected]
Luis Felipe Tabera Universidad de Cantabria Spain [email protected]
Carlo Traverso Universita di Pisa Italy [email protected]
M. Pilar Velez Universidad Antonio de Nebrija Spain [email protected]
Carlos Villarino Cabellos Universidad de Alcala Spain [email protected]
Julien Wintz GALAAD, INRIA Sophia Antipolis France [email protected]
Santiago Zarzuela Universitat de Barcelona Spain [email protected]
Contents
Part I Invited talks
1 Applications of µ-Bases in Geometric ModelingFalai Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Spatial Pythagorean Hodographs, Quaternions, and Rotations in R3 and R4
Rida T. Farouki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Basic Concepts for Geometric Constraint SolvingRobert Joan-Arinyo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Evolution-Based Fitting of Curves and SurfacesBert Juttler, Martin Aigner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 Surfaces with Rational Offsets and Their Blending ApplicationsRimvydas Krasauskas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6 Linear Systems of Plane CurvesJosef Schicho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7 Parametrizing Algebraic Curves Under Different Optimality CriteriaJ. Rafael Sendra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Part II Contributed talks
8 A New Method for Representation of Polyhedra of Surface MoleculesHesham Abdelmoez Mohamed, E. F. Alfred . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
9 Formulae for Arithmetic on non-Hyperelliptic CurvesAbdolali Basiri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
10 Improving the Computation of the Topology for an Arrangement of Plane CubicsJorge Caravantes, Laureano Gonzalez–Vega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
11 Using Symbolic and Numerical Techniques for Solving the Offset Sectioning Problem:The Implicit CaseFernando Carreras, Laureano Gonzalez–Vega, Jaime Puig–Pey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
12 Change of Ordering for Regular Chains in Positive DimensionXavier Dahan, Xin Jin, Marc Moreno Maza, Eric Schost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
13 Intersection Problem, Bivariate Resultant and Bernstein-Bezoutian MatrixMohamed Elkadi, Andre Galligo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6 Contents
14 Camera Self-Calibration with Planar MotionFerran Espuny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
15 A closed Formulae for the Separation of two Ellipsoids Involving Only Six PolynomialsLaureano Gonzalez-Vega, Esmeralda Mainar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
16 Composite Surfaces for Discrete Object Boundary ReconstructionStefka Gueorguieva, Pascal Desbarats, Remi Synave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
17 Geometric Aspects of Parametrization of Interpolating Bezier CurvesImre Juhasz, Miklos Hoffmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
18 Rational Parametrized Curves and Surfaces with Rational ConvolutionsMiroslav Lavicka, Bohumır Bastl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
19 An Alternative Representation of Helicoids and CatenoidsEsmeralda Mainar Maza, Juan Manuel Pena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
20 Implicitizing Rational Curves by Using Bernstein-Bezoutian MatricesAna Marco, Jose-Javier Martınez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
21 Approximating Clothoids by Bezier CurvesNicolas Montes, Josep Tornero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
22 Parallelization of Triangular DecompositionsMarc Moreno Maza, Yuzhen Xie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
23 The Geometry of FlagsBoris Odehnal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
24 Sphere-Geometric Aspects of Bisector SurfacesMartin Peternell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
25 Fast Computation of the Implicit Ideal of a HypercircleTomas Recio, J. Rafael Sendra, Luis Felipe Tabera, Carlos Villarino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
26 Subresultants and Implicit Equations of Rational CurvesIgnacio Fernandez Rua and Laureano Gonzalez–Vega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
27 Detecting Real Singularities of a Curve from a Rational ParametrizationR. Rubio, J.M. Serradilla, M. Pilar Velez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
28 Computing Minkowski Sums via Support Function RepresentationZbynek Sır . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
29 Morse theory, Mayer-Vietoris sequence and the Computation of the Topology ofHypersurfaces of dimension ≤ 3Carlo Traverso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
30 Subdivision Method for Computing an Arrangement of Implicit Planar CurvesJulien Wintz, Bernard Mourrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
31 An Evolution–Based Approach for the Approximate Parameterization of ImplicitlyDefined Curves by Parametric Spline CurvesHuaiping Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Part I
Invited talks
1
Applications of µ-Bases in Geometric Modeling
Falai Chen
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, [email protected]
This is a joint work with Wenping Wang, Jiansong Deng and Yang Liu.
Summary. The concept of a µ-basis was first introduced by Cox, Sederberg, and Chen in [7] in the case of parame-trized curves. It was then generalized to the rational ruled surface case by Chen, Zheng, Sederberg and Wang in [1, 4],and to the general rational surface case by Chen, Cox and Liu[5]. The µ-basis can be used to recover the parametricequation as well as to derive the implicit equation of a rational curve or surface. Thus it serves as a connection be-tween the parametric form and the implicit form of a rational curve or surface. In this talk, I will discuss some furtherapplications of µ-basis in geometric modeling. Specifically, I will talk about computing singular points and detectingimproper parametrization of planar rational parametric curves using µ-basis. Two conjectures regarding the singularpoints of a planar rational curve are proposed, and problems for further research are also discussed.
1.1 Introduction
LetP(t) = (a(t), b(t), c(t)) (1.1)
be a planar rational curve in homogenous form, where a(t), b(t), c(t) are relatively prime polynomials, themaximum degree of which is n.
A moving line L(x, y, w; t) is a family of lines with each parameter t corresponding to a line (Sederbergand Chen, 1995):
L(x, y, w; t) := A(t)x+B(t)y + C(t)w = 0. (1.2)
Sometimes we write a moving line in a vector form L(t) = (A(t), B(t), C(t)). A moving line is said to followthe rational curve (1.1) if
L(t) ·P(t) = A(t)a(t) +B(t)b(t) + C(t)c(t) ≡ 0. (1.3)
One important result about moving lines is the following
Proposition 1. (Cox, et al., 1998) There exist two moving lines p(t) and q(t) of degree µ (µ ≤ [n/2]) andn− µ respectively, such that
P(t) = κp(t)× q(t).
Here κ is some nonzero constant. p(t) and q(t) are called the µ-basis of the rational curve P(t) (Sometimes,we also call p(x, y, w; t) = p(t) ·X and q(x, y, w; t) = q(t) ·X the µ-basis, where X = (x, y, w)).
The µ-basis can be computed very efficiently (See[3] for an algorithm). The µ-basis has the followingproperties:
Proposition 2. (Chen and Wang, 2002)
1. The µ-basis forms a basis for the syzygy module syz(a, b, c) := (A,B,C) ∈ R[t]3 |Aa+Bb+ Cc ≡ 0..2. Let p(x, y, w; t) and q(x, y, w; t) be a µ-basis of the rational curve P(t). Then the implicit equation of P(t)
is given by the resultant of p and q with respect to t.
10 Falai Chen
By Proposition 2, the implicit equation of the rational curve P(t) can be written as the determinantof an (n − µ)× (n− µ) matrix, whereas the previous resultant technique writes the implicit equation as ann× n determinant. On the other hand, by Proposition 1, the µ-basis can recover the parametric equationof P(t) as well. Thus the µ-basis serves as a connection between the parametric form and the implicit formof a rational curve.
The concept of a µ-basis idea was subsequently generalized to rational ruled surfaces [1, 4] and arbi-trary rational parametric surfaces[5]. The µ-basis of a rational surface is defined to be three polynomialsp(x, y, z, s, t), q(x, y, z, s, t) and r(x, y, z, s, t) which are linear in x, y, z, and the intersection of the threeplanes p = 0, q = 0 and r = 0 gives exactly the parametric equation of the rational surface P(s, t). Theµ-basis can be used not only to recover the parametric equation but also to derive the implicit equation ofthe rational surface. Thus the µ-basis serve as a connection between the parametric form and the implicitform of a rational curve or surface.
Besides implicitization, µ-bases have found some other applications in geometric modeling. For example,the µ-basis provides a simple way to reparameterize a rational ruled surface [2]. As another application,the µ-basis can derive a simple inversion formula for a rational curve or surface. In this talk, I will presentmore applications of µ-bases in geometric modeling, specifically computing the singular points and detectingimproper parameterization of a planar rational curves.
1.2 Computing the singular points of planar rational curves
A singular point of a curve is a point on the curve where the tangent line of the curve is not unique. Thesingularities of a curve result in shape features known as cusps and self-intersections. Thus the singularitiescan help to determine the geometric shape and topology of curves, which has wide-ranging applicationsin computer aided geometric design and geometric modelling. There exist several algorithms to computesingular points of a planar rational curve in the literature. However, these algorithms either have to solve anon-linear system of equations or can’t find all the singular points and their multiplicities. Recently, a moredirect approach has been given by Chionh and Sederberg (2001). However, this approach suffers from seriousnumerical problem. Furthermore, it doesn’t explicitly provide the multiplicity of the singular points. In thispaper, we improve the method by Chionh and Sederberg to give an efficient and numerical stable algorithm tocompute the singular points of a planar rational curve. We start from the implicitization matrix derived fromthe µ-basis of the rational curve, and then compute the elementary divisors of the implicitization matrix.The elementary divisors contain all the information about the singular points, such as the parameter valuesof the singular points and their multiplicities. Based on the result, a numerically more stable algorithm isthen presented to compute the singular points. Furthermore, some relationship between the singular pointsand µ-basis is discussed; Inversion formula for the singular points are derived; a conjecture in (Chionh andSederberg, 2001) regarding the multiplicity of singular points is also proved.
To allow for t =∞, we use homogenous parameters (t : u) instead of t. Thus the curve P(t) becomes
P(t, u) = (a(t, u), b(t, u), c(t, u)), (1.4)
where a(t, u), b(t, u) and c(t, u) are homogenized from a(t), b(t) and c(t) by a(t/u)un, b(t/u)un and c(t/u)un
respectively. Similarly, the µ-basis p(t) and q(t) are homogenized to p(t, u) and q(t, u). We summarize themain results below.
Theorem 3. P0 = (x0, y0, w0) is an order r singular point of the rational curve P(t, u) if and only ifrank(B(x0, y0, w0)) = n− µ− r. Here matrix B(x, y, w) is the Bezout matrix of p(t, u) and q(t, u).
Thus to compute the singular points of a rational curve P(t, u), we have to explore the matrix
B(t, u) := B(a(t, u), b(t, u), c(t, u)). (1.5)
We still call B(t, u) the Bezout matrix or implicitization matrix derived from the µ-basis.
Theorem 4. Let Dk(t, u)Let be the determinant factor of order k. Then there exist homogeneous polynomialsdi(t, u), i = 2, . . . , n− µ such that
1 Applications of µ-Bases in Geometric Modeling 11
Dk(t, u) = dn−µ(t, u)kdn−µ−1(t, u)k−1 · · · dn−µ−k+2(t, u)
2dn−µ−k+1(t, u), (1.6)
k = 1, 2, . . . , n− µ− 1, and
rank(B(t, u)) = rank(diag(dn−µ(t, u), dn−µ(t, u)dn−µ−1(t, u), . . . , dn−µ(t, u) . . . d2(t, u))). (1.7)
di(t, u), i = 2, 3, . . . , n− µ are called the elementary divisors of B(t, u). Furthermore, P(t0, u0) is a singularpoint of order r if and only if dr(t0, u0) = 0 and di(t0, u0) 6= 0, i = r + 1, . . . , n− µ.
Remark 5. The elementary divisors of B(t, u) can be obtained by computing the Smith form of B(t, u).
If the singular points are ordinary, we have the following result– a conjecture proposed in (Chionh andSederberg 2001).
Theorem 6. Suppose all the singular points of P(t, u) are ordinary, and P(t0, u0) is a singular point oforder r such that t0 : u0 is a simple root of the inversion formula, then t0 : u0 is a root of Dn−µ−1(t, u) withmultiplicity r − 1.
Conjecture 7. Suppose P(t, u) has mr singular points of order r, in which mir singular points are the order r
neighborhood singular points from order i(i ≥ r) singular points. Then
dr(t, u) = hr(t, u)
n−µ∏
i=r
ψir(t, u),
where ψir(t, u) is a factor corresponding to the mi
r neighborhood singular points of order i, and deg(ψir(t, u)) =
rmir. hr(t, u) is the product of all the inversion formulas of order r singular points, and dr(t, u is the elementary
divisor.
For l = n − µ, n − µ − 1, . . . , 3, eliminate the common factors of dl(t, u) and di(t, u) from di(t, u), i =l − 1, . . . , 2. The modified elementary divisors are denoted by di, i = 2, . . . , n− µ.
Theorem 8. P(t0, u0) is a singular point of order r if and only if dr(t0, u0) = 0.
Conjecture 9. Suppose hr(t, u) =∏mr
i=1 hir(t, u), where hi
r(t, u) is the inversion formula for some order rsingular point. Then
dr(t, u) =
mr∏
i=1
hir(t, u)
li ,
where li, i = 1, 2, . . . ,mr are positive integers.
Now we describe the algorithm to compute the singular points. For simplicity, we use d(t, u) to denote themodified elementary divisor which determines the singular points of some order r. The number of singularpoints of order r ism. To simplify the computation, we use the reduced (or square-free) polynomial dred(t, u) ofd(t, u) instead of d(t, u) in the computation. Here dred(t, u) = d(t, u)/GCD(dt(t, u), du(t, u)) is the polynomialthat strips away the repeated factors of d(t, u).
The singular point (x, y, w) can be solved from the following system of equations
dred(t, u) = 0, c(t, u)x− a(t, u)w = 0, c(t, u)y − b(t, u)w = 0. (1.8)
Letf(x,w) := res(dred(t, u), c(t, u)x− a(t, u)w),
g(y, w) := res(dred(t, u), c(t, u)y − b(t, u)w).(1.9)
Solving f(x, 1) = 0 and g(y, 1) = 0 should give the x-coordinates and y-coordinates of the singular pointsrespectively (The case w = 0 can be treated similarly). Let xi, 1 ≤ i ≤ r and yj , 1 ≤ j ≤ s be the roots offred(x) = 0 and gred(y) = 0 respectively. For each point (xi, yj), 1 ≤ i ≤ r, 1 ≤ j ≤ s, we need further tocheck if it is a singular point on the curve or not. The implicit equation of the curve can be used to finishthe task.
12 Falai Chen
1.3 Detecting improper parameterization
Given a parametric equation of a planar curve in inhomogeneous form,
P(t) =
(a(t)
c(t),b(t)
c(t)
)(1.10)
it is called a proper parameterization if there is a one-to-one correspondence between parameter valuest and points on the curve (with the exception of singular points). Otherwise, it is called improper. Animproper paramterization of the rational curve P(t) can be reparameterized to a proper representation by aparameter substitution s = R(t). Here R(t) is a rational function of t. The problem of detecting improperparametrization for a rational curve can be stated as follows: given a rational parametric curve P(t), find aproper parametrization Q(s) and a rational function R(t) such that P(t) = Q(R(t)).
There exist several algorithms to compute a proper parameterization of a rational curve[8, 9]. Previousalgorithms generally need to compute GCD of several polynomials and to solve linear systems of equations.Here we provide an efficient approach to compute proper parameterization based on the µ-basis.
Theorem 10. Suppose p(t) = (a1(t), b1(t), c1(t)) and q(t) are a µ-basis of the rational curve P(t) withdeg p 6 deg q. Let P1(t) = (a1(t)/c1(t), b1(t)/c1(t)). If there exists a rational function R(t) such that P(t) =Q(R(t)), where Q(t) is proper parameterized, then there exists Q1(s) such that P1(t) = Q1(R(t)).
From the above theorem, we immediately have the following corollary.
Corollary 11. If gcd(deg p,deg P) = 1, then P(t) is proper parameterized.
Since 2 deg P1 6 deg P, it is more easily to find the rational function R(t) in P1(t) than in P(t). Thisfactorization process can be repeated until some low degree curve P(t) is reached.
Lemma 12. Given a rational curve P(t), which is improper parameterized with a rational function R(t)of degree n, P(t) (which usually has degree n) is the rational curve coming from the above factorizationprocedure. Then P(t) is also improper parameterized with R(t), and
1. deg P(t)|deg P(t), and2. rankM = 2, where P(t) = M(1, t, . . . , tn)T . Here M is called the coefficient matrix of P(t).
If P(t) and R(t) have the same degree, then we can find a candidate of R(t) as follows. Since rankM = 2,there exists a regular 3× 3 matrix N such that
M =
0bc
N.
Let b(t) = b(1, t, . . . , tn)T , c(t) = c(1, t, . . . , tn)T . Then we can select b(t)/c(t) as a candidate for parametersubstitution.
We also need to check the validity of the selected parameter substitution. This can be done by usingundetermined coefficient method proposed in [9].
References
1. Chen, F., Zheng, J. and Sederberg, T.W. (2001). The µ-basis of a rational ruled surface. Comput. Aided Geom.Design, 18, 61–72.
2. Chen, F. (2003). Reparametrization of a rational ruled surface using the µ-basis, Computer Aided GeometricDesign, 20, 11–17.
3. Chen, F. and Wang, W. (2002). The µ-basis of a planar rational curve-properties and computation, GraphicalModels, 65, 368–381.
4. Chen, F. and Wang, W. (2003). Revisiting the µ-basis of a rational ruled surface, J. Symbolic Computation,36(2003), 699A¡Aa716.
1 Applications of µ-Bases in Geometric Modeling 13
5. Chen, F., Cox, D. and Liu, Y. The µ-basis and implicitization of a rational parametric surface, J. SymbolicComputation, 2005.
6. Chionh, E.W. and Sederberg, T.W., 2001. On the minors of the implicitization matrix for a rational plane curve.Computer Aided Geometric Design 18, 21-36.
7. Cox, D., Sederberg, T.W. and Chen, F. (1998b). The moving line ideal basis of planar rational curves. Comput.Aided Geom. Design, 15, 803–827.
8. Gutierrez, J., Rubio, R., Sevilla, D., On multivariate rational decomposition. J. Symbolic Comput. Vol.33, 545–562, 2002.
9. Sederberg, T.W., Improperly parametrized rational curves, Computer Aided Geometric Design, Vol.3, 67–75,1986.
2
Spatial Pythagorean Hodographs,Quaternions, and Rotations in R3 and R4
Rida T. Farouki
Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616, [email protected]
Summary. Quaternions offer elegant characterizations of rotations in R3 and R4 that are fundamental to the con-struction and analysis of spatial Pythagorean–hodograph curves.
2.1 Algebra of quaternions
The quaternions are “four–dimensional numbers” of the formA = a+axi+ayj+azk and B = b+bxi+byj+bzk,where the basis elements 1, i, j, k satisfy the relations i2 = j2 = k2 = i j k = −1. Preserving the order ofterms in products, one may deduce that
i j = − j i = k , j k = − k j = i , k i = − i k = j . (2.1)
Thus, quaternion products are non–commutative: AB 6= BA in general (but they are associative). The sumof two quaternions is performed component–wise, and by relations (2.1) the product is
AB = (ab− axbx − ayby − azbz) + (abx + bax + aybz − azby) i
+ (aby + bay + azbx − axbz) j + (abz + baz + axby − aybx)k .
The notations of vector analysis offer a useful shorthand for quaternion operations. Considering i, j, k as aunit basis in R3, we regard A as comprising “scalar” and “vector” parts, a and a = axi + ayj + azk. WritingA = (a,a) and B = (b,b) the sum and product may be compactly expressed as
A + B = ( a+ b , a + b ) , AB = ( ab− a · b , ab + ba + a× b) .
Each quaternion A has a conjugate A∗ = (a,−a) and a magnitude defined by |A|2 = A∗A = a2 + |a|2. Theconjugates of quaternion products satisfy (AB)∗ = B∗A∗. We can also define an inverse for each quaternionA 6= 0 by A−1 = A∗/|A|, so that A−1A = AA−1 = 1.
2.2 Quaternions and spatial rotations
A is a unit quaternion if |A| = 1. Unit quaternions may be identified with points of the “3–sphere” definedin the Euclidean space R4 with coordinates (p, px, py, pz) by the equation
p2 + p2x + p2
y + p2z = 1 . (2.2)
Now if A and B are unit quaternions, C = AB is also unit, and identifies a point on (2.2). Thus, points on the3–sphere have the structure of a group1 under quaternion multiplication. Unit quaternions offer an elegant
1This is not true of the familiar “2–sphere” in R3. The only other generalized spheres whose points admit such agroup structure are the “0–sphere” in R1 (comprising the two points ±1) and the “1–sphere” in R2 (the unit circle).
2 Spatial Pythagorean Hodographs, Quaternions, and Rotations in R3 and R4 15
means of describing rotations in R3. They can be written as U = (cos 12θ, sin
12θ n) for some angle 1
2θ andunit vector n. If V = (0,v) is a pure vector, the quaternion product
U V U∗ = ( 0 , (n · v)n + sin θ n× v + cos θ (n× v)× n )
also defines a pure vector, corresponding to a rotation of v through angle θ about the axis n. Prior to therotation, v can be decomposed into components parallel and perpendicular to n as
v = (n · v)n + (n× v)× n .
The rotation leaves the parallel component unchanged, while the orthogonal component becomes sin θ n ×v + cos θ (n × v) × n. The correspondence between unit quaternions and rotations is not quite unique:U = (cos 1
2θ, sin12θ n) and −U = (− cos 1
2θ,− sin 12θ n) describe exactly the same rotation.
Any number of rotations can be replaced by a single “equivalent” rotation specified by a unique angleand axis. Quaternions provide an elegant illustration of this — applying consecutive rotations defined byU1 = (cos 1
2θ1, sin12θ1n1) and U2 = (cos 1
2θ2, sin12θ2n2) to V = (0,v) gives
U2 (U1V U∗1 ) U∗
2 .
Now since U∗1 U∗
2 = (U2 U1)∗ this can be written as U V U∗, where U = U2 U1. Thus, the outcome of applying
rotation U1 followed by rotation U2 is equivalent to a single rotation represented by U = U2 U1. The non–commutative nature, U2 U1 6= U1 U2, of quaternion products reflects the fact that the final outcome of asequence of rotations depends on the order in which they are applied. For the product U = (cos 1
2θ, sin12θ n)
of U2 = (cos 12θ2, sin
12θ2n2) and U1 = (cos 1
2θ1, sin12θ1n1) the equivalent angle θ and axis n for the compound
rotation are given (with cosα = n1 · n2 ) by
12θ = ± cos−1(cos 1
2θ1 cos 12θ2 − sin 1
2θ1 sin 12θ2 cosα) ,
n = ± sin 12θ1 cos 1
2θ2 n1 + cos 12θ1 sin 1
2θ2 n2 − sin 12θ1 sin 1
2θ2 n1 × n2√1− (cos 1
2θ1 cos 12θ2 − sin 1
2θ1 sin 12θ2 cosα)2
.
These formulae were obtained prior to Hamilton’s discovery of quaternions by Olinde Rodrigues.
2.3 Spatial Pythagorean hodographs
A polynomial curve r(t) = (x(t), y(t), z(t)) in R3 has a Pythagorean hodograph if the components of itsderivative r′(t) = (x′(t), y′(t), z′(t)) satisfy
x′2(t) + y′2(t) + z′2(t) = σ2(t) , (2.3)
for some polynomial σ(t). A sufficient–and–necessary condition [2] for satisfaction of (2.3) is that the hodo-graph components must be expressible in terms of four polynomials u(t), v(t), p(t), q(t) as
x′(t) = u2(t) + v2(t)− p2(t)− q2(t), y′(t) = 2[u(t)q(t) + v(t)p(t)], z′(t) = 2[v(t)q(t)− u(t)p(t)]
with σ(t) = u2(t) + v2(t) + p2(t) + q2(t). This form admits an elegant representation [1] as a pure vectorquaternion by writing
r′(t) = A(t) iA∗(t) . (2.4)
where we introduce the quaternion polynomial
A(t) = u(t) + v(t) i + p(t) j + q(t)k . (2.5)
Note that, for a given Pythagorean hodograph r′(t), the quaternion polynomial (2.5) is not unique. If Q isany quaternion satisfying
Q iQ∗ = i , (2.6)
16 Rida T. Farouki
then the quaternion polynomial defined by A(t) = A(t)Q generates the same hodograph, since
A(t) i A∗(t) = (A(t)Q) i (A(t)Q)∗ = A(t)Q iQ∗A∗(t) = A(t) iA∗(t) .
The solutions to (2.6) can be parameterized in terms of an angular variable φ as Q(φ) = cosφ+ sinφ i.An important feature of the form (2.4) is its rotation invariance [4]. Suppose r′(t) transforms to r′(t) =
U r′(t)U∗ under a rotation by angle θ about the vector n = (nx, ny, nz) — represented by the unit quaternionU = (cos 1
2θ, sin12θ n). Then the transformed hodograph can be written as
r′(t) = UA(t) iA∗(t)U∗ = UA(t) i [UA(t) ]∗ = A(t) i A∗(t) ,
where A(t) = UA(t). The rotated hodograph r′(t) has the Pythagorean form (2.4), with A(t) replaced byA(t). The components of A(t) are related to those of A(t) by
u
v
p
q
=
cos 12θ −nx sin 1
2θ −ny sin 12θ −nz sin 1
2θ
nx sin 12θ cos 1
2θ −nz sin 12θ ny sin 1
2θ
ny sin 12θ nz sin 1
2θ cos 12θ −nx sin 1
2θ
nz sin 12θ −ny sin 1
2θ nx sin 12θ cos 1
2θ
u
v
p
q
. (2.7)
The above 4 × 4 matrix is orthogonal and has determinant 1, and thus belongs to the group SO(4). Itrepresents a (special type of) four–dimensional rotation — see §2.5. The ambiguity arising from equation(2.6) incurs two free parameters in the construction of spatial PH quintic interpolants to Hermite data. Theoptimal selection of these parameters is an open problem.
2.4 Families of spatial rotations
Given two unit vectors in the plane, the problem of finding the rotation that maps one to the other is trivial.The equivalent problem in R3 is more subtle. The “obvious” solution is to rotate in the plane defined by thetwo vectors, in which case the solution to the planar problem holds — corresponding to motion along a greatcircle on the unit sphere. However, there is also a one–parameter family of spatial rotations that achieve thedesired result, for which the motion of one vector into the other departs from their common plane: thesecorrespond to motions along small circles of the unit sphere. Unit quaternions provide an elegant means ofcharacterizing such families of spatial rotations.
We choose coordinates such that the first vector coincides with i and the second vector v has an arbitraryorientation. We are interested in quaternion solutions U to the equation U iU ∗ = v that defines a spatialrotation of i into v. Writing U = u0 + uxi + uyj + uzk and v = λi + µj + νk, this equation is equivalent tothe system of three quadratic scalar equations
u20 + u2
x − u2y − u2
z = λ , 2(u0uz + uxuy) = µ , 2(uxuz − u0uy) = ν
in the four unknowns u0, ux, uy, uz (summing the squares on both sides gives (u20 + u2
x + u2y + u2
z)2 = λ2 +
µ2 + ν2 = 1, so the condition that U be a unit quaternion is automatically satisfied). Since we have threeequations in four unknowns, the solutions possess one degree of freedom:
U(φ) =√
12 (1 + λ)
(− sinφ + cosφ i +
µ cosφ+ ν sinφ
1 + λj +
ν cosφ− µ sinφ
1 + λk
).
Writing U = (cos 12θ, sin
12θ n) the rotation axis n = (nx, ny, nz) and angle θ must then satisfy
n2x(1− cos θ) + cos θ = λ , nxny(1− cos θ) + nz sin θ = µ , nznx(1− cos θ)− ny sin θ = ν .
With α = cos−1 λ, this has (for α ≤ θ ≤ 2π − α) the general solution
(nx, ny, nz) =±√
cos2 12α− cos2 1
2θ
sin 12θ
(1,
µ
1 + λ,
ν
1 + λ
)+
cos 12θ
sin 12θ
(0,−ν
1 + λ,
µ
1 + λ
),
This parameterizes the family of spatial rotations that map the unit vector i into another unit vector v byspecifying the rotation axis as a function n(θ) of the rotation angle θ, over the restricted domain θ ∈ [α, 2π−α ]where α is the angle between i and v (measured in their common plane).
2 Spatial Pythagorean Hodographs, Quaternions, and Rotations in R3 and R4 17
2.5 Four–dimensional rotations
Quaternions live in R4, a realm which admits possibilities that appear to defy our “common sense” geometricalintuition based on experience in R2 and R3. In R4 one discovers [7] that, for example, a sphere made of flexiblematerial may be turned inside out without tearing the material; a prisoner in a locked cell may escape withoutpenetrating its walls; and a knot in a string may be untied without moving its ends. Such possibilities arisefrom the extra “maneuvering freedom” in R4.
Quaternions can describe rotations in R4, as well as R3. Regarding a quaternion A as a vector in R4, ageneral rotation of it is specified [3, 6] using two unit quaternions U1, U2 by the map
A → U1A U∗2 . (2.8)
The correspondence between pairs of unit quaternions U1,U2 and rotations in R4 is not one–to–one, however,since the same rotation is defined by −U1,−U2. Special instances of (2.8) are the mappings A → UA andA → AU known as a right screw and left screw [3].
A rotation in R2 can be specified by a unit complex number eiθ and has one free parameter, the angle θ. Arotation in R3, being specified by a unit quaternion U = cos 1
2θ+ sin 12θ n, exhibits three degrees of freedom:
the rotation angle θ and direction cosines of the axis n. Since a general rotation in R4 is specified by two unitquaternions U1, U2 it has six degrees of freedom [3, 6]. To understand the geometrical significance of this, weneed to review some basic ideas from the geometry of R4 [7].
A line, plane, and hyperplane in R4 are point sets linearly dependent on two, three, and four points “ingeneral position” — alternately, they are the point sets satisfying three, two, and one linear equations inthe Cartesian coordinates of R4. A hyperplane in R4 is a copy of the familiar Euclidean space R3 and itdivides R4 into two disjoint regions: it is not possible to move from one to the other without crossing thehyperplane (as with a plane in R3, and a line in R2). The following incidence relations follow directly fromthese definitions: (1) two hyperplanes intersect in a plane; (2) three hyperplanes intersect in a line; (3) fourhyperplanes intersect in a point. Case (1) amounts to two linear equations in four unknowns, leaving twodegrees of freedom. Case (2) yields three linear equations in four unknowns, leaving one degree of freedom:equivalently, a plane and a hyperplane intersect in a line. Case (3) corresponds to four linear equations in fourunknowns, and thus admits a single point as its solution — “two planes intersect in a point” is an alternativephrasing.
Consider two planes Π1, Π2 in R4 with p as their intersection point. The planes are said to be absolutelyperpendicular if each line through p on Π1 is orthogonal to each line through p on Π2. This is a strictlyfour–dimensional phenomenon (in R4 it is possible to circumnavigate a plane without crossing it, just as itis possible to circumnavigate a line in R3 but not in R2). At each point p of a given plane in R4, there is aunique plane absolutely perpendicular to it.
A key geometrical feature of a rotation in Rn is its stationary set, i.e., the set of points that do not moveunder the rotation. In R2 the stationary set is a point (the center of the rotation), while in R3 it is a line (theaxis line of the rotation). These are simple rotations, characterized by the property that in Rn the stationaryset has dimension n − 2. Now in R4, one of two absolutely perpendicular planes may rotate on itself abouttheir common point while the other remains stationary. This is a simple rotation in R4 — the stationary set,the stationary axis plane, is of dimension n− 2.
However, a new possibility — a double rotation — arises in R4. This involves both of the planes absolutelyperpendicular to each other rotating on themselves about their common point. Such rotations of absolutelyperpendicular planes are commutative, i.e., the outcome is independent of the order in which they are per-formed, and the stationary set comprises the common point of the two absolutely perpendicular planes. Thesix parameters of a general (double) rotation in R4 can be understood as follows. Without loss of generality,take the origin of R4 as the common point of the absolutely perpendicular planes. We need specify only oneof these planes, and the other is then be uniquely determined. Each plane has a rotation angle associatedwith it, accounting for two parameters. The remaining four parameters specify one of the two planes: theplane is determined by two additional points, not collinear with the origin — each point has four coordinates,but also two freedoms of motion within the plane, so only four parameters are required to fix the plane.
Under a continuous rotation at angular speed ω in R2 or R3, every point (other than points of thestationary set) executes a periodic path — namely, a circle — and will return to its initial position, at timet = 0, every integer multiple of the motion period T = 2π/ω. In R4, however, we first encounter the strange
18 Rida T. Farouki
phenomenon of rotations incurring non–periodic motions of points. Consider a double rotation involvingangular velocities ω1 and ω2 about two absolutely perpendicular planes Π1 and Π2 with common point p. Asnoted above, these two rotations commute, and their angular velocities ω1 and ω2 are therefore completelyindependent. If a point in R4 is to return exactly to its initial position at t = 0, there must be a precisecoincidence of multiples jT1 = j2π/ω1 and kT2 = k2π/ω2 of the rotation periods for integers j and k, i.e., theangular velocity ratio must be a rational number, ω2/ω1 = j/k. If the ratio ω2/ω1 is irrational, the motionsof points in R4 induced by the double rotation are non–periodic — their paths are not closed curves!
References
1. H. I. Choi, D. S. Lee, and H. P. Moon (2002), Clifford algebra, spin representation, and rational parameterizationof curves and surfaces, Adv. Comput. Math. 17, 5–48.
2. R. Dietz, J. Hoschek, and B. Juttler (1993), An algebraic approach to curves and surfaces on the sphere and onother quadrics, Comput. Aided Geom. Design 10, 211–229.
3. P. Du Val (1964), Homographies, Quaternions, and Rotations, Clarendon Press, Oxford.4. R. T. Farouki, M. al–Kandari, and T. Sakkalis (2002), Structural invariance of spatial Pythagorean hodographs,
Comput. Aided Geom. Design 19, 395–407.5. R. T. Farouki, M. al–Kandari, and T. Sakkalis (2002), Hermite interpolation by rotation–invariant spatial
Pythagorean–hodograph curves, Adv. Comput. Math. 17, 369–383.6. P. Lounesto (1997), Clifford Algebras and Spinors, Cambridge University Press.7. H. P. Manning (1914), Geometry of Four Dimensions, Dover, New York.
3
Basic Concepts for Geometric Constraint Solving
Robert Joan-Arinyo
Escola Tecnica Superior d’Enginyeria Industrial. Univ. Politecnica de Catalunya, Diagonal 647, 8a, 08028 [email protected]
Summary. We survey the current state of the art in geometric constraint solving. Both 2D and 3D constraint solvingare considered, and different approaches are characterized.
3.1 Introduction
2D geometric constraint solving is arguably a core technology of Computer-Aided Design (CAD) and, byextension, of managing product design data. Since the introduction of parametric design by Pro/Enginyer inthe 1980s, every major CAD system has adopted geometric constraint solving into its design interface. Mostprominently, 2D constraint solving has became an integral component of sketchers on which most systemsbase feature design.
Beyond applications in CAD and, by extension, in manufacturing, geometric constraint solving is alsoapplicable in virtual reality, and is closely related in a technical sense to geometric theorem proving. Forsolution techniques, geometric constraint solving also borrows heavily from symbolic algebraic computationand matroid theory.
In this work, we review basic techniques that are widely available for solving 2D and 3D geometricconstraint problems. We focus primarily on the basics of 2D solving and touch lightly on spatial constraintsolving and the various ways in which geometric constraint solvers can be extended with relations, externalvariables and parameter value enclosures. These and other extensions and variants have been published in theliterature. You can download an extended manuscript from http://www.lsi.upc.edu/~ robert/AGGM06.pdf
It is recomended to the interested reader as follow-on material for study.
4
Evolution-Based Fitting of Curves and Surfaces
Bert Juttler and Martin Aigner
Institute of Applied Geometry, Johannes Kepler University, Altenberger Str. 69, A–4040 Linz, [email protected],[email protected]
Summary. Fitting of curves and surfaces to scattered data is a standard problem in Computer Aided GeometricDesign. Due the influence of the parameterization, it leads to a non-linear optimization problem. The talk will formulatethe problem as an evolution process, where a curve or surface is modified continuously and adapts itself to the givendata.
Fitting a curve or surface to a given set of unorganized points is an important problem in fields such asgeometric modeling and computer vision. It can be formulates as a non–linear optimization problem [6]. Inthe case of parametric curve and surfaces, this is due to the influence of the parameterization. In the case ofimplicitly defined curves and surfaces, it is due to the fact that the ‘algebraic distance’ is different from hetrue geometric distance.
Different approaches for dealing with the effects of this non–linearity have been developed, such as ‘pa-rameter correction’ or the use of quasi–Newton methods [1, 3, 6, 7, 8, 10, 11, 13, 15]. Clearly the choice of agood initial solution is of outmost importance for the success of the optimization. Geometrically motivatedoptimization strategies [8, 9, 10, 13, 15], where the initial solution is replaced by an initial curve and theformulation of the problem uses some geometric insights, may lead to more robust techniques.
Due to the iterative nature of the techniques for non–linear optimization, it appears to be natural toview the intermediate results as a time–dependent curve which tries to adapt itself to the target shapedefined by the unorganized point data [9, 13]. This is related to the concept of ‘active curves’ used for imagesegmentation in Computer Vision [12], and to so–called Level Set methods [4, 5], where shapes are definedby time–dependent discretizations of (approximations to) the signed distance function.
This talk presents a general framework for fitting an element of a given manifold of shapes (e.g., implicitlydefined or parametric curves or surfaces) to unorganized point data, by defining an evolution process whichgenerates a time–dependent shape. More precisely, we define a flow on the manifold of shapes, whose sta-tionary points correspond to (local) solutions of the curve or surface fitting problem. We analyze the relationof the new framework to existing Gauß–Newton methods and analyze the dependency of the results on thechoice of the parameters describing the manifold of shapes.
References
1. M. Alhanaty and M. Bercovier, Curve and surface fitting and design by optimal control methods, Computer–AidedDesign 33 (2001), 167–182
2. A. Blake and M. Isard, Active contours, Springer, 2000.3. J. Hoschek and D. Lasser, Fundamentals of Computer Aided Geometric Design, AK Peters, Wellesley Mass., 1996.4. S. Osher and J. Sethian, Fronts propagating with curvature dependent speed, algorithms based on a Hamilton-
Jacobi formulation, J. Comp. Phys. 79 (1988), 12–49.5. S. Osher and R. P. Fedkiw, Level set methods and dynamic implicit surfaces, Springer, 20036. D. F. Rogers and N. G. Fog, Constrained B-spline curve and surface fitting, Computer Aided Design 21 (1989),
641–648.7. B. Sarkar and C.-H. Menq, Parameter optimization in approximating curves and surfaces to measurement data,
Comp. Aided Geom. Design 8 (1991), 267–280
4 Evolution-Based Fitting of Curves and Surfaces 21
8. H. Pottmann and S. Leopoldseder, A concept for parametric surface fitting which avoids the parametrizationproblem. Comp. Aided Geom. Design 20 (2003), 343-362.
9. H. Pottmann, S. Leopoldseder, and M. Hofer. Approximation with active B-spline curves and surfaces. Proc.Pacific Graphics 2002, IEEE Press, 8–25.
10. H. Pottmann et al., Industrial geometry: recent advances and applications in CAD, Computer-Aided Design 37(2005), 751–766.
11. T. Speer, M. Kuppe, and J. Hoschek, Global reparametrization for curve approximation, Comput. Aided Geom.Design 15 (1998), 869–877.
12. M. Kass, A. Witkin and D. Terzopoulos, Snakes: active contour models. Int. J. Comp. Vision 1.4 (1987), 321–331.13. W. Wang, H. Pottmann and Y. Liu, Fitting B-spline curves to point clouds by squared distance minimization.
ACM Transactions on Graphics 25.2 (2006).14. H. Yang et al., Evolution of T-spline Level Sets with Distance Field Constraints for Geometry Reconstruction
and Image Segmentation, Shape Modeling International, IEEE Press, 2006.15. Z. Yang, J. Deng and F. Chen, Fitting unorganized point clouds with active implicit B-spline curves. The Visual
Computer 21 (2005), 831-839.
5
Surfaces with Rational Offsets and Their Blending Applications
Rimvydas Krasauskas
Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225 Vilnius, [email protected]
Summary. General theory of surfaces with rational offsets (PN-surfaces) is developed using dual Laguerre geometryand a universal rational parametrization of the Blaschke cylinder. A review about recent results on minimal rationalparametrizations of canal surfaces is presented and their applications to rolling ball blends between natural quadricsare discussed. The proposed approach allows us to construct similar blending solutions in the wider class of PN-surfacesthat have improved shape and lower parametrization degree.
5.1 Introduction
In current CAD systems curves and surfaces are represented in a standard NURBS form, i.e. they areparametrized by rational B-splines. Unfortunately, offsets of rational surfaces arising in practical applicationsare in general not rational and need to be approximated. On the other hand traditional 3d modeling primitiveslike natural quadrics (sphere, circular cylinder/cone) or torus surfaces admit rational offsets. According to[10] about 99% of mechanical parts can be represented by combinations of planes and natural quadrics withthe possibility of representing blends between them. Rolling ball blends are the most popular, since they haverather intuitive shape. In many cases the smoothness of the blend is more important than its exact shape.
This situation in CAD industry motivates us to search for rational surfaces with rational offsets that canserve as blending surfaces between natural quadrics.
A rational surface F (t, u) = (F1(t, u), F2(t, u), F3(t, u)) with a unit normal
N(t, u) =
(∂F
∂t× ∂F
∂u
)/∣∣∣∣∣∣∣∣∂F
∂t× ∂F
∂u
∣∣∣∣∣∣∣∣
is called a PN-surface (PN = Pitagorean Normal) if the normal N(t, u) is rational. Notice that an offset ofsuch PN-surface in the distance h can be parametrized as F (t, u) + hN(t, u) and it is rational as well.
5.2 Blaschke model of Laguerre geometry
The main idea of Laguerre geometry is to consider oriented spheres in R3 as points in R4. A sphere c witha center (x1, x2, x3) and radius x4 is mapped to a point ζ(c) = (x1, x2, x3, x4) ∈ R4. Each oriented plane e:e0 + e1x1 + e2x2 + e3x3 = 0, e21 + e22 + e23 = 1, in R3 is mapped to the hyperplane ζ∗(e) in (R4)∗, defined ase0 +e1x1 +e2x2 +e3x3 +x4 = 0, i.e. with homegeneous coordinates (e0, . . . , e4, 1). A Laguerre transformationof R4 is an affine transformation f(x) = λA(x) + b, where A is a linear transformation that preserves thepseudo-euclidean metrics 〈v, w〉 = v1w1 + v2w2 + v3w3 − v4w4. In particular, offsetting corresponds to thesimple Laguerre transformation – translation in the x4-axis direction.
Pottmann and Peternell [9] proposed to use the Blaschke model of Laguerre geometry for PN-surfacemodeling. The Blaschke map δ : (R4)∗ → R4 maps hyperplanes to points. For an appropriate coordinatesystem δ(ζ∗(e)) = (1, e1, . . . , e4, e0). Therefore the image is lying in a quadric x2
1 + x22 + x2
3 = 1 in R4 whichis called the Blaschke cylinder B ⊂ R4.
The main observation of [9] is the following 1–1 correspondence:
5 Surfaces with Rational Offsets and Their Blending Applications 23
duals of PN-surfaces in R3 ↔ rational surfaces in the Blaschke cylinder B (5.1)
The unit sphere S: x21 + x2
2 + x23 = x2
0 in RP 3 can be parametrized by homogeneous coordinates of CP 1:
PS(z0, z1) = (|z0|2 + |z1|2, 2Re(z0z1), 2Im(z0z1), |z0|2 − |z1|2). (5.2)
The map PS : C2 → R4 is homogeneous in the sense that PS(λz0, λz1) = |λ|2PS(z0, z1). Bezier curves andpatches on S can be lifted to C2 uniquely up to constant multiplier. Therefore PS is called a universal rationalparametrization of S (see [1, 5] for details).
From the projective point of view the Blaschke cylinder is a cone over a sphere, i.e. it is a toric threefold.Its universal rational parametrization is slightly more complicated (cf. [1]):
PB(s, d, z0, z1) = (s, 2dRe(z0z1), 2dIm(z0z1), d(|z0|2 − |z1|2), d(|z0|2 + |z1|2)). (5.3)
The map PB : R2 × C2 → R5 is homogeneous, PB(ρ|λ|2s, ρd, λz0, λz1) = ρ|λ|2PB(s, d, z0, z1), and can beuseful for studying rational surfaces in the Blaschke cylinder. According to (5.1), this is equivalent to studyingdual surfaces of NP-surfaces in R3.
5.3 Rational canal surfaces
A canal surface is an envelope of 1-parameter family of spheres in R3, defined by a spine curve s(t) =(s1(t), s2(t), s3(t)) and a radius function r(t). For the envelope to be real the condition on derivatives ‖s(t)‖2−r2(t) ≥ 0 is necessary. Let us collect this data in a curve γ(t) = (s(t), r(t)) ∈ R4 and denote the canal surfaceby Cγ . Peternell and Pottmann [8] proved that a canal surface Cγ defined by a rational curve γ of degree kcan be rationally parametrized with bidegree (5k − 6, 2).
Under a mild condition ‖s(t)‖2 − r2(t) > 0 (for example, in case of a pipe surface) the degree bound of[8] was be improved to (3k − 2, 2) [4], which is in general optimal. A parametrization algorithm of a canalsurface patch with given boundary curves was also presented in [4].
All these results enables us to develop a teory of rational variable radius rolling ball blends betweennatural quadrics.
Example 1. Let Qa and Qb be two cylinders defined by equations x21 + x2
2 = r21 and x12 + x2
3 = r22, where0 < r1 < r2. The conditions that a sphere touches both cylinders Qa and Qb define a quartic surface V ⊂ R4.Any curve on V define a canal surface touching both cylinders, i.e. a rolling ball blend. Unfortunately a fixedradius case corresponds to irrational curve on V . Nevertheless, a certain rational quartic curve γ ⊂ V canbe found [6]. This construction generates a canal surface Cγ of bidegree (10, 2) which is minimal possible [4].One can make this blending solution more flexible using quartic spline curves on the surface V [6], but stillit is impossible to construct a blending with a boundary circle on the cylinder Qa using this approach.
Any pair of natural quadrics in an arbitrary position can be similarly blended using rational canal surfaces[2].
5.4 New blendings between natural quadrics with PN-surfaces
The general scheme of the proposed method consists of three steps: (1) construct a Gaussian map; (2) definea support function; (3) convert from dual to point representation.
Here we illustrate the new blending method in the case of two cylinders considered in Example 1.
Example 2. Our goal is to generate a ring shaped PN-surface bounded by a circle Ca, x3 = h, on the verticalcylinder Qa and certain rational curve Cb (which will be determined later) on the upper side of the horizontalcylinder Qb (see Example 1).
Step 1. Normals along Ca and Cb define the following curves on the unit sphere: a circle C ′a on the equator
and a circular arc C ′b on the plane section x1 = 0. We do not fix endpoints of C ′
b yet: we keep them symmetricw.r.t. the plane x2 = 0. In order to build a symmetric gaussian map it remains to find a Bezier representation
24 Rimvydas Krasauskas
of the spherical quarter, x1, x3 ≥ 0, with the fixed quadratic and quartic parametrizations on the oppositeboundary curves C ′
a and C ′b, respectively. This is a standard situation where URP of a sphere can be used
as described in [5]. The result is in fact a unique such parametrization N(t, u) of bidegree (4, 2).Step 2. Combining Eq. (5.2), Eq. (5.3) and taking d = 1, the following tangent plane formula for a
PN-surface x = x(t, u) with the gaussian map N(t, u) has the following form (assume x4 = 0)
s(t, u) +N1(t, u)x1 +N2(t, u)x2 +N3(t, u)x3 +N0(t, u)x4 = 0. (5.4)
A support function sa(t, u) of a sphere touching the vertical cylinder along the circle Ca is derived from thesame Eq. (5.4) by substituting x = (0, 0, h, r1). The other substitution x = (0, 0, 0, r2) will define a supportfunction sb(t) of the horizontal cylinder along the curve Cb. Finally we define a support function for ourblending surface as s(t, u) = sa(t, u) + u2(sb(t)− sa(t, 1)).
Step 3. From the plane representation A + Bx1 + Cx2 + Dx3 = 0, with A = S(t, u), B = N1(t, u),C = N2(t, u), D = N3(t, u), we obtain point (x1, x2, x3) by solving the linear system
A+Bx1 + Cx2 +Dx3 = 0,At +Btx1 + Ctx2 +Dtx3 = 0,Au +Bux1 + Cux2 +Dux3 = 0.
If bidegree of (A,B,C,D) is (dt, du) then bidegree of the solution (x1, x2, x3) is (3dt − 2, 3du − 2) in general.Since (dt, du) = (4, 2) we can expect a solution of bidegree (10, 4). Now it is time to remember that we stillhave one free parameter which controls endpoints of the arc C ′
b. There exists a unique value of this parameterthat enables us to drop bidegree down to (6, 3). This is the minimal possible bidegree for Laguerre invariantsolution.
The proposed method in combination with Laguerre transformations allows us to build similar blend-ings between cylinders or cones with skew or non-perpendicular axes. Investigations of blendings in case ofarbitrary positions of natural quadrics are not finished yet.
References
1. Cox, D., Krasauskas, R., Mustata, M. (2003), Universal rational parametrizations and toric varieties, Topics inAlgebraic Geometry and Geometric Modeling, Contemporary Mathematics, Vol. 334, pp. 241–265.
2. Kazakeviciute, M. (2005), Blending of natural quadrics with rational canal surfaces, PhD Thesis, Vilnius Univer-sity.
3. Kazakeviciute, M., Krasauskas, R. (2000), Blending cylinders and cones using canal surfaces, Nonlinear Analysis:Modelling and Control. Vilnius, IMI, Vol. 5, pp. 77–89.
4. Krasauskas, R. (2006), Minimal rational parametrizations of canal surfaces, Computing, to appear.5. Krasauskas, R. (2006), Bezier patches on almost toric surfaces, Proc. of AGGM’04, Nice.6. Krasauskas, R., Kazakeviciute, M. (2005) Universal rational parametrizations and spline curves on toric surfaces,
in: Computational Methods for Algebraic Spline Surfaces, ESF Exploratory Workshop, Springer, pp. 213–231.7. Pottmann, H. (1993), Rational curves and surfaces with rational offsets, Computer Aided Geometric Design, Vol.
12, pp. 175–192.8. Peternell, M. and Pottmann, H. (1997), Computing rational parametrizations of canal surfaces, J. Symbolic
Computation, Vol. 23, pp. 255–266.9. Pottmann, H. and Peternell, M. (1998), Application of Laguerre geometry in CAGD, Computer Aided Geometric
Design, Vol. 15, pp. 165–186.10. Rossignac, J. R. (1987), Constraints in Constructive Solid Geometry, Proceedings of the 1986 workshop on Inter-
active 3D graphics, Chapel Hill, North Carolina, 1987, pp. 93 - 110.
6
Linear Systems of Plane Curves
Josef Schicho
Johann Radon Institute, Austrian Academy of Sciences, Linz, [email protected]
6.1 Abstract
A family of plane curves is called a linear system if their equations form a vector space. The classical Italianschool of algebraic geometry studied linear systems of plane curves from a birational point of view (e.g. in[4, 3, 6, 7]); this theory contains complete classifications of linear system of curves of genus up to 4, whichwere used as cornerstones in their theory of rational surfaces (see [5]), and re-stated in modern language by[8], who also gave rigorous proofs for many details. The revival of adjunction theory in the 1980’s brought newinsight, exploring the connections with Mori’s Minimal Model Program (see [1, 2] and the references there).Application are the estimation of the degree of the smallest possible parametrisation of a rational surfacein [9] (for the complex case) and [12] (for the general case), and the algorithm for simplifying parametricrational surfaces in [10, 11].
In this first part of the talk, we revisit the classical geometric theory of linear systems and their associatedrational maps, e.g. the relations between the birational invariants genus, self-intersection number, and genus(Theorem of Riemann-Roch). Then we introduce adjoint systems, as well as the level and keel of a linearsystem, and show how these tools can be used.
References
1. M. Andreatta, E. Ballico, and J. Wisniewski. Vector bundles and adjunction. Internat. J. Math., 3(3):331–340,1992.
2. Mauro C. Beltrametti and Andrew J. Sommese. The adjunction theory of complex projective varieties, volume 16of de Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin, 1995.
3. G. Castelnuovo. Un osservazione sul grado massimo dei sistemi lineari di curve piane algebriche. Annali di Mat.,2, 1890.
4. G. Castelnuovo. Ricerche generali sopra sui sistemi lineari di curve piane. In Memorie scelte, pages 137–187.Zanichelli, 1891.
5. F. Conforto. Le superficie razionali. Zanichelli, 1939.6. G. Jung. Ricerche sui sistemi lineari di curve algebriche di genere qualunque. Annali di Mat., 2:277–312, 1888.7. G. Jung. Un’ osservazione sul grado massimo dei sistemi lineari di curve piane algebriche. Annali di Mat.,
2:129–130, 1890.8. M. Nagata. Rational surfaces I + II. Mem. Coll. Sci. Kyoto, 32 and 33:351–370+271–293, 1960.9. J. Schicho. A degree bound for the parameterization of a rational surface. J. Pure Appl. Alg., 145:91–105, 1999.
10. J. Schicho. Simplification of surface parametrizations. In Proc. ISSAC 2002, pages 229–237. ACM Press, 2002.11. J. Schicho. Simplification of surface parametrizations – a lattice polygon approach. J. Symb. Comp., 36:535–554,
2003.12. J. Schicho. The parametric degree of a rational surface. Math. Z., 254:185–198, 2006.
7
Parametrizing Algebraic Curves Under Different OptimalityCriteria
J. Rafael Sendra∗
Departamento de Matematicas, Universidad de Alcala, Ap. de Correos 20, 28871 Alcala de Henares, Madrid, [email protected]
Summary. In this talk we plan to review some algorithmic methods to either parametrize or reparametrize (rationallyand globally) algebraic curves under different optimality criteria. More precisely, we will see how to compute rationalparametrizations of algebraic curves being optimal from the geometric, algebraic and normality point of view.
7.1 Introduction
Conversion algorithms (i.e. implicitization/parametrization algorithms) play an important role in many ma-nipulations of algebraic varieties. The implicitization problem can be approached by means of eliminationtheory techniques (see e.g. [1],[7]). However, the parametrization problem requires first to solve a decisionproblem, since not all algebraic varieties can be parametrized rationally.
A (rational) parametrization of an affine variety V ⊂ Fn, where we take F being an algebraically closed fieldof characteristic zero, is a n–tuple of rational functions P( t ) ∈ F(t1, . . . , ts)
n such that the set P( t ) | t ∈ Fsis dense in V , with the Zariski topology. If V has a rational parametrization, we say that V is unirational. Arational parametrization P( t ) induces a rational map
P : Fs −→ V ⊂ Fn : t 7−→ P( t ).
Note that because of this fact, a first requirement for being unirational is to be irreducible; nevertheless notall irreducible varieties are unirational. If this rational map P is birational, then the parametrization P( t ) iscalled proper and the variety V is called rational. Here a natural question arises: are the concepts of rationalityand unirationality equivalent? For curves over every field, and because of Luroth’s theorem, the answer isyes; for surfaces over algebraically closed fields the two notions are also the same (Castelnuovo’s theorem);however, in general, the equivalence is not true.
There exist algorithms for checking the rationality, and in fact computing a rational parametrization,for curves (see e.g. [10], [25]) and surfaces (see [16]). Nevertheless, even for these two cases, many otheradditional questions related to parametric representations appear. This is the central topic of the talk. Inorder to motivate the problem, let me propose the reader an easy example. Assume we are given the parabolaC defined over C by the equation y = x2 (note that, even though the parabola is defined over C, the implicitequation of C is expressed over the field Q of the rational number), and we are asked to provide a rationalparametrization of C. Probably, all of us will answer
P(t) = (t, t2).
Nevertheless, there exist many other possible parametrizations of C as for instance:
(a) (tn, t2n) with n ∈ N, n > 1,(b)(
1tn ,
1t2n
), with n ∈ N,
∗Supported by the Spanish “ Ministerio de Educacion y Ciencia” under the Project MTM2005-08690-C02-01 andby the “Direccion General de Universidades de la Consejerıa de Educacion de la CAM y la Universidad de Alcala”under the project CAM-UAH2005/053.
7 Parametrizing Algebraic Curves Under Different Optimality Criteria 27
(c) (it,−t2), where i2 = −1,(d) (√
2 t, 2t2),(e) (106t, 1012t2),(f) etc.
However, our first choice is indeed the best one. The parametrization (a) has higher degree and the inducedrational map is not injective. The parametrization (b), with n = 1, is as good as (t, t2) in the degree sense,and it also provides injectivity. However, the parametrization ( 1
t, 1
t2) does not cover all the parabola when
giving values to t; note that the origin is not reachable. Thus the induced rational map, although injective,is not surjective. Parametrization (b), with n > 1, is neither injective nor surjective. Option (c) has thementioned properties of (t, t2) but it involves complex coefficients, unnecessarily. Option (d), although withreal coefficients, needs a degree two field extension of Q. Option (e) is again as good as (t, t2) in terms ofinjectivity, surjectivity and field extensions, but now the integers involved in the parametrization are of muchhigher length, etc.
Depending on the different properties required to the parametrization, different optimality criteria appear:
• Properness Optimality: it requires injectivity, i.e. properness.• Normal Optimality: it requires surjectivity.• Algebraic Optimality: it requires to express the parametrization over the smallest possible field extension
of the ground field.• Arithmetic Optimality: when Q is a field of parametrization, it requires the smallest possible integer length
in the coefficients.• Reality questions: If the starting field is F = C, in practical applications one desires to have answers
expressed over R. How to decide whether this is possible? can we achieve the above optimality requirementsbut now over R?
In addition, when approaching the problem, two different formulations of the problem are considered. Thefirst one assumes that the variety is given by means of its implicit equations (Implicit Version). The secondone assumes that the input data is provided parametrically (Parametric Version), and one wants to solve theproblem without implicitizing the parametric representation.
For algebraic curves, most of these questions have been addressed satisfactorily, with the exception of thearithmetic optimality that (at least for my knowledge) remains unsolved. For surfaces, some achievementshave been done (see e.g. [5], [12], [17], [18], [19], [20]). In this talk, we plan to see how to solve some of thesequestions for the case of algebraic curves.
For further reading on these topic we refer to: for properness optimality [12], [21], [28]; for normal optimal-ity [2], [6], [8], [22]; for algebraic optimality (implicit version) [9], [11], [26]; for algebraic optimality (parametricversion) [3], [4], [14], [15], [23],[24]; for reality questions [27], [28] (implicit version), [13] (parametric version).
References
1. Adams, W.W., Loustaunau, P., (1994): An Introduction to Grobner Bases. AMS, Providence, RI, Graduate studiesin Mathematics 3.
2. Andradas, C., Recio (2006): Plotting missing points and branches of real parametric curves. Special issue onalgebraic curves of Applicable Algebra in Engineering, Communication and Computing (To appear).
3. Andradas, C., Recio, T., Sendra, J. R., (1997): A Relatively Optimal Rational Space Curves ReparametrizationAlgorithm through Canonical Divisors. Proc. ISSAC ’97, Kchlin W. (ed.): 349-356. ACM Press, New York.
4. Andradas, C., Recio, T., Sendra, J. R., (1999): Base Field Restriction Techniques for Parametric Curves. Proc.ISSAC ’99, Dooley S. (ed.): 17-22. ACM Press, New York.
5. Andradas, C., Recio, T., Sendra, J. R., (2004): La variedad de Weil para variedades unirracionales. ContribucionesMatematicas: Homenaje al Profesor Enrique Outerelo Domınguez. Editorial de la Universidad Complutense deMadrid, pp. 33-51.
6. Bajaj C.L., Royappa A.V. (1995): Finite Representation of Real Parametric Curves and Surfaces. InternationalJournal on Computational Geometry and Applications, vol. 5. no. 3, pp. 313-326.
7. Cox D., Little J. and O’Shea D. (1997): Ideals, Varieties, and Algorithms. Springer-Verlag, New York.8. Chou S.C., Gao. X.S, (1991): On the Normal Parametrization of Curves and Surfaces. International Journal on
Computational Geometry and Applications, vol. 1. no. 2, pp. 125-136.
28 J. Rafael Sendra
9. Hilbert, D., Hurwitz, A., (1890): Uber die Diophantischen Gleichungen vom Geschlecht Null. Acta math. 14:217-224.
10. van Hoeij, M., (1994): Computing Parametrizations of Rational Algebraic Curves. Proc. ISSAC ’94, von zurGathen J. (ed.): 187–190. ACM Press, New York.
11. van Hoeij, M., (1996): Rational parametrization of curves using canonical divisors. Journal of Symbolic Compu-tation 23: 209–227.
12. Perez–Dıaz S., (2006): On the Problem of Proper Reparametrization for Rational Curves and Surfaces. ComputerAided Geometric Design 23, pp. 307-323.
13. Recio, T., Sendra, J.R., (1997): Real Reparametrizations of Real Curves. J. Symbolic Computation 23: 241-254.14. Recio, T., Sendra, J.R., Tabera, L.F., Villarino C. (2006): Fast computation of the implicit ideal of a hypercircle.
In this volume.15. Recio, T., Sendra, J.R., Villarino, C., (2004): From hypercircles to units. Proc. Internat. Symp. on Symbolic and
Algebraic Computation: 258-265. J. Gutierrez (ed.). ACM Press.16. Schicho J. (1998): Rational Parametrization of Surfaces. Journal of Symbolic Computation 26: 1-9.17. Schicho J. (1998). Rational Parametrization of Real Algebraic Surfaces. Proc. ISSAC ’98, Gloor O. (ed), pp.
302–308, ACM Press, New York.18. Schicho J. (2000). Proper Parametrization of Real Tubular Surfaces. Journal of Symbolic Computation vol. 30,
pp. 583–593.19. Schicho J. (2000). Proper Parametrization of Surfaces with a Rational Pencil. Proc. ISSAC 2000, Traverso C.
(ed.), pp. 292–299, ACM Press, New York.20. Schicho J. (2002). Symplification of Surface Parametrizations. Proc. ISSAC 2002, Mora T. (ed.), pp. 229–237,
ACM Press, New York.21. Sederberg, T.W., (1986): Improperly parametrized rational curves. Computer Aided Geometric Design 3: 67–75.22. Sendra, J. R., (2002): Normal Parametrizations of Algebraic Plane Curves. Journal of Symbolic Computation 33:
863–885.23. Sendra, J. R., Villarino, C., (2001): Optimal Reparametrization of Polynomial Algebraic Curves. International
Journal of Computational Geometry and Applications 11, no. 4: 439–453.24. Sendra, J. R., Villarino C.,(2002): Algebraically Optimal Reparametrizations of Quasi-Polynomial Algebraic
Curves. Journal of Algebra and Its Applications 1, no. 1: 51–74.25. Sendra, J.R., Winkler, F., (1991): Symbolic Parametrization of Curves. Journal of Symbolic Computation 12:
607–631.26. Sendra, J.R., Winkler, F., (1997): Parametrization of Algebraic Curves over Optimal Field Extensions. Journal
of Symbolic Computation 23/2&3: 191–207.27. Sendra, J. R., Winkler, F., (1999): Algorithms for Rational Real Algebraic Curves. Fundamenta Informaticae 39,
no. 1–2: 211–228.28. Sendra J.R., Winkler F., (2001): Tracing Index of Rational Curve Parametrizations. Computer Aided Geometric
Design Vol. 18/8, pp. 771-795.
Part II
Contributed talks
8
A New Method for Representation of Polyhedra of SurfaceMolecules
Hesham Abdelmoez Mohamed1 and E. F. Alfred2
1 Physics and Engineering Mathematics Dept., Mataria Faculty of Engineering, Helwan University, Cairo, [email protected]
2 Physics and Engineering Mathematics Dept., Mataria Faculty of Engineering, Helwan University, Cairo, [email protected]
Summary. The representation of the Voronoi diagram for a set of points is given in this paper in order to achievethe polyherdra of molecules in the bulk. Using a periodic boundary conditions, the polyhedra of surface molecules canbe either included or eliminated. This construction is of interest in Astronomy, Biology, Chemistry, Materials science,as well as in Physics (with points representing atoms, molecules, ions, etc.). The present method is more efficient thanother procedures described in the literature.
Keywords: Voronoi Diagram, Computational Geometry, Computational Complexity, Divide and Conquer.
8.1 Introduction
Consider a physical system consisting of a number of distinct entities. Typically, the entities are molecules,but they can also be ions, atoms, polymer segments, radicals, and so on. We know that equilibrium and otherproperties of the system depend on spatial distribution of the entities, and the question is how to represent thisdistribution conveniently ? A method known for a long time, but which has recently been strongly increasingin popularity, consists in dividing the three-dimensional space between entities. Each entity ”owns” a certainportion of the space in the shape of a polyhedron. Thus, each physical entity is principally characterized bythe location of its geometrical center (to be shortly called center throughout this paper) and by the size andshape of the surrounding polyhedron.
The polyhedra in question were first defined by the mathematicians Dirichlet and Voronoi and thenrediscovered several times by physicists. Consequently, they are variously known as Dirichlet regions andVoronoi polyhedra.
Although procedures for the construction of the polyhedra were developed by several authors [2, 3, 4, 5,6, 7], they were not efficiently enough calculated. In these circumstances, we developed a new method, whichis described in the present paper.
In section 1 we define the basic notions. In section 2 we show how to construct the Voronoi diagram. Onceproposed, our method is compared in section 3 with earlier approaches. Section 4 contains some algebraicdetails pertinent for the users of our method. In the final section we discuss briefly the present and potentialapplications of the method.
8.2 Basic Definitions
Consider a set of points P1, P2, ..., Pn in L-dimensional Euclidean space E. The Voronoi Polyhedron Vi arounda given center Pi, is the set of points in E closer to Pi than to any Pj : more formally,
Vi = x ∈ E : d(x, Pi) ≤ d(x, Pj), j = 1, 2, ..., n. (8.1)
where d denotes distance. Thus, the polyhedra are intersections of half-spaces; they are convex but notnecessarily bounded. The polyhedra is a partition of E in a unique way. The set of Voronoi polyhedracorresponding to a given configuration of centers is called the Voronoi diagram.
32 Hesham Abdelmoez Mohamed and E. F. Alfred
For obvious physical reasons we consider mainly the case L = 3. Given a center Pi and its neighbor Pj ,the line PiPj is cut perpendicularly at its midpoint yij by the plane hij . We call Hij the half-space generatedby hij that consists of the subset of E on the same side of hij as Pi; that is,
Vi = IjHij (8.2)
Vi is bounded by faces, with each face fij belonging to a distinct plane hij . Each face is characterized bylisting its vertices and edges in cyclic order. It is pertinent to distinguish various possible kinds of neighborsof Pi. We define the following classes of neighbors :
• (i) direct neighbors : if yij belongs to Vi, then Pj is a direct neighbor.• (ii) indirect neighbors if a subset of hij is a factor of Vi but yij does not belong to Vi, then Pj is an
indirect neighbor; or Pj is an indirect neighbor if fij ∩ yij = φ.• (iii) Degenerate neighbors : if the intersection of hij and Vi is just a vertex or an edge, then Pj is a
degenerate neighbor.• (iv) quasi− direct neighbor : if Pj is a direct neighbor or if Pj would be a direct neighbor in the absence
of all indirect neighbors, then Pi is a quasi-direct neighbor of Pi.
Clearly, all direct neighbors are also quasi-direct. Examples of neighbors representing classes defined aboveare shown in Fig. 8.1 in two dimensions; extension to three dimensions is obvious. The quasi-direct neighborsgenerate a direct polyhedron D, in the same way that direct and indirect neighbors generate V. Clearly :
Vi ⊂ Di, for each i. (8.3)
We define the geometric coordination number fi as the number of nondegenerate direct or indirect neighbors
Fig. 8.1. Example of various classes of neighbors of a Voronoi polygon.
of Pi. The number fi has to be distinguished from the structural coordination number zi. The latter is definedin terms of the binary radial distribution function g(R), that is, in terms of the probability of finding anothermolecule at a distance R from a given molecule. The average value of f for random models seems to be 15,but Voronoi polyhedra with f = 20 have been constructed., and arbitrarily high values of f are possible [7].By contrast, if only appropriate integration limits are used in the evaluation of z, the highest value of z is 12in crystals and 11 in liquids [2].
8 A New Method for Representation of Polyhedra of Surface Molecules 33
We embed Pi in a large ”boundary” cube Ci, where the closest face of Ci to Pi is relatively far away fromthe farthest quasi-direct neighbor of Pi. Except for dealing with surface phenomena, it would be convenientto use a single cube C for the whole system. If a face of Ci cuts Di, we call Di virtually unbounded. Avirtually unbounded polyhedron may be either bound or unbounded. The appearance of virtually unboundedpolyhedra is intimately related to surface effects. Therefore, detection of the presence of these polyhedrais important. Our method finds them, and the corresponding Vi are not constructed. We believe that thismathematical minus is actually a physical plus in relating the Voronoi diagram to real physical systems.
It is worth noting that Ci can be very large. Extremely large Di, while mathematically possible, simply donot occur in Voronoi diagrams representing the interior of systems of interest in astronomy, biology, chemistry,or physics. For values of Di of realistic relative dimensions with respect to Ci, the size of Ci has little -if any-effect on the work of constructing the Voronoi diagram.
We also have the option of using a periodic boundary condition. This can be done as follows. Put the givenconfiguration of points in a box. Now put this box in the center of a stack of congruent boxes, each containingthe same configuration of points as the original. For each point in the center box, and only these, Voronoipolyhedra are constructed taking account of all points in all boxes. By definition of the periodic boundarycondition, the resulting collection of Voronoi polyhedra constitutes the Voronoi diagram. When this optionis taken, all Di in the original box are automatically bounded, and a boundary cube Ci is not needed.
8.3 Construction of the Polyhedra
There seems to be a growing consensus among users of geometry that many geometrical problems have to berevised and considered from the point of view of computing facilities now available. This is due to the factthat geometry developed and flourished in a period when fast algorithms were of little -if any- importance.In the present section we discuss these properties of the polyhedra which we found useful in the quest for anefficient procedure of constructing them.
We begin with a simple observation : ON the average, direct polyhedra D have simpler shapes, that is,fewer faces, than Voronoi polyhedra V. This and relation (3) motivate our key idea : begin by constructingthe polyhedron D, and only then proceed towards V. Given a center Pi and its bounded direct polyhedronDi, we can circumscribe a sphere of diameter di around Di (even when not explicitly stated, we use squareddistances to avoid calculating square roots). Then any point more than di away from Pi cannot be a neighbor;this simple criterion eliminates most of the candidates for indirect neighbors. Those which are not eliminatedserve to obtain Vi from Di; vertices, edges, and sometimes entire faces of Di are cut off by planes generatedby indirect neighbors. Note that by definition di is twice the distance from Pi to the farthest vertex of Di.
In many if not most geometrical problems one begins with locating vertices, then joins appropriate pairsof vertices by edges, and finally constructs faces or planes. Studies of random models [3, 4, 5] indicate that theaverage number of edges per face is ≥ 5. In constructing both Di and Vi we chose to find first the respectivefaces, from these to find the edges, and then finally to find the vertices. A careful analysis of the problem (cf.Section 5 and the end of Section 6) indicates that any other order would be much less efficient. A computerprocedure called FACFIN for finding faces of the direct polyhedra is shown in algorithm 1.
• 1. i← 1.• 2. Sort the squared distances d(Pi , yij) in increasing order. Let the indices of the sorted list be [1],
[2],...,[n-1], in that order.• 3. Fi ← [1].• 4. j ← 1.• 5. If y[j] is in Him for all m in Fi, set Fi ← Fi ∪ [1].• 6. If j = n-1, go to 7. Else, j ← j+1 and go to 5.• 7. If i = n, STOP, else, i ← i+1 and go to 2.
Algorithm 1: FACFIN computer procedure
We devised this procedure with the objective of satisfying:
Lemma 1. FACFIN terminates with Fi containing exactly the indices of the faces of Di .
34 Hesham Abdelmoez Mohamed and E. F. Alfred
Proof.We use induction. Clearly [1] is the index of a face. Assume that the algorithm is correct up to andincluding j-1. If y[j] is in Him for all m in Fi, then y[j] is in Di because it cannot be cut off by a planegenerated by a quasi-direct neighbor, since we consider the planes in the order specified in step 2. On theother hand, if y[j] is not in Him for some m in Fi, it is not a quasi-direct neighbor by definition.
As already noted, one checks for intersections of Di with the boundary cube Ci. If there are nonemptysuch intersections, Di, is eliminated as virtually unbounded. If Di remains, di is computed. A procedurewhich constructs and checks direct polyhedra called DIRPOL is shown in Algorithm 2. Given Di, one findsall indirect neighbors and proceeds towards Vi. A procedure which produces Vi called VORPOL is shown inAlgorithm 3.
• 1. i ← 1.• 2. With the set of edges associated with fi denoted by Ai and analogously for the vertices Sij , compute
: Ai = yj Aij , and Si = yj Sij .• 3. Given faces cij of Ci, sort the distances d(Pi, cij) in increasing order. Sort outer half-spaces Cij , which
contain each of the respective cij but not Pi, in the same order.• 4. t ← 1.• 5. If Cij ∩ Si 6= φ, set flag. (Di is virtually unbounded).• 6. If t = 6, go to 7; else t ← t+1 and go to 5.• 7. Compute di.• 8. If i = n, STOP. Else, i ← i + 1 and go to 2.
Algorithm 2: DIRPOL procedure to construct and check direct polyhedra
• 1. i ← 1.• 2. Number the faces of Di as 1, 2, ..., fd. For every center within di of Pi, construct its hij . Number the
new candidate faces fd + 1, ..., r.• 3. j ← fd + 1, Vi ← Di. (Note that Fi, Ai, Aij , Si, and Sij are initialized to the values produced by
FACFIN and DIRPOL except that Fi is the set of faces of Di, not just their indices).• 4. If hij cuts off one or more vertices of Si:• a. Vi ← Vi ∩Hij . (If fik remains but yik is cut off, then fik becomes indirect).• b. Fi ← fij ∪ (Fi ∩Hij).• c. Ai ← Aij ∪ (Ai ∩Hij).• d. Si ← Sij ∪ (Si ∩Hij).• 5. If j = fd + r, go to 6; else, j← j + 1 and go to 4.• 6. If i = n, STOP; else, i ← i + 1 and go to 2.
Algorithm 3: VORPOL procedure to produce Vj
Since efficiency and exactness is thought, we do not construct each polyhedron in the Voronoi diagram fromscratch. Results from the polyhedron already constructed are used, provided that there is sufficient computermemory to access this information quickly. Of course, if there is enough high-speed memory available in agiven computer, each face can be computed twice, once for each polyhedron involved. We also observed thatif Pj is a neighbor of Pi, then Pi is a neighbor of Pj . This also holds for any qualification of a neighbor suchas direct, indirect or degenerate.
Lemma 2. For any face fij, each vertex of fij is a vertex of both polyhedra Vi and Vj. Thus fij = fji andits edges and vertices have to be calculated just once.Proof.Suppose that x is a vertex of fij but not of fji. On the plane hij , draw a circle of radius e around x. Forsufficiently small positive e, some points in this circle are in Vi and some are not; simultaneously, either everypoint in the circle is in Vj or none are. However, every point in hij (particularly in the circle) is equidistantfrom Pi and Pj . Hence, all such points are in both Vi and Vj , or in neither. Combining our remarks, we havea contradiction.
8 A New Method for Representation of Polyhedra of Surface Molecules 35
8.4 Relative Efficiency
The overall work of step 2 of FACFIN is O(n2logn). The work for step 5 is at worst O(n[|F1|+|F2|+...+|Fn|]).As most planes will be eliminated well before all m in Fi are checked. The implicit proportionality constantshould be small.
Suppose that each Vi has k faces and that the number of centers less than di away from Pi is O(k). OnceDi is found, the work to construct Vi is O(k3). Thus, except for the face-finder routine (FACFIN), the overallwork is O(k3n). As noted in Section 1, large values of f, even if possible in principle, rarely occur in practice.Therefore, k is generally significantly less than n. If k does not grow with n, for large n the dominant termfor our procedure is O(n2logn). This is more efficient than that mentioned in [2, 3, 7].
8.5 Algebraic Details
For completeness of this work, we indicate how to interpret our geometrical constructions in terms of standardsolid analytic geometry. A plane in 3-space has the form
h(x) = b (8.4)
where:h(x) = a1x1 + a2x2 + a3x3 (8.5)
Points c and d are on the same side of this plane if and only if h(c) and h(d)≤ b or h(c) and h(d)≥ b.Obviously, point e is on the plane if and only if h(e) = b.
Consider two (nonparallel) planes :
ai1x1 + ai2x2 + ai3x3 = bi (8.6)
aj1x1 + aj2x2 + aj3x3 = bj (8.7)
Define A, B, and C by the determinant :
∣∣∣∣∣∣
α β γai1 ai2 ai3
aj1 aj2 aj3
∣∣∣∣∣∣= Aα+Bβ + Cγ (8.8)
Suppose that the point (k1, k2, k3) is on the line formed by the intersection of these planes. Then
B(x1 − k1) = A(x2 − k2), (8.9)
C(x1 − k1) = A(x3 − k3), (8.10)
determine this line. Cyclic permutation of parameters A, B, and C along with the indices 1, 2, and 3 can bemade to avoid zero denominators in subsequent equations. To determine a suitable point (k1, k2, k3) easily,note that at least one of the planes
x1 = 0, x2 = 0, x3 = 0 (11)
must intersect both the given planes. For example, suppose x3 = 0 works. Then k3 = 0 and (k1, k2) is thesolution to the system :
αi1x1 + αi2x2 = bi, (8.12)
αj1x1 + αj2x2 = bj , (8.13)
which is easily solved by the usual method of determinants, with the value of C calculated only once.For a given face i, varying j gives all the candidate edges of this face. Use primes to denote quantities
associated with the second candidate edge, and put α = A/B and α′ = A′/B′. If α = α′ (for every cyclicpermutation of A, B, C along with the indices 1, 2, and 3), the two edges are parallel; otherwise, we have :
36 Hesham Abdelmoez Mohamed and E. F. Alfred
x1 = α(x2 − k2) + k1, (8.14)
x2 = (αk2 − α′k′2 + k1 − k′1)/(α− α′), (8.15)
x3 = C(x1 − k1)/A+ k3, (8.16)
from which we get (in order) x2, x1, x3. The pairwise intersection of these candidate edges give the candidatevertices. The real edges and vertices are selected from these candidates using cutoff criteria discussed inSection 1. If we had tried to find the candidate vertices on a given face directly, we would have had to solve(i−11
)systems of three simultaneous equations. Finding the edges first is also an O(
(11
)) scheme, but the
proportionality constant is smaller.
8.6 Conclusion
Throughout this paper we have talked about individual physical entities represented by the Voronoi polyhedra.We know that most molecules are polyatomic, but this by no means prevents the use of the present method.Molecules of chemical compounds may be represented by graphs, and such graphs are directly useful forpredicting thermodynamic properties of liquid phases [2]. Atoms, groups of atoms, or polymeric segmentsmay be represented by graph points, and these points may serve as centers of the Voronoi polyhedra. Thus,it is only necessary to indicate connectedness between the centers.
Physical entities of different size can be treated also. Solid state physicists use the Voronoi polyhedramostly to describe crystalline materials, but the polyhedra are clearly even more useful for dealing withirregular structures. The computer program VORDIAG which constructs the Voronoi diagram is availablefrom us on request.
References
1. Abdelmoez H. M., Abbas Y. A., On the Bisectors of Weakly Separable Sets, Journal for Geometry and GraphicsJGG, International Society for Geometry and Graphics ISGG, Heldermann Verlag, Germany, vol. 4, No. 1, 2000.
2. Dmitrii N. R., Victor S. L., and Dimitris K. A., Nonlinear Mapping of Massive Data Sets by Fuzzy Clustering andNeural Networks, Journal of Computational Chemistry, Vol. 22, No. 4, pp. 373-398, 2001.
3. Gernot S., Michael M. H., Kinetic and Dynamic Delaunay tetrahedralization in three Dimensions, Computer PhysicsCommunications, Vol. 162, pp. 9-23, 2004.
4. Imma B., Narcis C., Narcis M., and Antoni J., Approximation of 3D Generalized Voronoi Diagram, 21st EuropeanWorkshop on Compuational Geometry, 2005.
5. Jeong W. Y., Brian M., and Jiun-Shyan C., Stabilized Conforming Nodal Integration in the Natural-Element Method,International Journal for Numerical Methods in Engineering, Vol. 60, pp. 861-890, 2004.
6. Lee D. T. and Drysdale R. L., Generalization of Voronoi Diagrams in the Plane, SIAM J. Comput., 10 (1981), pp.73-87.
7. Rudolf K, Friederike N., Martin N., Paul S., and Wolfgang S., Voronoi Polyhedra Analysis of Optimized ArterialTree Models, Annals of Biomedical Engineering, Vol. 31, pp. 548-563, 2003.
9
Formulae for Arithmetic on non-Hyperelliptic Curves
Abdolali Basiri
Mathematics Department, Damghan University of Basic Sciences, Damghan, [email protected] and [email protected]
Summary. In this paper we present an efficient method for addition in the divisor class group of a genus fournon-heperelliptic curve (as well as hyperelliptic curves) over a field of any characteristic. In the case of the curvedefined over a finite field of characteristic greater than 5, this method can add two distinct divisors with less than400 multiplications.
9.1 Introduction
In 1976 Diffie and Hellman introduced the concept of public-key cryptography [7]. Their key exchange protocolis based on the difficulty of solving the discrete logarithm problem (DLP) over a finite field. There are severalgeneral purpose algorithms for Jacobian arithmetic via effective versions of the Riemann–Roch theorem[12, 11]. For superelliptic, Cab and CA curves, more efficient algorithms are described in [1, 9, 10, 2]. In[6, 3, 4], we described a new algorithm for realizing the arithmetic in the Jacobians of superelliptic cubics. Wehereby concentrate on typical ideals as introduced in [6]. This approach follows the framework of Algorithm1 of [3]. Representing ideals by their lexicographic Grobner bases, we use FGLM algorithm [8] to find theCab minimum. In [5], we obtained closed formulae for the reduced ideal in the case of C34 . It is our aimin the present article to obtain efficient formulae to perform calculation in the divisor class group of a non-hyperelliptic genus 4 curve.
We first compute a Grobner basis for the composition ideal and then we compute a Grobner basis fora typical reduced ideal [6, 4] and we develop explicit formulae for it, counting precisely the number of fieldmultiplications and inversions.
We use two known little tricks to speed up the computations. First, by using two linear changes ofvariables, we can assume that
C := Y 3 + C1Y + C0
where C1 and C0 are some polynomials in K[X] of degree 3 and 5 and the coefficient of X4 in C0 is zero.Second, the composition step involves the extended Euclidian algorithm, and the resulting greatest commondivisor is normalized to 1. This normalization step requires an inversion, which can be saved by modifyingthe output of the composition to be (u, v, d) with d in the base field such that the real ideal product is givenby (u, Y − d−1v).
9.2 Computation
In the first Theorem we compute a Grobner basis for the multiplication ideal.
Theorem 1. Let a1 = 〈u1, Y − v1〉 and a2 = 〈u2, Y − v2〉 be two ideals in K[X,Y ] where ui divides v3i +
viC1 +C0 and u1 and u2, co-prime polynomials in K[X]. Then in the general case, polynomials u and v anda field element d such that a1a2 = 〈u, Y − d−1v〉 can be computed with 63 multiplications.
Here we obtain the reduced ideal associated with ideal a of K[X,Y ]/ id(C) where C is a C35 curve. Inthis section we denote the coefficient of a polynomial in front of X i by a subscript i and keep the followingnotations:
38 Abdolali Basiri
• K is a field of characteristic greater than 5 and K is its algebraic closure.• C0 := C00+C01X+C02X
2+C03X3+X5 and C1 a polynomial of degree 3 inK[X] and C := Y 3+C1Y +C0.
• v a polynomial of degree 7 and u a monis polynomial of degree 8 in K[X] and v3 + C1v + C0 = wu.• q = Y 2 + vY + v2 + C1.• b := [id(u,C) : a] = id(C, u, q) = id(C) + [u, uY, q]K[X].
Remark 2. We consider the general case where deg(u) = 8 and deg(v) = 7, the other cases are easier.
Now, we compute a Grobner basis for the reduced ideal of (a) with respect to ≺Lex order in the generalcase (where there is no division by zero). Let α be a polynomial of K[X] such that minC35
(b) = φ(αq) = e.Put
e1(Y ) = Y 3 + C1Y + C0,e2(Y ) = αw + (δ(αv)(Y + v) + δ(ϕv))(Y − v)e(Y ) = αY 2 + ϕY + ψ,
ϕ := φ(αv) = αv mod u be of degree 5, and ψ := φ(αv2 + αC1) = φ(ϕv) + αC1 of degree 6. In this case
e := minC35
(b) = φ(αq) = αY 2 + ϕY + ψ (9.1)
u′ = monic(((αC2
0 + C1(C1ψ − C0ϕ))α+ ψ(3C0ϕ− 2C1ψ))α+ ϕ2(C1ψ − C0ϕ) + ψ3
u2) (9.2)
and put
λ :=α2C0 + ϕψ
u(9.3)
µ :=α(αC1 − ψ) + ϕ2
u
v′ := −µ−1λ mod u′
Here, all divisions by u are exact, that is with remainder zero.
Theorem 3. u′, Y − v′ is a grobner basis for the reduced ideal of (a) which is equal to a′ = e
ua.
The next Theorem computes a Grobner basis for the reduced ideal, counting precisely the number of fieldmultiplications and inversions.
Theorem 4. Let a = 〈u, Y −d−1v〉 where u divides (d−1v)3+d−1vC1+C0, u monic of degree 8, v of degree 7,the reduced representative a
′ = 〈u′, Y − v′〉 in the ideal class of a can be computed with 282 multiplicationsand 2 inversions. In the superelliptic case, 24 multiplications may be saved.
So two distinct elements in the Jacobian of a C35 curve can be added with about 344 multiplications and2 inversions in the field of definition of the curve (Theorems 1 and 4). In the case of a genus 4 superellipticcurves, 24 multiplications can be saved. This method can be generalized for finding the same formulas on theJacobian of a CA curves.
Acknowledgement
I would like to thank Damghan University of Basic Sciences for supporting this research.
9 Formulae for Arithmetic on non-Hyperelliptic Curves 39
References
1. S. Arita. Algorithms for computations in Jacobian group of Cab curve and their application to discrete-log basedpublic key cryptosystems. IEICE Transactions, J82-A(8):1291–1299, 1999. In Japanese. English translation inthe proceedings of the Conference on The Mathematics of Public Key Cryptography, Toronto 1999.
2. S. Arita, S. Miura, and T. Sekiguchi. An addition algorithm on the jacobian varieties of curves. Journal of theRamanujan Mathematical Society, 19(4):235–251, December 2004.
3. A. Basiri. Fast arithmetic for cab curves. submitted.4. A. Basiri. Bases de Grobner et LLL; Arithmetique rapide des courbes Cab. PhD thesis, Laboratoire d’Informatique
de Paris 6, November 2003.5. A. Basiri, A. Enge, J.-C. Faugere, and N. Gurel. Implementing the arithmetic of c3,4 curves. In Lecture Notes in
Computer Science, Proceedings of ANTS, pages 87–101. Springer-Verlag, June 2004.6. A. Basiri, A. Enge, J.-C. Faugere, and N. Gurel. The arithmetic of jacobian groups of superelliptic cubics. Math.
Comp., 74:389–410, 2005.7. W. Diffie and M.-E. Hellman. New directions in cryptography. IEEE Transactions on Information Theory,
IT-22(6):644–654, 1976.8. J.-C. Faugere, P. Gianni, D. Lazard, and T. Mora. Efficient computation of zero-dimensional Grobner bases by
change of ordering. Journal of Symbolic Computation, 16:329–344, 1993.9. S.-D. Galbraith, S. Paulus, and N.-P. Smart. Arithmetic on superelliptic curves. Mathematics of Computation,
71(237):393–405, 2002.10. R. Harasawa and J. Suzuki. Fast Jacobian group arithmetic on Cab curves. In Wieb Bosma, editor, Algorithmic
Number Theory — ANTS-IV, volume 1838 of Lecture Notes in Computer Science, pages 359–376, Berlin, 2000.Springer-Verlag.
11. F. Heß. Computing Riemann–Roch spaces in algebraic function fields and related topics. To appear in Journalof Symbolic Computation, 2001.
12. M.-D. Huang and D. Ierardi. Efficient algorithms for the Riemann–Roch problem and for addition in the Jacobianof a curve. Journal of Symbolic Computation, 18:519–539, 1994.
10
Improving the Computation of the Topology for an Arrangementof Plane Cubics
Jorge Caravantes and Laureano Gonzalez–Vega∗
Departamento de Matematicas, Estadıstica y Computacion, Universidad de Cantabria, Santander, [email protected], [email protected]
Summary. A modification is proposed for the algorithm in [3] in order to compute the topology of a plane cubic curvetaking advantage from the geometry of the degenerate cases. This will be very helpful for improving the efficiency ofthe algorithms dealing with the computation of the topology of an arrangement of plane cubics.
10.1 Introduction
In [3], the authors provide a method for computing the topology of an arrangement of real plane cubic curves.In this brief note, we develop a small modification for each possible case of a (maybe unlikely to be useda lot) step in the study of a single cubic. The choice of the case and our alternative method might be fastenough to make our method faster than the old one. The first section sketches the unchanged part of theoriginal method. The second one describes the subroutine for which we propose the modification and thethird section presents our alternative.
10.2 The Original Method
The method presented in [3] to compute the topology of an arrangement of plane cubics begins with theanalysis of each curve separately. Since our proposal is just for this step, we will just give a sketch of thispart of the whole method.
The algorithm in presented in [3] computes the geometrical analysis of a plane cubic f ∈ Q[x, y] withthe conditions of y–regularity, square–freeness, no two points of VC(F ) ∩ VC(fy) being covertical, no verticalflexes on f and no vertical singularities on f ; or detects infractions to these conditions if they exist.
10.2.1 General set up
The first condition is easy to check and the third one comes for free due to the degree of f . We now computethe resultant Rf :=res(f, fy; y). If Rf = 0 then the second condition is not satisfied, so we exit the algorithm.Otherwise, the roots xi of Rf are ordered and we choose rational numbers ri such that:
r1 < x1 < r2 < x2 < ... < rn < xn < rn−1.
We also find the multiplicities mi of all xi through a squarefree decomposition of
Rf :=∏
Rmfm.
We know that, by the implicit function theorem and the y-regularity, everywhere but in the xi, VR(f)has the shape of one or three disjoint graphs of functions, with this number being constant in (xi−1, xi).Therefore, we order the ki (by definition) roots of f(ri, y). We finish checking that there are not vertical
∗Partially supported by the Spanish Ministerio de Educacion y Ciencia grant MTM2005-08690-C02-02.
10 Improving the Computation of the Topology for an Arrangement of Plane Cubics 41
flexes v ∈ f = 0 ∩ fy = 0 ∩ fyy = 0\fx = 0. We have to check that the solutions xi of the quadratic equationres(fy, fyy; y) = 0 together with the multiple root yi of fy(xi, y) satisfy fx(xi, yi) = 0. If the equation isidentically zero (i.e. fy = 1
2f2yy), checking that the real roots of Rf2 are roots of res(fx, fyy; y).
We call a point (xi, yi) ∈ f = 0∩ fy = 0 an event point, and an arc of f = 0 containing it, involved in theevent. We now have two possibilities for the events:
• mi = 1, so the event point is a x-extreme point.• mi > 1, so (since vertical flexes are already discarded) the event point is a singularity.
10.2.2 x–extreme points
In this case, |ki − ki+1| = 2, and we know if our point is a left x-extreme or right x-extreme just by decidingwhich of ki or ki+1 is greater. To check if the uninvolved arc passes above or below the event, we choosel ∈ i, i+1 such that kl =minki, ki+1 and compute the second derivative fyy(r′l, y) = ay−b (with a, b ∈ Q,a 6= 0), where r′l is between rl and xi and is closer to xi than any event x coordinate of f or fy. In this caseit is well known that the uninvolved arc is above the event if
sign(f(b/a)) 6= sign(l(f))
(where l(f) is the leading coefficient of f) and below otherwise. Let this way of working be called the secondderivative trick.
10.2.3 Locatable singularities
When our event is a singularity and the squarefree component Rf,mihas degree 1, both xi and yi are rational
numbers and easily locatable (xi is trivially computed from Rf,mi, yi is computed through the greatest
common divisor of f(xi, y) and fy(xi, y)). Therefore we know the factorization of the polynomial
f(xi, y) = a(y − yi)2(y − y′i)
which, together with f(x− xi, y − yi), gives all the needed information.
10.3 Non Locatable Singularities
This case is presented separately into this section because here is where our modification is introduced.When degRf,mi
> 1, we cannot guarantee that our singularity is a rational point, so we work withoutknowing explicitly the event point (xi, yi). We know that (y − yi)
2 divides f(xi, y); let y′i be the other root(which must be real). To know if the uninvolved arc is above or below the singularity (i.e. the sign of yi−y′i),we define a polynomial δ(x) whose value in xi is precisely y′i − yi.
In [3], the authors give a method to calculate δ. We introduce here a new one, more geometrical. Let
Sres1(f, fy; y) = a1(x)y + a2(x) = u(x, y)
be the index 1 subresultant of f and fy with respect to y (see [1]): therefore,
u(xi, y) = gcd(f(xi, y), fy(xi, y)) = y − yi.
Then,
H(x, y) :=f(x, y)
l(f)(y − a2(x)a1(x) )
2
(which can be computed applying twice Ruffini, ignoring the remainders) satisfies the property H(xi, y) =y − yi for every xi vanishing Rf2. So what we need is
δ(x) := H(x,a2(x)
a1(x)
).
We finish by finding the sign of δ in the roots of Rf2 by different efficient methods depending on the valueof degRf2.
42 Jorge Caravantes and Laureano Gonzalez–Vega
10.4 A More Geometric Method for Non Locatable Singularities
Our approach starts from a fact that was mentioned in the previous section: degRf,mi∈ 2, 3 (if it is not
1) and the shape of the cubic is determined by this choice.
10.4.1 deg Rf,mi= 3: three lines
3 lines, 1 real root: If xi is the only real root of Rf,mithen it is an acnode and we can use the second
derivative trick with any of both sides (or just check if the real roots of fy(r′i, y) are above or below the onlyreal root of f(r′i, y)), since (xi, yi) is both left and right x–extreme point3 lines, 3 real roots: Then we have that the only event points are the three singularities. It is easy to checkthat the middle point in VR(f) ∩ x = r1 and the middle point in VR(f) ∩ x = r3 are in the same lineL ⊂ VR(f) (let the others be L′ and L′′). Consider the second partial derivative fyy. Then the line fyy = 0cuts L (if they are not parallel, otherwise see below) to the left or right of all the three singularities (i.e.when L is the middle arc) and the other two lines in the segments between singularities. So we compare ifthe only real root of res(f, fyy) that is not in (x1, x3) is to the left or to the right side of the interval. Let ussuppose that it is to the left (since the other case is symmetric). Now we go to the infinity line (just takingthe homogeneous component of degree three f3 of f). We dehomogeneize giving x the value 1 (we can discardzero because there cannot be a vertical line due to y–regularity) and compare the three real roots of f3 (theslopes of L′, L and L′′, from lower to higher) with the slope of fy. There are two possibilities:
• The slope of L is greater than the slope of fyy. Since L cuts fy to the left of the singularities, it meansthat L is above fyy during the events. Since L is the uninvolved arc in the middle singularity, we get thatthe uninvolved arc is above for x2 and below for x1 and x3.
• The slope of L is smaller than the slope of fyy. Symmetric to the previous case.
Example 1. Let us consider the cubic
f(x, y) = 9y3 − 6xy2 − 5x2y + 2x3 − 27y2 + 45xy − 18x2 − 36y + 36x.
From the common part of the algorithm, we have
fy(x, y) = 27y2 − 12xy − 5x2 − 54y + 45x− 36fyy(x, y) = 54y − 12x− 54
We now compute and decompose Rf = 729(10x3 − 59x2 + 9x + 180)2, so Rf2 = 10x3 − 59x2 + 9x + 180with three real roots x1, x2 and x3 (so m1 = m2 = m3 = 2). We choose r1, r2, r3 and r4 so that r1 <x1 < r2 < x2 < r3 < x3 < r4 and get k1 = k2 = k3 = k4 = 3. So we know our cubic is actually three linesand all the three singularities are crunodes. Our method starts by substituting y = 2
9x + 1 (obtained fromfyy = 0) in f and get g := 56
81x3 − 47
3 x2 + 61x − 54. It is easy to check that the first root of g is positive
while the first one of Rf is negative (use Descartes method for Rf (−x) and g(−x)). So fyy cuts L to theright of the singularities. Now we go to the infinity and compare the slope of fyy, which is 2
9 with the rootsof f3(1, y) = 9y3 − 6y2 − 5y + 2. We substitute y = 2
9 and get 5681 , which is possitive. Therefore there are two
roots of f3(1, y) over 29 , which means that L is over fyy after the cut, so under fyy before (i.e. during the
events). Therefore, yi − y′i is positive for i = 2 and negative for i = 1, 3.
3 lines, 3 roots, parallel case: If L and fyy are parallel, we have that there exist an unique c ∈ R so thatfyy + c divides f . It is
c = −29a2,0 − 3a1,1a1,2 − 3a0,2a2,1 + 2a0,2a
21,2
(a21,2 − 3a2,1)
(10.1)
wheref =
∑ai,jx
iyj
and a0,3 = 1. It is clear that L 6= fyy, so c is obviously nonzero. Therefore, if c > 0, L is below fyy. Hencethe middle singularity is above L (which is the uninvolved arc) and the two extremal singularities are belowthe uninvolved arc. The case c < 0 is symmetric.
10 Improving the Computation of the Topology for an Arrangement of Plane Cubics 43
10.4.2 deg Rf,mi= 2: the union of a smooth conic (maybe two parallel lines) and a line
First of all, we have to check that ki = ki+1 = 3, because in other case, it would be a vertical singularityand we would exit at this point. In other case, it is well known that both singularities are crunodes. Todecide where the uninvolved arc is, we shall work with the solutions of the quadratic equation Rf,mi
= 0 andperform the second derivative trick. To avoid the use irrational numbers, we consider the following cases:The cubic has two x–extremal points: In this case, we will use the x–extremal points we studied before.There is again a distinction: Consider the interval (a, b) whose extremal points are the x–coordinates of thex–extremal points. The possibilities are:
• Both singularities have x–coordinates in (a, b) (line and ellipse)• Both singularities have x–coordinates to the left (or both to the right) of (a, b) (line and hyperbola, both
singularities in the same branch).• The interval defined by the x–coordinates of the singularities contains (a, b) (line and hyperbola, one
singularity in each branch)
We explain the case of the ellipse since the other two are similar. We have an ellipse Q and a line L. Betweenthe singularities, L is the center arc and near the extremal points it is the uninvolved one (so the upperor the lower one). Therefore, if the left x–extremal point has the uninvolved arc (i.e. L) above, in the leftsingularity, the arcs crossing are the upper ones (so the uninvolved arc is below the singularity). The othercases (L is below and the right singularity) are symmetric.The cubic has one x–extremal point: Then the cubic consists in a parabola Q and a line L. This case isthe same as the previous one. We know the other extremal point, which is x =∞ (let us use formally +∞).So we solve this (locatable) extremal point and decide where is the uninvolved arc. Now, if the parabola is⊂, we work as in the ellipse case. If it is ⊃, we work as in the case of two intersections in the left branch ofa hyperbola.The cubic has no x–extremal point: This means that the conic is a hyperbola (two parallel lines as adegenerate case). There are two possibilities: the line L cuts two branches of the hyperbola (once each) or justone (twice). In the first case, the line fyy = 0 cuts L undoubtedly when L is the middle arc, i.e. between thesingularities and the hyperbola (if it does) out of the inteval between them. In the second case, the situationis the reciprocal. So we compute the resultant res(f, fyy; y) to see what the case is and after that we use againthe “intersection at the infinity” argument. Considering the division of f by fyy + c (and c is again given by(10.1)) if fyy cuts L at the infinity (it is impossible for fyy to be the asymptote of the conic) as in the threelines case.
10.5 Conclusions
A new approach for computing in a very efficient way the shape (or characteristics) of a given plane cubichas been presented with the objective of analyzing the topology of an arrangement of a finite family of planecubics. The first experiments performed in Maple (see Figure 10.1 to see the output of the current Maple
implementation) show an excellent behavior producing the shape or topology of 1000 cubics (randomlygenerated) in less than one minute. The approach here presented has been currently extended to the case ofthe arrangement of a finite family of plane quartics (see [2]).
44 Jorge Caravantes and Laureano Gonzalez–Vega
Fig. 10.1. The shape of y3 − y2 − xy = 0 as produced by the Maple implementation.
References
1. S. Basu, R. Pollack, M.–F. Roy: Algorithms in Real Algebraic Geometry. Algorithms and Computation in Math-ematics 10, Springer–Verlag, 2003.
2. J. Caravantes, L. Gonzalez–Vega: On the computation of the topology of (an arrangements of) quartics. Preprint,2006.
3. A. Eigenwillig, L. Kettner, E. Schomer, N. Wolpert: Exact, efficient and complete arrangement computation forcubic curves. Computational Geometry 35, 36–73, 2006.
11
Using Symbolic and Numerical Techniques for Solving the OffsetSectioning Problem: The Implicit Case
Fernando Carreras1, Laureano Gonzalez–Vega1∗, and Jaime Puig–Pey2
1 Dep. de Matematicas, Estadıstica y Computacion, Universidad de Cantabria, Santander, Spain.fernando.carreras, [email protected]
2 Dep. de Matematica Aplicada y Ciencias de la Computacion, Universidad de Cantabria, Santander, Spain.E mail: [email protected]
Summary. A new seminumerical algorithm for computing the intersection curve between a plane and the offset of animplicit surface is presented. The corresponding implementation and the performed experimentation are also shown.
11.1 Introduction
Given the implicit surface f(x, y, z) = 0, a point (s1, s2, s3) is in its offset surface at a distance d (see [4]) ifthere exists (x, y, z) ∈ R3 such that:
(x− s1)2 + (y − s2)2 + (z − s3)2 − d2 = 0 (11.1)
f(x, y, z) = 0 (11.2)x− s1
fx(x, y, z)=
y − s2fy(x, y, z)
=z − s3
fz(x, y, z)(11.3)
where the subscripts denote partial differentiation.The purpose of this paper is to present a fast and efficient method to compute the sectioning of an offset
to an implicit surface by a plane: given an implicitely presented surface f(x, y, z) = 0, a plane Π and adistance d > 0, to determine the intersection curve C between the offset of the surface f(x, y, z) = 0 at adistance d and the plane Π. This is a critical problem in Computer Aided Design arising in many practicalsituations such as tool path generation, 3D NC machining, etc (see for example [5, 6]).
The algorithm here presented includes the use of several symbolic tools (like polynomial manipulationor univariate resultants) together with several seminumerical techniques such as the determination of thetopology of a real algebraic plane curve presented implicitely. The generation of the points in the intersectioncurve is performed numerically by applying a Runge–Kutta scheme on very controlled way (in the sense thatthe shape of the final result is known in advance).
A similar algorithm for solving the same problem than here has been presented in [2] but where theconsidered initial surface is presented parametrically.
11.2 The Algorithm
The offset points in the intersection curve to be determined are also points of the plane: if Π(u, v) is presentedby its parametric description
X(u, v) := a1u+ a2v + a3
Y (u, v) := b1u+ b2v + b3Z(u, v) := c1u+ c2v + c3
(ai, bi, ci ∈ R) then the point (X(u, v), Y (u, v), Z(u, v)) is in the intersection curve C between the plane andthe offset surface if there exists (x, y, z) ∈ R3 such that:
∗Partially supported by the spanish Ministerio de Educacion y Ciencia grant MTM2005-08690-C02-02.
46 Fernando Carreras, Laureano Gonzalez–Vega, and Jaime Puig–Pey
(x−X(u, v))2 + (y − Y (u, v))2 + (z − Z(u, v))2 − d2 = 0 (11.4)
f(x, y, z) = 0 (11.5)
fy(x, y, z)(x−X(u, v))− fx(x, y, z)(y − Y (u, v)) = 0 (11.6)
fz(x, y, z)(y − Y (u, v))− fy(x, y, z)(z − Z(u, v)) = 0 (11.7)
fx(x, y, z)(z − Z(u, v))− fz(x, y, z)(x−X(u, v)) = 0 (11.8)
Only two of the last three equations are independent. Note that the considered points in the offset tof(x, y, z) = 0 belong to the two components of the offset to f(x, y, z) = 0 depending on which normaldirection to f(x, y, z) = 0 is taken.
Due to the linearity of u and v in equations (11.6), (11.7) and (11.8), with two of those equations, both uand v can be described in most cases in terms of (x, y, z) (see Subsection 11.2.1 for the analysis of the caseswhen this is not possible). Substituting u = u(x, y, z) and v = v(x, y, z) into (11.4) we obtain an expressiong(x, y, z) = 0. This new equation together with (11.5) allows to eliminate one of the variables, for example z,by computing the resultant polynomial of both equations. At this point, we have a plane curve W (x, y) = 0whose lifting to f(x, y, z) = 0 and g(x, y, z) = 0 provides the so called “footpoint curve”: i.e. the curve whoselifting to distance d following the normal direction to f(x, y, z) = 0 produce the desired intersection curvebetween the plane and the offset to the surface f(x, y, z) = 0.
In practice, once the curve W (x, y) = 0 has been determined and lifted to f(x, y, z) = 0 and g(x, y, z) = 0,the intersection curve is determined by evaluating u = u(x, y, z) and v = v(x, y, z) (as determined from (11.6),(11.7) and (11.8)) at the footpoint curve.
11.2.1 Pathological situations where this approach fails and their treatment
The before mentioned method can not be applied if one of the next two situations happens: when u and vcan not be described in terms of x, y and z or when the resultant of f(x, y, z) and g(x, y, z) with respect toz vanishes identically.
In the first case, to ensure that we can obtain (u, v) in terms of (x, y, z) we must examine the equations(11.6), (11.7) and (11.8). They can be rewritten as:
fy(x, y, z) −fx(x, y, z) 0
0 fz(x, y, z) −fy(x, y, z)−fz(x, y, z) 0 fx(x, y, z)
·
a1u+ a2v + a3 − xb1u+ b2v + b3 − yc1u+ c2v + c3 − z
=
000
.
Manipulating the terms, the previous matrix equality can be rewritten in the following way:fy(x, y, z) −fx(x, y, z) 0
0 fz(x, y, z) −fy(x, y, z)−fz(x, y, z) 0 fx(x, y, z)
·
a1 a2
b1 b2c1 c2
·[uv
]=
=
fy(x, y, z) −fx(x, y, z) 0
0 fz(x, y, z) −fy(x, y, z)−fz(x, y, z) 0 fx(x, y, z)
·
x− a3
y − b3z − c3
or in a more compact way
A ·B[u v]>
= C
where A and C are 3 × 3 matrices involving x, y and z and B is a 3 × 2 numerical matrix. To check theconditions when it is possible to describe u and v in terms of (x, y, z), the matrix A ·B, where rank(B) = 2and rank(A) ≤ 2 for any (x, y, z), must be analyzed.
One easy to avoid case arise when A ·B = 0 since it corresponds to the case when the vectors defining theplane Π are normal to the surface f(x, y, z) = 0: in this case the considered intersection problem is reducedto the easier problem of computing the offset of a planar curve (the one given by the intersection betweenf(x, y, z) = 0 and Π). In the general case, it is enough to check that generically rank(A ·B) = rank(A ·B|C)in order to describe u and v in terms of x, y and z.
In the second case, if the resultant of f(x, y, z) and g(x, y, z) with respect to z vanishes identically thenthis means that the greatest common divisor of these two polynomials is not trivial, i.e. h(x, y, z). Assuming
11 Using Symbolic and Numerical Techniques for Solving the Offset Sectioning Problem: The Implicit Case 47
that f(x, y, z) is irreducible, then the only possibility is that f(x, y, z) = h(x, y, z) and the process must berepeated between f(x, y, z) and g(x, y, z)/f(x, y, z) in order to get the curve W (x, y) = 0.
Example 1. Letx2
9+y2
4+z2
9− 1 = 0,
the plane z = 1 and d = 2. In this case:
W (x, y) =−884736x2 + 2985984 y2 + 102400x4 − 933120 y4 − 1105920 y2x2
+115200 y2x4 + 32400 y4x4 − 583200 y6 + 164025 y8 + 145800 y6x2 (11.9)
Difficulties arise when analyzing the curve defined by W (x, y) = 0. An exact computation would requirea big amount of time with the usual software due to the need of handling singularities like critical or isolatedpoints. The algorithm introduced in [3] to compute the topology of W (x, y) = 0 will be used for determiningthe points of the curve W (x, y) = 0.
11.3 Implementation and Experimentation
The first step of the algorithm manipulates equations (11.4)-(11.8) to get the polynomial W (x, y). Thisis a completely symbolic task. Second step computes the topological graph of W (x, y) = 0 by using theseminumerical procedure described in [3]. Third step concerns with the numerical integration of the branchesof W (x, y) = 0 by using the outgoing information from the topological graph W (x, y) = 0: this is done byusing a Runge–Kutta like method on the differential equations:
∂W
∂x
/∂W∂y
+dy
dx= 0 or
∂W
∂y
/∂W∂x
+dx
dy= 0
depending if in the integration interval there is a vertical or a horizontal critical point.It is worth to remark here that this is a very fast and simple procedure because all the points inside the
integration interval are free of singularities of any kind (due to the performed topological analysis). Forth stepmoves the points (x, y(x)) or (x(y), y)) (the curve W (x, y) = 0) to R3 by finding the common real roots off(x, y, z) = 0 and g(x, y, z) = 0 for each given point in W (x, y) = 0. Fifth, and final step, moves these pointsin f(x, y, z) = 0 to distance d following the direction given by the normal vector of f at those points: thisis merely done by evaluating, for each (x, y, z), in the footpoint curve, first, u = u(x, y, z) and v = v(x, y, z)and, second, X(u, v), Y (u, v) and Z(u, v).
The described algorithm has been implemented in the Computer Algebra System Maple. Two examplesare presented and Table 11.1 shows the computing time and how it is distributed between the topologicalgraph computation (steps 1 and 2), the integration process (step 3) and the final manipulations to get theoffset intersection curve (steps 4 and 5).
Example Topological Graph (1, 2) Integration (3) Manipulation (4, 5)
2 0.349 0.983 3.139
3 0.962 2.799 10.615
Table 11.1. Computing times (in seconds).
Example 2. Surface: x2 + y2 − z2
4 − 1. Plane: z = 3. Distance: d = 1.
W (x, y) = 10− 109x2 − 109 y2 − 525 y4x2 + 330x4 + 330 y4 + 660 y2x2 − 525 y2x4
+150 y4x4 − 175 y6 + 25 y8 + 100x6y2 + 100 y6x2 + 25x8 − 175x6
Figure 11.1 shows the topological graph of W (x, y) = 0, the surface f(x, y, z) = 0 (one hyperboloid), theplane Π and the intersection curves.
48 Fernando Carreras, Laureano Gonzalez–Vega, and Jaime Puig–Pey
Fig. 11.1. Topological graph of W (x, y) = 0 (left) and the intersection curves between the plane and the offset(right). Red curve corresponds to the outer intersection and green curve to the inner one.
Example 3. Surface: x2
9 + y2
4 + z2
9 − 1. Plane: z = 1. Distance: d = 2.
W (x, y) = −884736x2 + 2985984 y2 + 102400x4 − 933120 y4 − 1105920 y2 x2
+115200 y2 x4 − 583200 y6 + 164025 y8 + 32400 y4 x4 + 145800 y6 x2
This example is the continuation of Example 1 and W (x, y) is the equation in (11.9). Figure 11.2 shows thetopological graph of W (x, y) = 0, the surface f(x, y, z) = 0, the plane Π and the intersection curves.
Fig. 11.2. Topological graph of W (x, y) = 0 (left) and the intersection curves between the plane and the offset(right). Red curve corresponds to the outer intersection and green curve to the inner one.
References
1. F. Carreras, L. Gonzalez-Vega and J. Puig–Pey: A Mixed (Numerical and Symbolic) Method for Sectioning Offsets.Abstracts Book of the Conference “Curves and Surfaces” (Avignon), 14, 2006.
2. F. Carreras, L. Gonzalez-Vega and J. Puig–Pey: A new approach for solving the offset sectioning problem. Preprint,2006.
3. L. Gonzalez-Vega and I. Necula: Efficient topology determination of implicitly defined algebraic plane curves. Com-puter Aided Geometric Design 19, 719–743, 2002.
4. C. M. Hoffmann: Algebraic and numerical techniques for offsets and blends. In Computation of curves and surfaces,499–528, Kluwer Acad. Publ., 1990.
5. T. Maekawa: An overview to offset curves and surfaces. Computer–Aided Design 31, 165–173, 1999.6. M. Sabin: Interrogation of subdivision surfaces. In The Handbook of Computer Aided Geometric Design (G. Farin,
J. Hoschek eds.), North-Holland, 327–342, 2002.
12
Change of Ordering for Regular Chains in Positive Dimension
Xavier Dahan1, Xin Jin2, Marc Moreno Maza2, and Eric Schost2
1 LIX, Ecole polytechnique, Palaiseau, France [email protected]
2 Computer Science Department, University of Western Ontario, London, Ontario, [email protected], [email protected], [email protected]
Summary. We discuss changing the variable ordering for a regular chain in positive dimension. This quite generalquestion has applications going from implicitization problems to the symbolic resolution of some systems of differentialalgebraic equations.
We propose a modular method, reducing the problem to dimension zero and using Newton-Hensel lifting tech-niques. The problems raised by the choice of the specialization points, the lack of the (crucial) information of whatare the free and algebraic variables for the new ordering, and the efficiency regarding the other methods are discussed.Strong hypotheses (but not unusual) for the initial regular chain are required. Change of ordering in dimension zerois taken as a subroutine.
12.1 Introduction
Lexicographic orders on polynomial rings are useful tools. Even if computing Grobner bases for such ordersis difficult, these Grobner bases are well suited to answer fast and easily to many problems. Lexicographicorders are also a key component to define regular chains (see Definition 2 and [3]), which are well establishedobjects for polynomial system solving [16, 14, 15].
Suppose that we are given a regular chain as input, as well as a target order on the variables; we areinterested in converting symbolically this input into a new regular chain with respect to the target order,while describing the same solutions. This is required by many applications, ranging from implicitizationproblems to invariant theory, as in the following example.
Example. Consider the polynomials P in Q[X1, X2] such that P (X1, X2) = P (−X1,−X2). Invariant theorytells us that any such polynomial can be written as a polynomial in X2
1 , X22 (the primary invariants π1 and
π2) and X1X2 (the secondary invariant σ); natural questions to ask are whether such a representation isunique, and how to perform the rewriting. This can be done by getting an expression of X1 and X2 infunction of π1 and π2, hence by changing the order of the following system from π1 > π2 > σ > X1 > X2 toX2 > X1 > σ > π1 > π2. Given
∣∣∣∣∣∣
π1 = X21
π2 = X22
σ = X1X2
or
∣∣∣∣∣∣
π1 −X21 = 0
π2 −X22 = 0
σ −X1X2 = 0,
we obtain ∣∣∣∣∣∣
σX2 − π1X1 = 0X2
1 − π1 = 0σ2 − π1π2 = 0
or
∣∣∣∣∣∣
X2 = σπ1
X1
X21 = π1
σ2 = π1π2
.
In this form, we observe the relation σ2 = π1π2 between our basic invariants, which establishes that therepresentation cannot be unique. Furthermore, the new form of the system can be used as a set of rewritingrules, so as to obtain a canonical form for any invariant polynomial.
Main results. To state our results, we will make the following assumption:
(H) the input is a regular chain whose saturated ideal is prime.
50 Xavier Dahan, Xin Jin, Marc Moreno Maza, and Eric Schost
Without this assumption, one may need several regular chains to describe our output; observe that theexample above, and more generally, implicitization problems, satisfy this assumption.
Theorem 1. Let K be a perfect field, let C be a regular chain in K[X] = K[X1, . . . , Xn], satisfying assumptionH, and let < be an order on the variables X. Suppose that the polynomials in C can be evaluated in Loperations, and that the saturated ideal of C has dimension r and degree δ. There exists an algorithm thatcomputes a regular chain C′ for the order <, that admits the same saturated ideal as C, and with the followingfeatures.
The complexity is polynomial in L, δ, n, and in the number(r+δ
δ
)of monomials in r variables of degree at
most δ. The algorithm chooses O(n) random values in K. Let d be the maximum degree of the polynomials inC, and suppose that all random choices are done in a finite set Γ ⊂ K, with the uniform distribution. Thenthe probability of failure is at most 2n(3dn + n2)d2n/|Γ |.Previous work. In this paper, we focus on the case of positive dimension. There already exist many algo-rithms to perform the change of order in this context, see for instance [7, 4, 19]. Further, for the implicitizationproblem, which is an important application of change of order, there exist many specialized algorithms, rely-ing on some form of resultant formalism or homological algebra techniques, see for instance [6, 2, 8] and thenumerous references therein.
Our specificity is the use of modular methods, reducing as much as possible the positive-dimensionalproblems to zero-dimensional ones, following the philosophy of [12]. To do so, we rely on a few well-identifiedsubroutines, such as change of order in dimension zero, and Newton-Hensel lifting to go back to positivedimension. Hence, most of the implementation effort is transferred to these central subroutines. Accordingly,though we do not do it for lack of space, one can state the complexity of our algorithm in terms of the costof these subroutines only.
12.2 Overview of the Algorithm
Definitions and first properties. Let K be a perfect field, let X be a set of n variables. Given a totalorder ≺ on X, every non-constant polynomial in K[X] can be viewed as univariate in its greatest variable;then, its initial is its leading coefficient.
Definition 2. Let C = C1, . . . , Cs be in K[X] with respective (pairwise distinct) main variables Xi1 ≺ · · · ≺Xis
. For all 1 ≤ i ≤ s the saturated ideal of C1, . . . , Ci is the ideal 〈C1, . . . , Ci〉 : (h1 · · ·hi)∞ where hi is
the initial of Ci. Then, the set C is a regular chain if for all 2 ≤ i ≤ s the initial hi is regular modulo thesaturated ideal of C1, . . . , Ci−1.
The main variables of the polynomials in C are its algebraic variables; the other variables are free. For
y ∈ Kn−s
, the specialization of C at y is obtained by evaluating the free variables at y in C. For a genericvalue of y, it is a zero-dimensional regular chain in K[Xi1 , . . . , Xis
].Many concepts used below are relevant from matroid theory. A matroid [20] is a combinatorial structure
that captures the notion of independence (generalizing linear independence in vector spaces), and studies itscombinatorial properties; it thus relates to notions of linear and algebraic independence, but also independencein graph theory.
Definition 3. A matroid M over a finite set X is given by a non-empty family B(M) of subsets of X withthe same cardinality r and satisfying the exchange property: for all e, f ∈ B(M), for every v ∈ e − f thereexists w ∈ f − e such that e − v + w ∈ B(M) holds. The elements of B(M) are called the bases of M andr is its rank. The family of the X− e, for all e ∈ B, is the set of the bases of a matroid M ∗ called the dualmatroid of M .
As an example used below, consider a matrix A over K, and suppose that the columns of A are indexed byX. Let B(A) be the set of all Z ⊆ X such that: (1) the columns of Z are linearly independent and (2) Zis maximal for inclusion. Then, the elements of B(A) are the bases of a matroid over X, which we call thelinear matroid generated by A over K.
Let now V ⊂ Kn
be an irreducible algebraic variety defined over K, and let P ⊂ K[X] be its definingideal. Let r be the dimension of V , with 0 < r < n, and define s = n− r. To such a variety, one can associatethe matroid of the so-called “maximal sets of free variables”.
12 Change of Ordering for Regular Chains in Positive Dimension 51
Definition 4. Let B(X) be the family of all Y ⊆ X such that P ∩K[Y] equals 〈0〉, and such that Y ismaximal for inclusion. The family B(X) is the collection of the bases of a matroid on X of rank r, denotedby Mcoord(V ) (the coordinate matroid of V ).
This result shows how bases of Mcoord(V ) are related to the descriptions of V by regular chains.
Theorem 5. Let Y be a subset of X of cardinal r. Then, Y is a basis of Mcoord(V ) if and only if thereexists a regular chain C in K[X] having P as saturated ideal and Y as free variables.
Overview of the algorithm. Suppose now that we are given a regular chain C0 having P as saturatedideal. Assuming that the variables of X are ordered by a target order <, we aim at computing a regular chainfor the target order, that has P as saturated ideal.
Our algorithm relies on the “specialize and lift” paradigm, in the following form. If C is any regularchain having P as saturated ideal, given a specialization c of C, one can reconstruct C itself by applyingthe Newton-Hensel operator [12, 13, 18] to c and C0. Thus, we will first aim at computing a specializationof the output regular chain, starting from a specialization of the input one. As intermediate steps, we willconsider a sequence of regular chains C0, . . . ,C`, where C` is our output. At step i, the regular chain Ci+1
is obtained from the current one Ci by exchanging the roles of two suitably chosen variables: a formerlyalgebraic variable vi becomes free, and conversely, a formerly free variable wi becomes algebraic. As above,these regular chains will be handled only through specializations ci.
As mentioned above, knowing a specialization c` of C`, the output C` can be recovered by Newton-Hensellifting. Until this last step, we will work only with varieties of dimension zero or one; this is the key to keepingthe complexity under control.
Algorithmic details. The algorithm is divided into two steps. First, we determine the sequences ofvariables vi, wi. Let Z1,Z2 be two distinct bases of the dual matroid M∗
coord(V ) of Mcoord(V ). We writeZ1 <lex Z2 if the largest element of (Z1 − Z2)∪ (Z2 − Z1) is in Z2.
Theorem 6. Let C′ be a regular chain for the ordering <, having P as saturated ideal in K[X]. The set ofthe main variables of C′ is the maximum basis of M∗
coord(V ) for the ordering <lex.
The matroid M∗coord(V ) is not easily accessible a priori. The following theorem, mostly a translation of the
Jacobian criterion, shows how to solve this problem by linearization. Let Jac(C0) be the Jacobian matrix ofC0; then, the columns of Jac(C0) are indexed by the variables of X.
Theorem 7. There is an non-empty open subset V ′ ⊂ V such that for any x ∈ V ′, the linear matroid on X,over the field K, defined by the specialization of Jac(C0) at x equals M∗
coord(V ).
Let Z0 be the algebraic variables of C0, and let y be a random point in Kr. Specializing C0 at y, we obtaina zero-dimensional regular chain c0 in K[Z0]. Let next J0 be the matrix obtained by specializing Jac(C0) aty. Using Theorem 7, and working modulo c0, we can use J0 to test linear independence inM∗
coord(V ). Then,we will compute two sequences v1, . . . , v` and w1, . . . , w` satisfying the following requirements: the elementsof the sequence (Zi)i=0,...,` defined by Zi = Zi−1 + vi − wi are in M∗
coord(V ), and satisfy Zi+1 >lex Zi. Theexchange property shows that Z` is indeed the maximal element inM∗
coord(V ); Theorem 6 shows that Z` arethe algebraic variables of our output regular chain.
It remains to do the change of variables at the level of regular chains. At step i, knowing a specializationci of Ci, we will compute a specialization ci+1 of Ci+1 through the following steps. First, we change the orderof the variables in ci, putting wi as least variable. Since ci has dimension zero, this can be done using severalalgorithms [4, 7, 11, 17]. Then, we lift the free variable vi using the Newton-Hensel operator, obtaining aone-dimensional regular chain c′i+1. Finally, we specialize wi at a random value in c′i+1, obtaining ci+1, backin dimension zero.
Complexity and error probability analysis. We give only a sketch of the complexity analysis, usingthe notation of Theorem 1. The first part (determining the vi and wi) consists in linear algebra operationsmodulo the zero-dimensional regular chain c0; its complexity is thus polynomial in n and δ. The second part(computing the ci) uses change of order in dimension zero, and Newton-Hensel lifting for a single variable ata time; its complexity is linear in the complexity of evaluation L of C0, and polynomial in n and δ.
The last part in the algorithm recovers the r-dimensional regular chain C` from its specialization c`;here, the complexity becomes polynomial in the number of monomials that can appear in C`, inducing a
52 Xavier Dahan, Xin Jin, Marc Moreno Maza, and Eric Schost
polynomial dependence in(r+δ
δ
). Observe that using the straight-line program encoding for the output, we
could make this cost polynomial in δ. However, actual implementations use the dense encoding, with the costgiven above.
Random choices are made to find the initial specialization point y, the specialization values for the variableswi, and in the stopping criterion for the Newton-Hensel operator. Keeping track of all possible degeneraciesleads to the bound in Theorem 1, which allows us to get the result with a probability as close to 1 as wanted.The proof relies on bounds of the size of coefficients of triangular sets [10], and on Bezout’s theorem. Thevalue reported in Theorem 1 essentially grows like d3n; we are currently investigating the question to reducethis dependence to d2n.
Worked example. We can describe the main steps of our algorithm in the previous simple example; allspecialization values will be 1. Here, Z0 is the set of algebraic variables π1, π2, σ. Through linear algebra,we determine the variables that we will have to change, (v1 = X2, w1 = π2) and (v2 = X1, w2 = π1), leadingto the dual bases Z1 = π1, X2, σ and Z2 = X1, X2, σ. Then, at the level of the regular chains themselves,the following operations take place.
Step 1.
π2 = 1π1 = 1σ = 1
variables π2, π1, σwith X1 ← 1, X2 ← 1
σ = 1π1 = 1π2 = 1
variables σ, π1, π2
with X1 ← 1, X2 ← 1
σ = X2
π1 = 1π2 = X2
2
variables σ, π1, π2, X2
with X1 ← 1
Change of order
Dimension zero
Lift X2
Step 2.
σ = X2
π1 = 1X2
2 = 1
Specialization π2 ← 1
variables σ, π1, X2
with X1 ← 1, π2 ← 1
X2 = σσ2 = 1π1 = 1
variables X2, σ, π1
with X1 ← 1, π2 ← 1
X2 = σX1
σ2 = X21
π1 = X21
variables X2, σ, π1, X1
with π2 ← 1
Change of order
Dimension zero
Lift X1
Step 3.
X2 = σX1
σ2 = 1X2
1 = 1
variables X2, σ,X1
with π1 ← 1, π2 ← 1
X2 = σX1
X21 = 1
σ2 = 1
variables X2, X1, σwith π1 ← 1, π2 ← 1
X2 = σX1
π1
X21 = π1
σ2 = π1π2
variables X2, X1, σ, π1, π2
Change of order
Dimension zero
Lift π1, π2
Specialization π1 ← 1
Conclusion. This work extends the scope of modular methods to the problem of change of ordering. For thisfirst insight, the strong primality hypothesis was required, but our results may open ways to a less restrictivesituation. A first implementation has been achieved in Maple’s RegularChains library [15]; a better-tunedimplementation and comparative results are works in progress.
12 Change of Ordering for Regular Chains in Positive Dimension 53
References
1. C. Alonso, J. Guiterrez and T. Recio. An implicitization algorithm with fewer variables. Comput. Aided Geom.Des., 12:251-258, Elsevier, 1995.
2. C. D’Andrea and A. Khetan. Implicitization of rational surfaces with toric varieties, 2003.3. P. Aubry, D. Lazard and M. Moreno Maza. On the Theories of Triangular Sets. J. Symb. Comp., 28:105–124,
1999.4. F. Boulier, F. Lemaire and M. Moreno Maza. PARDI !, In ISSAC’01, pp. 38–47, ACM, 2001.5. F. Boulier, F. Lemaire and M. Moreno Maza. Well known theorems on triangular systems and the D5 principle,
In Transgressive Computing 2006, University of Granada, Spain, 2006.6. L. Buse and M. Chardin. Implicitizing rational hypersurfaces using approximation complexes. J. Symb. Comp.,
40:1150-1168 (2005).7. S. Collart, M. Kalkbrener and D. Mall. Converting Bases with the Grobner Walk. J. Symb. Comp., 24(3-4):465–470,
1997.8. D. Cox. Curves, surfaces, and syzygies. Topics in algebraic geometry and geometric modeling, Contemp. Math.
334:131–150, Amer. Math. Soc., 2003.9. X. Dahan, M. Moreno Maza, E. Schost, W. Wu and Y. Xie. Lifting techniques for triangular decompositions. In
ISSAC’05, pp. 108–115, ACM, 2005.10. X. Dahan and E. Schost. Sharp Estimates for Triangular Sets. In ISSAC’04, pp. 103–110, ACM, 2004.11. J.-C. Faugere, P. Gianni, D. Lazard and T. Mora. Efficient Computation of Zero-Dimensional Grobner Bases by
Change of Ordering J. Symb. Comp., 16(4):329–344, 199312. M. Giusti, J. Heintz, J. E. Morais, and L. M. Pardo. When polynomial equation systems can be solved fast? In
AAECC-11, vol. 948 of Lect. Not. in Comp. Sci., pp. 205–231. Springer, 1995.13. M. Giusti, G. Lecerf, and B. Salvy. A Grobner free alternative for polynomial system solving. J. Complexity,
17(1):154–211, 2001.14. M. Kalkbrener. A generalized Euclidean algorithm for computing triangular representations of algebraic varieties.
J. Symb. Comp., 15:143–167, 1993.15. F. Lemaire, M. Moreno Maza and Y. Xie. The RegularChains library. In Maple Conference 2005, 355-368, I.
Kotsireas Ed., 2005.16. M. Moreno Maza. On triangular decompositions of algebraic varieties. MEGA-2000 Conference, Bath, 2000.17. C. Pascal and E. Schost. Change of order for bivariate triangular sets. In ISSAC’06, pp. 277–284, ACM, 2006.18. E. Schost. Degree Bounds and Lifting Techniques for Triangular Sets. In ISSAC’02, pp. 238–245, ACM, 2002.19. Q.-N. Tran. Efficient Groebner walk conversion for implicitization of geometric objects. Comput. Aided Geom.
Des., 21(9):837-857, Elsevier, 2004.20. D. J. A. Welsh. Matroid Theory. Academic Press, 1976.
13
Intersection Problem, Bivariate Resultant andBernstein-Bezoutian Matrix
Mohamed Elkadi and Andre Galligo
Universite de Nice Sophia Antipolis, Laboratoire J-A. Dieudonne, Parc Valrose, 06108 Nice Cedex2, France.
Summary. Our aim is to study the intersection problem of geometric objects, in the mulivariate case, given by Bezierparameterizations, avoiding the ill-conditionned conversion process between Bernstein bases and monomial bases.
13.1 Introduction
In Computer Aided Geometric Design (CAGD) curves and surfaces are given parametrically in Bernsteinform. Some basic manipulations (intersection, self-intersection, implicitization) can be handled using al-gebraic techniques [3, 4] based on univariate and bivariate resultants computations. These resultants areusually expressed in monomial bases, but in numerical analysis it is widely admited that the conversionbetween Bernstein bases and monomial bases is not well-conditionned and must be avoided. Moreover re-sultant computations tend to increase the sizes of expressions and are generally not well suited for floatingpoint computations (i.e. with fixed precision). To adress these problems several authors including [1, 2] haveintroduced and studied the univariate resultant in Bernstein basis. They report improved results by usingthis technique on the computation of the intersection of two planar curves.
We aim to extend this approach to the multivariate case and amplify its applications in CAGD. Forthis purpose we rely on one of our recent joint-paper [3], where we studied the resultants of polynomialsystems with separated variables and their applications to intersection and self-intersection of surfaces. Here,we concentrate on the very first step of our program, which is to treat the Bezier curve-surface intersectionproblem, (the curve is given by a Bezier parameterization and the surface is given by the tensor product of twoBezier parameterizations). This approach will be generalized to other intersection problems. We implementedan algorithm in Maple, and we will present a commented example.
In the next two sections, we briefly recall the previous results on which we rely.
13.2 Univariate Case
The univariate resultant is used to eliminate one variable (say z) between two polynomials p(z) (of degree n)and q(z) (of degree m). It is equal to the determinant of the Bezoutian matrix computed from the coefficientsof p and q in the monomial basis. In [1], it is shown that this resultant is also equal, up to a constant, to thedeterminant of another Bezoutian square matrix B(p, q) computed from the coefficients pi of p and qi of q inthe Bernstein basis.
The Bernstein basis of the univariate polynomials of degree at most n in z is formed by the n + 1polynomials Bn,i(z) with Bn,i(z) =
(ni
)zi(1− z)n−i, i = 0 . . . n The (univariate) Bernstein-Bezoutian matrix
B(p, q) = (bi,j) which is described in [1] when n = m can be computed in O(n2) operations applying thefollowing rules
13 Intersection Problem, Bivariate Resultant and Bernstein-Bezoutian Matrix 55
bi,1 =n
i(piq0 − p0qi) , 1 ≤ i ≤ n ,
bi,j+1 =n2
i(n− i) (piqj − pjqi) +j(n− i)i(n− j)bi+1,j , 1 ≤ i, j ≤ n− 1 ,
bn,j+1 =n
n− j (pnqj − pjqn) , 1 ≤ j ≤ n− 1. (13.1)
This is easily generalised to n 6= m. These rules were applied in [2] to study the curve-curve intersectionproblem by eliminating one variable between two bivariate polynomials. In that case the coefficients of thesepolynomials in the other variable are also given in Bernstein Bases and all intermediate operations areperformed in that format. The authors used a variant of Bareiss process to compute the determinant ofthe obtained matrix. They also relayed on the structure-preserving property of the Schur complement of aBezoutian matrix to reduce the complexity of the computations. As a numerical application of their method,they computed several examples of intersections of two planar rational Bezier curves. Their program calledthe software BPOLY and was tested using the 53-bits IEEE floating point arithmetic. They reported goodresults in all their examples while other classical resultant computations failed.
13.3 Bivariate Resultant
There is a wide literature on multivariate resultants and in particular on bivariate resultants and theirapplications to CAGD [5, 6, 4].
Very recently [3] produced simplified Bezoutian square matrices whose determinant is equal to the resul-tant of polynomials with separated variables. They also studied its use for the computation of intersectionproblems in CAGD but, although the theory is general, the examples were computed in the monomial bases.Here we aim to adapt these results to polynomial equations expressed in Bernstein bases.
The bivariate resultant of a system of 3 polynomials with separated variables hi(x, y) := fi(x)−gi(y), i =1 . . . 3, is given by the determinant of the square Bezoutian matrix constructed from the polynomialBez(x, y, x1, y1) of h1, h2, h3, which is equal in this particular case to
Bez(x, y, x1, y1) = det
(f(x)− g(y), f(x)− f(x1)
x− x1,g(y)− g(y1)y − y1
)(13.2)
=
3∑
i=1
Bez(gi, 1)(y, y1)Bezf [i](x, x1) +
3∑
i=1
Bez(fi, 1)(x, x1)Bezg[i](y, y1),
where Bezf [i] denotes the Bezoutian of (f1, f2, f3) without the component fi. This Bezoutian is usually givenin the monomial bases, in the next section we analyse it in order to express it in the Bernstein bases.
This formulation of the resultant will be applied to the determination of the Bezier curve-surface inter-section problem in order to get more robust floating point computations.
The paper [3] also provided more general formulae, they should be treated later following the same patternto address the more ambitious rational Bezier surface-surface intersection problem with numerical data.
13.4 Algorithm of Bezier Curve-Surface Intersection
We describe an algorithm for computing Bezier curve-surface intersection. The curve is given by a parameteri-zationG(t) =
(G1(t), G2(t), G3(t)
)and the surface by a parameterization F (s, v) =
(F1(s, v), F2(s, v), F3(s, v)
)
in Bernstein bases.With the previous notations, we compute the simplified (bivariate) Bernstein-Bezoutian matrix of F (s, v)−
G(t), where F is viewed as a polynomial in s wich coefficients are polynomials in v expressed in Bernsteinbasis.
Using (13.1) and (13.2),
Bez(s, t, s1, t1) =∑
i,j,k,l
cijkl(v)Bn−1,i(s)Bn−1,j(t)Bm−1,k(s1)Bm−1,l(t1)
56 Mohamed Elkadi and Andre Galligo
in the tensor product Bernstein bases, which defines a square matrix (with entries depending on v) of sizemn. The determinant of this matrix is a polynomial in v equals to the bivariate resultant of the equationsF1 −G1, F2 −G2, F3 −G3 as polynomials in s and t.
One of the main difficulties is the computation with a good approximation of this determinant. Thetechnique used in [2] that we adapt consists in performing a variant of Bareiss algorithm in the Bernsteinbases for the variable v. Doing so, we get a polynomial in the Bernstein basis of degree mn that we solve(with a good precision) in the interval [0, 1]. Then each obtained potential solution is checked to see if itactually corresponds to a solution in [0, 1]3.
13.5 An Example
We consider the cubic Bezier space curve described by the 4 following control points (given by their coordinatesin R3)
[[0.5, 1.72,−1.03], [−2.96,−3.63,−1.72], [−2.11,−6.22,−1.96], [0.27, 3.09,−1.12]],
and the bicubic Bezier surface described by the 4× 4 array of control points
[[−0.81,−2.97,−2.43], [1.77,−3.39,−1.62], [3.75,−1.47,−3.75], [4.92, 4.08,−7.80]],
[[0.87,−0.81, 2.34], [1.14, 1.50, 3.39], [0.600, 4.20, 4.86], [0.51, 6.39, 7.38]],
[1.50,−1.65,−1.92], [4.38, 0.360, 0.630], [5.31, 2.76, 5.82], [3.54, 4.65, 13.1]],
[1.44,−0.36,−2.64], [−1.47,−0.360,−1.44], [−1.65, 0.720,−2.82], [3.15, 0.18,−4.38]].
We draw the curve and the surface on the same picture and remark 6 intersection points: when the curve go tothe right and return, it crosses 3 visible sheets of the surface. This is a good example for evaluating a resultantmethod because the situation is geometrically net and a sampling or subdivision method provides here a goodand robust complete solution. Of course it is not always the case when the situation is geometrically moreintricated, and this justifies the development of elimination methods for CAGD.
For this example, we can also use a numerical solver: we tried with the one developed by J-P. Pavoneet al.[7], which uses the properties of Bernstein bases. We found the 6 intersection points, so we know themwith a good accuracy for our comparisons.
–2
–1
0
1
2
–3
–2
–1
0
1
2
3
0.5
1
1.5
2
2.5
We first performed, in Maple, the computation with the Bezoutians taken in the monomial bases and thedeterminant computed with the usual command det (improved by the option sparse). We gave several values
13 Intersection Problem, Bivariate Resultant and Bernstein-Bezoutian Matrix 57
to the variable Digits, which controls the precision of the used bigfloats. With Digits = 120, or greater, wegot a good answer: 16 real roots for the computed bivariate resultant. With Digits = 100, or less, we gota completely wrong answer: 6 real roots (and only the 4 last ones were correct) for the computed bivariateresultant.
Then we implemented in Maple the Bernstein-Bezoutian algorithm and got a good result withDigits = 30.
13.6 Conclusion
We studied a simplified bivariate Bernstein-Bezoutian resultant, adapted to polynomial systems with sepa-rated variables and applied it to the solution of the Bezier curve-surface intersection problem. The resultsseem promising but need to be tuned and further analysed. This is a continuation to the bivariate case of[2], and is also the first step of our program. The work is under progress, we aim to extend this approach tomore general surface-surface intersection problems.
References
1. D. A. Bini and L. Gemignani. Bernstein-Bezoutian matrices. Theoret. Comput. Sci., 315(2-3):319–333, 2004.2. Dario A. Bini, Luca Gemignani, and Joab R. Winkler. Structured matrix methods for CAGD: an application to
computing the resultant of polynomials in the Bernstein basis. Numer. Linear Algebra Appl., 12(8):685–698, 2005.3. L. Buse, M. Elkadi, and A. Galligo. Intersection and self-intersection of algebraic surfaces by means of bezoutain
matrix. preprint, 2006.4. C. D’Andrea and A. Khetan. Implicitization of rational surfaces with toric varieties. preprint, 2006.5. Amit Khetan. The resultant of an unmixed bivariate system. J. Symbolic Comput., 36(3-4):425–442, 2003. Inter-
national Symposium on Symbolic and Algebraic Computation (ISSAC’2002) (Lille).6. Amit Khetan, Ning Song, and Ron Goldman. Sylvester A-resultants for bivariate polynomials with planar Newton
polygons (extended abstract). In ISSAC 2004, pages 205–212. ACM, New York, 2004.7. B. Mourrain and J-P. Pavone. Subdivision methods for solving polynomial equations. Technical Report 5658,
INRIA Sophia-Antipolis, 2005.
14
Camera Self-Calibration with Planar Motion
Ferran Espuny
Dep. d’Algebra i Geometria, Univ. de Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, [email protected]
Summary. We consider the self-calibration (affine and metric reconstruction) problem from images acquired with acamera with unchanging internal parameters undergoing planar motion. After introducing the general self-calibrationproblem, we review our linear method presented in [2]. In that work, we used two linear constraints on the coordinatesof the plane at infinity that are linear versions of the quartic modulus constraint. Now we propose a non-linear methodto improve the linear solution. The geometric self-calibration elements given by our method can be used in vehiclenavigation to estimate the vehicle pose and orientation along its trajectory.
14.1 Camera Self-Calibration Framework
In this Section we give a partial introduction to Self-Calibration, which only requires standard ProjectiveGeometry knowledge. Its main purpose is to set the definitions and notations necessary to understand thelater section, in which we will present a reffinement of our Self-Calibration algorithm with Planar Motion [2].
14.1.1 One View Geometry
A pinhole camera is given by a point C, the camera centre, together with a plane R, called the image plane[3, 5]. The image of a 3D point is obtained by intersecting the line joining the point and the camera centrewith the image plane (Fig. 14.1, left). Therefore, a pinhole camera in a projective frame is characterised bya rank 3 quasi-projective mapping P : P3 \ C → P2.
X
C
R
x
θ
c
u
v
uº
x
y
Z=f
vº
O
Fig. 14.1. Left: a pinhole camera viewing a point X in 3D space. Right: the pixel coordinate system taken on theimage plane.
If we fix a Euclidean projective frame, and call R and t the camera orientation and pose in this frame,then the camera can be decomposed as P = K(R|t), where K is is the calibration matrix of the camera,which encodes the camera pixel geometry (Fig. 14.1, right):
K :=
fku −fku cot θ u0
fkv/ sin θ v01
(14.1)
14 Camera Self-Calibration with Planar Motion 59
14.1.2 Multiple View Geometry
R
e
X
x
CC’
R’x’
e’base line
Fig. 14.2. Two cameras, their base line and epipoles; the image of a point X.
If we have two (or more) cameras the epipoles are the imaged camera centres; i.e. the intersection of theline joining these points, base line, with the image planes (Fig. 14.2). A point x on the first image back-projects on a space line, whose image by the second camera is a line l′, called the epipolar line of x. This isthe line of possible corresponding points to x on the second image. The mapping between the image planes
R \ e −→ (R′)∨
x 7→ l′ 3 e′ (14.2)
is given by a 3× 3 matrix F , which is called the fundamental matrix of the pair of cameras. It is a rank twomatrix (Fe = 0) such that every two corresponding points x and x′ must satisfy the epipolar constraint
x′tFx = 0 . (14.3)
This constraint can be used to compute F from at least 7 correspondences. For multiple views, tensors andconstraints are defined in a similar way.
14.1.3 3D Reconstruction
Let us assume that we know a set of image points corresponding to (two) images of a common scene acquiredby unknown cameras. A projective reconstruction of the cameras and the scene corresponds to the real camerasand scene via a projective transformation. In general, if we know enough correspondences between the images,then it is possible to compute a projective reconstruction by estimating multiple view tensors.
An affine reconstruction is a projective reconstruction where π∞ is known; in that case we can takethis plane to be the set of space points with las projective coordinate equal to zero. A Euclidean or metricreconstruction of the cameras will be characterised by the fact that the absolute conic Ω∞, which encodes
the metric in the associated affine frame, is given by Ω∞ '(
Id 00t 0
). If we denote by R, t the relative
displacement, taking the second camera to the first one, then a Euclidean reconstruction is given by
PM ' K(Id 0
), P ′
M ' K ′(R t). (14.4)
Note that the knowledge of the calibration matrices is necessary to achieve a Euclidean reconstruction; infact, there exists a link between K and Ω∞: the image ω by PM of the absolute conic Ω∞ is ω ' K−tK−1.
14.1.4 Camera (Self-)Calibration
The calibration of a camera is the process of determining its calibration matrix K. The calibration-from-pattern methods use images of objects with known geometry (calibrating patterns) to calibrate a camera.In order to obtain good results, usually the imaged pattern must cover as much part of the image plane
60 Ferran Espuny
as possible. The calibration under this approach is an offline extra process, independent of the process ofacquisition of images for scene reconstruction.
By Affine/Metric Self-Calibration we mean the determination of affine/metric properties of the camerasand/or the scene from uncalibrated images, without assuming any knowledge of the scene. The self-calibrationmethods use the rigidness of the relative displacements between views, optionally combined with some knowl-edge about the camera motion or its parameters. Given a projective reconstruction, the direct self-calibrationmethods try to solve directly for Ω∞, whereas the stratiffied methods solve first for π∞ and then linearly forΩ∞ ⊂ π∞.
14.1.5 Modulus and Schaffalitzky’s Constraints on π∞
Let us asume that we know a projective reconstruction of two identically calibrated cameras in a projectiveframe:
P 1 = (Id | 0), P 2 = (A | a). (14.5)
We denote by π∞ = ( pt : 1 ) the projective coordinates of the plane at infinity in said frame. We defineH∞ the infinite homography as the planar homography between the image planes induced by π∞. Since ahomography H transforming the projective reconstruction (14.5) into the metric one (14.4), that is
P 1M = P 1H, µP 2
M = P 2H, (14.6)
takes the form
H =
(K 0−ptK τ
), (14.7)
where τ 6= 0 is uniquely determined by (14.4), it follows that H∞ must satisfy
H∞ := A− apt = µKRK−1 . (14.8)
For any p ∈ R3, we denote by σj(p) the coefficients of the characteristic polynomial of the matrix A−apt:
det(A− apt − xId) = −x3 + σ1(p)x2 − σ2(p)x+ σ3(p) . (14.9)
By (14.8) the matrix H∞ is conjugate to a rotation matrix, from what it follows the modulus constraint[6]:
σ2(p)3 = σ3(p)σ1(p)
3, (14.10)
which is a quartic polynomial equation in the π∞ components.Given three general images, one can intersect the corresponding three quartic surfaces to obtain 43 = 64
possible solutions for π∞.The horopter curve is defined to be the set of space points that project in image points with the same
coordinates in two views. In general, it is a twisted cubic passing through the camera centres. In [7] a newrelation between three images was deduced from the corresponding horopter curves by Schaffalitzky, reducingthe number of possible solutions for π∞ to 21.
14.2 Linear Self-Calibration with Camera Planar Motion
Let us consider three images acquired by a camera with unchanging internal parameters undergoing planarmotion. We require the three camera centres not to be aligned. The modulus constraints are not sufficient toachieve affine self-calibration [8]: any plane parallel to the motion plane satisfies said constraints. In [2] werevisited Schaffalitzky’s constraint showing that planar camera motion is also critical for this constraint.
We define the apex as the image of the infinity point of the screw axes. We also define the circular pointsI, J as the complex conjugate points that form the intersection of the absolute conic with the motion plane[1, 4]. These three points must lie on the imaged horopters, given by the symmetric parts of fundamentalmatrices. In the case of planar motion, the imaged horopters split into two lines: the horizon and the imagesof the screw axes. Thus, the apex can be found by intersecting the images of the screw axes (see Fig. 14.1,
14 Camera Self-Calibration with Planar Motion 61
apex
imagedimaged
imagedscrew
screwscrew
horizon
reconstructed horizon
C1
C C23
Fig. 14.3. Left: the apex on the image can be generally obtained by intersecting the imaged horopters. Right: wereconstruct the horizon on the plane of motion using the points at infinity of the baselines.
left). The imaged circular points were computed in [1] as the fixed points for the trifocal 2d-tensor, and in[4] as the fixed points for the trifocal 1d-tensor, which relates the three horizons.
We showed in [2] that in the planar motion case the modulus constraint quartic surface contains a plane,which can be computed in several ways from a projective reconstruction, and gave us a new linear constrainton π∞. Geometrically speaking we could recover the point at infinity of the base line for any pair of views.Thus, we were able to linearly reconstruct the horizon line, common to the pencil of planes parallel to theplane of motion, using three views (see Figure 14.3). Finally, by imposing that the reconstructed apex lieson π∞, we could determine this plane linearly.
Remark 1. With the plane at infinity determined, we can compute the imaged circular points, and can estab-lish the metric structure on any plane parallel to the motion one. In particular, we are able to determine, upto an unknown scale factor, the camera poses and orientations. Vehicle navigation is the main aplication ofthis work.
14.3 Non-linear Optimisation
The starting point of our linear algorithm is a projective reconstruction obtained from a set of correspondencesin at least three images. Since the fundamental matrices are defined for pairs of views, in order to have a setof matrices consistent with the camera planar motion in [2] we first computed separately the fundamentalmatrices corresponding to any different pair of views, and then computed in a standard way [1, 4] an initialvalue for the epipoles ei, e′i, the apex w and the horizon line h. We then used the following parametrisationfor the fundamental matrices:
F i = [e′i]× · [si ∧ w]× · [ei]× withei, e′i, si ∈ h, (14.11)
to minimise the first order approximation of the sum of squares of the distances of the image points to theepipolar lines (given v ∈ R3, we denote by [v]× the only skew-symmetric matrix such that [v]×x = v∧x, ∀x ∈R3). We used 4 parameters to describe the apex and imaged horizon, which were common to all the views.Since the epipoles ei, e′i and the singular point si of the symetric part of F i must lie on the imaged horizonline h, we needed 3 parameters to describe said points in each view. Therefore, we conducted the minimisationprocess over a total of 3× 3 + 4 = 13 parameters.
It follows form (14.8) that the infinite homographies satisfy
Hj∞ '
(I J v
)e−iθj
eiθj
1
( I J v
)−1. (14.12)
Moreover, since they are induced by a plane, it holds:
F j ' [e′j ]×Hj∞ . (14.13)
62 Ferran Espuny
Therefore, we obtain a parametrisation of the fundamental matrices from θk,j , I, J , v, which are the relevantelements in planar motion navigation, and the epipoles ek,j . Note that we can impose some restrictions onthese elements: ‖v‖ = 1, J = I, θ3,2 = θ3− θ2, one component of I can be fixed to be 1, and if I = Ire + iIimthen the epipoles must lie on the line given by Irre ∧ Iim (the imaged horizon line). Therefore, we can use 11parameters to find the fundamental matrices F j that minimise the first order approximation of the sum ofsquares of the distances of the image points to the epipolar lines. We have proved by numerical simulationsthat the obtained solution gives us a estimation of the self-calibration parameters that improves the obtainedwith the linear algorithm.
Acknowledgements
This work was partly supported by project BFM2003-02914 from the Ministerio de Ciencia y Tecnologıa(Spain).
References
1. M. Armstrong, A. Zisserman, and R. Hartley, “Self-Calibration from Image Triplets,” in Proc. European Conf. onComputer Vision, Vol. A, pp. 3–16, 1996.
2. F. Espuny, “A New Linear Method for Camera Self-Calibration with Planar Motion,” to appear in Journal ofMathematical Imaging and Vision.
3. O. Faugeras and Q.-T. Luong, The Geometry of Multiple Images, MIT Press, Cambridge, 2001.4. O. Faugeras, L. Quan, and P. Sturm, “Self-Calibration of a 1D Projective Camera and its Application to the
Self-Calibration of a 2D Projective Camera,” In Proc. European Conf. on Computer Vision, Vol. A, pp. 36–52,1998.
5. R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, Cambridge University Press, 2000.6. M. Pollefeys, L. Van Gool, and A. Oosterlinck, “The Modulus Constraint: A New Constraint for Self-Calibration,”
In Proc. International Conf. on Pattern Recognition, pages 349–353, 1996.7. F. Schaffalitzky, “Direct Solution of Modulus Constraints,” In Proc. Indian Conf. on Computer Vision, Graphics
and Image Processing, pages 314–321, 2000.8. P. Sturm, Vision 3D non calibree : contributions a la reconstruction projective et etude des mouvements critiques
pour l’auto-calibrage, PhD thesis, Institut National Polytechnique de Grenoble, 1997.
15
A closed Formulae for the Separation of two EllipsoidsInvolving Only Six Polynomials
Laureano Gonzalez-Vega∗ and Esmeralda Mainar†
Departamento de Matematicas, Estadıstica y Computacion, Universidad de Cantabria, [email protected], [email protected]
Summary. By using several tools coming from Real Algebraic Geometry and Computer Algebra (Sturm–Habichtsequences), a new condition for the separation of two ellipsoids in three-dimensional Euclidean space is introduced.This condition is characterized by a set of equalities and inequalities depending only on the matrices defining thetwo considered ellipsoids and does not require in advance the computation (or knowledge) of the intersection pointsbetween them. Moreover, this characterization is well adapted for computationally treating the case where the twoellpsoids depend on one or several parameters.
15.1 Introduction
The problem of detecting collisions or overlap of two ellipsoids is of interest to robotics, CAD/CAM, computeranimation, etc., where ellipsoids are often used for modelling (or enclosing) the shape of the objects underconsideration. The problem to be considered here is obtaining closed formulae characterizing the separationof two ellipsoids in the three dimensional real affine space by using several tools coming from Real AlgebraicGeometry and Computer Algebra. Moreover this characterization can provide easily the manipulation of theformulae for exact collision detection of two ellipsoids under rational motions.
Note that the problem considered in this paper is not the computation of the intersection points betweenthe two considered ellipsoids. This intersection problem can be solved by any numerical nonlinear solver orby “ad–hoc” methods. Nevertheless, the results later described can be used as a preprocessing step since anyintersection problem is highly simplified if the structure of the intersection set is known in advance.
Let a11x2 + a22y
2 + a33z2 + 2a12xy+ 2a13xz + 2a23yz + 2a14x+ 2a24y+ 2a34y+ a44 = 0 be the equation
of the ellipsoid A. As usual it can be rewritten as XTAX = 0, where XT = (x, y, z, 1) and A = (aij)4×4 isthe symmetric matrix of coefficients normalized so that XT
0 AX0 < 0 for the interior points of A. Consideringtwo ellipsoids A and B given by XTAX = 0 and XTBX = 0 and, following the notation in [5] and [4], thedegree four polynomial, f(λ) = det(λA + B), is called the characteristic polynomial of the pencil λA + B.In [5] and [4] the authors give some partial results about how two ellipsoids intersect (without computingthe intersection points), obtaining a complete characterization, in terms of the sign of the real roots of thecharacteristic polynomial, of the separation case:
1. The characteristic equation f(λ) = 0 always has at least two negative roots.2. The two ellipsoids are separated by a plane if and only if f(λ) = 0 has two distinct positive roots.3. The two ellipsoids touch externally if and only if f(λ) = 0 has a positive double root.
It is important to notice that in these characterization conditions only the signs of the real roots are importantand that their exact value is not needed. As soon as two distinct positive roots are detected, one concludesthat the two ellipsoids are separated.
By using Sturm–Habicht sequences (as done in [3] for the ellipses case), the conditions the coefficients off(λ) must verify in order to have exactly two positive real roots are determined. These conditions providethe searched closed formulae depending only on the entrees of matrices A and B and characterizing when
∗Partially supported by the Spanish Ministerio de Educacion y Ciencia grant MTM2005-08690-C02-02.†Partially supported by the Spanish Ministerio de Educacion y Ciencia grant BFM2003-03510.
64 Laureano Gonzalez-Vega and Esmeralda Mainar
two ellipsoids are separated. The main difference with the approach in [2, 6] is the fact that, when the twoellipsoids depend on a parameter t, the curve f(t;λ) = 0 does not need to be analyzed: only the study of thereal roots of six polynomials in t is required.
The approach presented in this paper is specially well suited for analyzing the relative position of twoellipsoids depending on a parameter t. For example, let A and B be two ellipsoids depending on a parametert: A(t) : XTA(t)X = 0 and B(t) : XTB(t)X = 0. In this case the characteristic polynomial
f(t;λ) = det(λA(t) +B(t))
is a degree four polynomial in λ whose coefficients depend on the parameter t. The sign of the real roots of thecharacteristic polynomial is the only information needed: the behaviour of the real roots of f(t;λ) by usingalgebraic techniques and without requiring the knowledge of any approximation of those roots will provideeasy to manipulate formulae in t specially suited in order to characterize when A(t) and B(t) are separatedin terms of t.
15.2 On the Signs of the Real Roots of f(λ) = det(λA + B)
In order to characterize when two ellipsoids are separated, the first step is the study of the sign of the realroots of its characteristic polynomial. The main tools (coming from Computer Algebra and Real AlgebraicGeometry) to solve the sign behaviour problem before described will be the Sturm–Habicht sequence andthe sign determination scheme (see, for example, [1, 3]). Thus the sign behaviour of the real roots of thepolynomial
P = x4 + ax3 + bx2 + cx+ d
is completely determined by the signs of the six polynomials
p1 = −8b+ 3a2,
p2 = −4b3 + a2b2 + 16bd+ 14cab− 6a2d− 3ca3 − 18c2,
p3 = −27d2a4 − 4a3c3 + 18a3dcb+ a2c2b2 + 144a2bd2 − 6a2c2d− 4a2b3d− 192ad2c− 80ab2cd
+18ac3b− 27c4 + 144c2bd+ 256d3 − 128d2b2 + 16db4 − 4b3c2,
q1 = a,
q2 = ba2 + 3ac− 4b2,
q3 = 4a3c2 − 3da3b+ 7dca2 − b2ca2 + 12dab2 − 18ac2b+ 16ad2 + 27c3 − 48dcb+ 4b3c.
In the concrete case considered here, the polynomial P represents the characteristic polynomial of the pencil
λA+B once it has been transformed into a monic polynomial, P (λ) = − f(λ)k
, with k > 0.There are 36 = 729 possibilities of sign conditions in the polynomial sequence p1, p2, p3, q1, q2, q3. The
sign determination scheme (see, for example, [1, 3]) produces a list [[a, b], [n]] 1 ≤ n ≤ 729, indicating thatin the n–th element of the list, P has a total of a different real roots and b of them are positive. Forexample, [[3, 1], [5]] means that the fifth case P has 3 different real roots and just one is positive. Taking intoaccount that the characteristic polynomial P of the pencil λA+ B has always two negative roots (countingmultiplicities) at least and that two ellipsoids are separated by a plane if and only if P has two distinctpositive roots, the cases to be considered are only those producing [[4, 2], [n]] and [[3, 2], [n]].
This process, completely automatized by using the Computer Algebra System Maple, produces the fol-lowing 28 possibilities (see Table 15.1) which completely characterize the separation of the two consideredellipsoids. A simple inspection allows to check that all elements of
[1, 1, 1, ?,−1, ?] := [1, 1, 1, n,−1,m] : n ∈ −1, 0, 1,m ∈ −1, 0, 1 ,[1, 1, 0,−1, 0, ?] := [1, 1, 0,−1, 0, n] : n ∈ −1, 0, 1 ,[1, 1, 0, ?,−1,−1] := [1, 1, 0, n,−1,−1] : n ∈ −1, 0, 1 ,
are included in Table 15.1. In fact, denoting [1, 1, 1, n 6= 0, 1, n 6= 0] := [1, 1, 1, n, 1, n], n 6= 0, and[1, 1, 1, n 6= 0, 0,m 6= −1] := [1, 1, 1, n, 0,m], n 6= 0,m 6= −1, the 28 cases are included in
15 A closed Formulae for the Separation of two Ellipsoids Involving Only Six Polynomials 65
1 [[4, 2], [1]] 2 [[4, 2], [4]] 3 [[4, 2], [5]] 4 [[4, 2], [7]]
5 [[4, 2], [8]] 6 [[4, 2], [9]] 7 [[4, 2], [11]] 8 [[4, 2], [13]]
9 [[4, 2], [14]] 10 [[4, 2], [15]] 11 [[4, 2], [16]] 12 [[4, 2], [17]]
13 [[4, 2], [18]] 14 [[4, 2], [21]] 15 [[4, 2], [23]] 16 [[4, 2], [24]]
17 [[4, 2], [25]] 18 [[4, 2], [26]] 19 [[4, 2], [27]] 20 [[3, 2], [36]]
21 [[3, 2], [37]] 22 [[3, 2], [42]] 23 [[3, 2], [45]] 24 [[3, 2], [48]]
25 [[3, 2], [49]] 26 [[3, 2], [50]] 27 [[3, 2], [51]] 28 [[3, 2], [54]]
1 [1,1,1,1,1,1] 2 [1,1,1,1,0,1] 3 [1,1,1,1,0,0] 4 [1,1,1,1,-1,1]
5 [1,1,1,1,-1,0] 6 [1,1,1,1,-1,-1] 7 [1,1,1,0,1,0] 8 [1,1,1,0,0,1]
9 [1,1,1,0,0,0] 10 [1,1,1,0,0,-1] 11 [1,1,1,0,-1,1] 12 [1,1,1,0,-1,0]
13 [1,1,1,0,-1,-1] 14 [1,1,1,-1,1,-1] 15 [1,1,1,-1,0,0] 16 [1,1,1,-1,0,-1]
17 [1,1,1,-1,-1,1] 18 [1,1,1,-1,-1,0] 19 [1,1,1,-1,-1,-1] 20 [1,1,0,1,-1,-1]
21 [1,1,0,0,1,1] 22 [1,1,0,0,0,-1] 23 [1,1,0,0,-1,-1] 24 [1,1,0,-1,1,-1]
25 [1,1,0,-1,0,1] 26 [1,1,0,-1,0,0] 27 [1,1,0,-1,0,-1] 28 [1,1,0,-1,-1,-1]
Table 15.1. Sign conditions for p1, p2, p3, q1, q2, q3 implying the separation of the ellipsoids.
[1, 1, 1, ?,−1, ?] [1, 1, 0,−1, 0, ?] [1, 1, 0, ?,−1,−1][1, 1, 1, ? 6= 0, 1, ? 6= 0] [1, 1, 1, ? 6= 0, 0, ? 6= −1] [1, 1, 0,−1, 1,−1]
In other words, if P = λ4 + aλ3 + bλ2 + cλ + d represents the characteristic polynomial of the pencilλA + B (once turned monic) then the ellipsoids A and B are separated if and only if (a, b, c, d) verifies oneof the following six possibilities:
p1 > 0, p2 > 0, p3 > 0, q2 < 0 or p1 > 0, p2 > 0, p3 = 0, q1 < 0, q2 = 0 orp1 > 0, p2 > 0, p3 = 0, q2 < 0, q3 > 0 or p1 > 0, p2 > 0, p3 > 0, q1 6= 0, q2 > 0, q3 6= 0 orp1 > 0, p2 > 0, p3 > 0, q1 6= 0, q2 = 0, q3 ≥ 0 or p1 > 0, p2 > 0, p3 = 0, q1 < 0, q2 > 0, q3 > 0
15.3 On the Relative Position of two Ellipsoids Depending on a Parameter
The results obtained in the previous section can be applied to study the case of two ellipsoids depending onone parameter. For example, given two moving ellipsoids A(t) : XTA(t)X = 0 and B(t) : XTB(t)X = 0 ,t > 0, respectively, A(t) and B(t) are said to be collision-free if A(t) and B(t) are separated for all t in agiven interval; otherwise A(t) and B(t) collide.
The characteristic equation of A(t) and B(t), f(λ; t) := det (λA(t) +B(t)) = 0 (t ≥ 0), is a degree fourpolynomial in λ with real coefficients depending on the parameter t. At any time t0 ≥ 0, if A(t0) and B(t0)are separated then f(λ; t0) has two distinct positive roots; otherwise A(t0) and B(t0) are either touchingexternally or overlapping, and f(λ; t0) has a double positive root or no positive roots, respectively.
In order to determine the relative position of the ellipsoids, the study of the sign behaviour of the roots ofthe characteristic polynomial for all the possible values of the parameter t is required. This is accomplishedby using the techniques presented in Section 15.2 where the analysis of the possible sign conditions verifiedby six polynomials in the coefficients of f(t;λ) (as polynomial in λ) produces in an automatic manner (andin terms of t) the behavior of the sign of the real roots of f(t;λ).
Example 1. Let A(t) and B(t) be two spheres, depending on t ∈ R, defined by the equations
x2(t2 + 1) + y2(t2 + 1) + z2 = 1, (x− t)2 + y2 + z2 = 1.
A(t) is the set of concentric spheres of radius less or equal to 1 and B(t) is a (radius 1) sphere whose centremoves along the axis x. The matrices associated to A(t) and B(t) are in this case:
A(t) =
t2 + 1 0 0 00 t2 + 1 0 00 0 1 00 0 0 −1
, B(t) =
1 0 0 −t0 1 0 00 0 1 0−t 0 0 t2 − 1
66 Laureano Gonzalez-Vega and Esmeralda Mainar
and the characteristic polynomial of the pencil λA(t) +B(t):
f(t;λ) = det(λA(t)+B(t)) = (−2t2−1− t4)λ4 +(t6−5t2−4)λ3 +(2t4−4t2−6+ t6)λ2 +(−4− t2 + t4)λ−1.
Turning f(t;λ) into a monic polynomial (with respect to λ) produces the following coefficients
a = t6−5t2−4−2t2−1−t4
b = 2t4−4t2−6+t6
−2t2−1−t4c = −4−t2+t4
−2t2−1−t4d = −1
−2t2−1−t4.
According to the results in subsection 15.2, the sign behaviour of the real roots of f(t;λ) is determined bythe sign conditions verified by the polynomials
p1 = t2(3t6+2t4−5t2−8)(t2+1)2 p2 = t6(t10+t8−3t6−7t4−7t2−4)
(t2+1)4
p3 = t14(t3+2t2+2t+2)(t3−2t2+2t−2)(t2+1)6 q1 = −(−4−t2+t4)
(t2+1)
q2 = −t2(t10+3t8−5t6−12t4−t2+8)(t2+1)3 q3 = t6(−4−t2+t4)(t10−5t6−7t4−4t2−2)
(t2+1)6
In the concrete problem considered here, once denominators and those factors without real roots and constantsign are removed the following six polynomials are obtained
3t6 + 2t4 − 5t2 − 8, t10 + t8 − 3t6 − 7t4 − 7t2 − 4, t3 − 2t2 + 2t− 2,4 + t2 − t4, −t10 − 3t8 + 5t6 + 12t4 + t2 − 8, (−4− t2 + t4)(t10 − 5t6 − 7t4 − 4t2 − 2).
Next the positive real roots of these polynomials are computed producing the following results:
Positive Real Roots(4 + t2 − t4) = 1.600485180,Positive Real Roots(t3 − 2t2 + 2t− 2) = 1.543689012,Positive Real Roots(t10 + t8 − 3t6 − 7t4 − 7t2 − 4) = 1.52066394,Positive Real Roots(3t6 + 2t4 − 5t2 − 8) = 1.240967508,Positive Real Roots(−t10 − 3t8 + 5t6 + 12t4 + t2 − 8) = 0.8540956701, 1.424253130,Positive Real Roots((−4− t2 + t4)(t10 − 5t6 − 7t4 − 4t2 − 2)) = 1.600485180, 1.684484014
They provide the following solution for the separation problem: A(t) and B(t) are separated if and only ift > 1.600485180. This information is obtained by determining the number of real roots of f(t;λ) when tbelongs to each of the intervals defined by the real roots previously determined.
15.4 Conclusions
A closed formulae requiring only the evaluation of six polynomials has been presented for characterizingthe separation of two ellipsoids which is specially well suited when the considered ellipsoids depend on oneparameter. Further analysis for the treatment of the involved polynomials here presented is required toconsider the case of two moving ellipsoids under rational motions since the size of the involved polynomialsrequire ”ad–hoc techniques” for their study (see [2, 6]).
References
1. S. Basu, R. Pollack, M.-F. Roy: Algorithms in Real Algebraic Geometry. Algorithms and Computations in Math-ematics 10, Springer–Verlag (2003).
2. Y.–K. Choi, M.–S. Kim, W. Wang: Exact collision detection of two moving ellipsoids under rational motions.Proceedings of the 2003 IEEE International Conference on Robotics & Automation, 349–354, 2003.
3. F. Etayo, L. Gonzalez–Vega, N. del Rio: A new approach to characterizing the relative position of two ellipsesdepending on one parameter . Computer Aided Geom. Design 23, 324–350 (2006).
4. W. Wang, R. Krasauskas: Interference analysis of conics and quadrics. Contemporary Mathematics 334, 25–36,AMS (2003).
5. W. Wang, J. Wang, M.-Soo Kim: An algebraic condition for the separation of two ellipsoids. Computer AidedGeometric Design 18, 531–539 (2001).
6. W. Wang, Y.-K. Choi, B. Chan, M.-S. Kim, J. Wang: Efficient collision detection for moving ellipsoids usingseparating planes. Geometric modelling. Computing 72, 1-2, 235–246 (2004).
16
Composite Surfaces for Discrete Object Boundary Reconstruction
Stefka Gueorguieva, Pascal Desbarats, and Remi Synave
LaBRI, Universite de Bordeaux I, 33405 Talence, [email protected], [email protected], [email protected]
Keywords: discrete objects, isosurface extraction, combined subdivision schemes, multiresolution
Boundary reconstruction of discrete objects based on isosurface extraction is a very common tool in Com-puter Aided Geometric Design. This applies typically when measurement data arises as in medical imaging.Starting with a coarse representation and taking into consideration some error function, finer resolutionrepresentations are constructed in areas of high interest.
The scope of this paper is to propose a method for composite surface construction that allows adaptiverefinement of the underlying surface while preserving important object features such as object surface areaand volume. The basic idea of the elaborated technique is the observation that using subdivision schemes as[4] and according of the type of the schemes, approximating vs. interpolating, the resulting surface is likelyto occupy smaller or bigger volume than the original control mesh. Thus applying subdivision schemes ofdifferent types locally on chosen parts of the initial surface and following the request of preservation as muchas possible of the original object area and volume we will produce a surface, we called ”composite surface”,that optimizes the representation of the initial object boundary.
At a preprocessing step, the object boundary reconstruction follows [3] and produces a valued triangularmesh. The intuition is to apply a marching cube like technique for the boundary surface reconstruction inparallel with a surface mesh valuation.
The valuation consists in associating to each triangle of the mesh an attribute: black, white or gray,depending on either it is included ”entirely in”, ”entirely out”, or ”partilally in” the initial discrete object.The valuation of the triangles supporting the different subdivision surfaces gives the criterion for choice ofthe subdivision scheme type for the final composite surface construction.
The main difficulty of the subdivision surface combination is how to avoid the gaps (”cracks”) in thefinal surface shape. This is achieved by the introduction of a neighbourhood, called ”direct composite neigh-bourhood”, surrounding the parts of the surface to be subdivided. A set of local topological operators isdevelopped to carry out a C0 continuous merging of the subdivision surfaces and the rest of the boundarysurface in a transition zone spread over the direct composite neighbourhoods of interest. The final compositesurface model is evaluated according common error metrics as in [1, 2]. Experiments are carried out on raw1
(cf Figure 16.2) and synthetic discrete objects2 (cf Figure 16.1).As one can see in Figure 16.2(d), the visual quality of the composite surface model is comparable with
Loop and Butterfly surface subdivision ones. This result is obtained with negligeable variation of the objectsurface area (less than 0.3%) while for Loop subdivision model, the initial surface area to subdivised surfacearea ratio is 5% and for Butterfly this ratio is 0.6%. These experiments are done using one subdivisioniteration.
The error measured in L∞ metric [1] is constant for the three subdivisions cf Figure 16.2(b,c,d) (Loop,Butterfly and Composite) while the error measured in L2 metric for the Composite subdivision is three timesless than for the other surface subdivision models.
1Segmented brain MR image obtained from IMF laboratory, http://www.imf.u-bordeaux2.fr2http://www.3dcafe.com
68 Stefka Gueorguieva, Pascal Desbarats, and Remi Synave
Fig. 16.1. T model. (a) Initial boundary valued mesh, (b) Loop subdivision, (c) Butterfly subdivision, (d) Compositesurface model.
Fig. 16.2. Brain model. (a) Initial boundary valued mesh, (b) Loop subdivision, (c) Butterfly subdivision, (d)Composite surface model.
References
1. Paolo Cignoni, Claudio Montani, and Roberto Scopigno. A comparison of mesh simplification algorithms. Com-puters & Graphics, 22(1):37–54, 1998.
2. Michael Garland. Multiresolution modeling: Survey and future opportunities. In EUROGRAPHICS’99, 1999.3. S. Gueorguieva and P. Desbarats. Reconstruction of topology valid boundary of discrete object from 3d range
images. In IASTED International Conference on Visualization, Imaging and Image Processing, pages 388–392,2005.
4. D. Zorin, T. DeRose, A. Levin, P. Schroder, L. Kobbelt, and W. Sweldens. Subdivision for modeling and animation.In SIGGRAPH Course Notes, pages 1–116, 2000.
17
Geometric Aspects of Parametrization of Interpolating BezierCurves
Imre Juhasz1 and Miklos Hoffmann2
1 Department of Descriptive Geometry, University of Miskolc, Miskolc-Egyetemvaros, H-3515 Miskolc, Hungary.2 Inst. of Mathematics and Computer Science, Karoly Eszterhazy University, Leanyka str. 4., H-3300 Eger, Hungary.
Summary. In interpolation by free-form curves users have to specify the parameter values numerically. In thispaper the geometric aspects of changing these parameters have been studied and an interactive, control point basedinterpolation scheme is suggested for Bezier curves.
17.1 Introduction
The curve interpolation in general is as follows. We are given points p0,p1, . . . ,pn and corresponding pa-rameter values u0, u1, . . . , un, and we want to find those curves g (u) , u ∈ [u0, un] for which g (ui) =pi, (i = 0, 1, . . . , n). Unless otherwise declared, throughout the paper we assume the curve g (u) to be ofBezier-type, i.e.
g (u) =
n∑
i=0
Bni (u)bi
where Bni (u) are the Bernstein polynomials and bi are control points. In order to calculate with numbers
of small absolute value, that decreases the error due to rounding, parameter values are normalized, i.e.,u0 = 0 < u1 < · · · < un = 1.
In an interpolation problem specification and influence of data points pi are straightforward, users of CADsystems can take this job, however the specification of the corresponding parameter values ui is not intuitive,although they also have a significant influence on the shape of the resulted curve. They can be specified ininfinite ways and there is no universally optimal solution. We do not know much about the nature of thisinfluence, hence in this paper our objective is to explore the geometric properties of this effect.
On the other hand, while control point reposition is a powerful interactive design tool in case of approx-imation, users can modify the interpolating curves purely by numerically altering the parameter values. Toovercome this restriction an interactive interpolation tool is presented in the paper, where control points ofa Bezier curve can be modified interactively meanwhile the curve interpolates the given data points.
17.2 Polynomial interpolation and paths
In this section we assume that the interpolating curve is of the form
g (u) =n∑
j=0
Fj (u)bj , u ∈ [u0, un] , (17.1)
i.e. conditionsg (ui) = pi, (i = 0, 1, . . . , n)
have to be fulfilled. On the grounds of these assumptions we obtain the system of equations
70 Imre Juhasz and Miklos Hoffmann
F0 (u0) F1 (u0) · · · Fn (u0)F0 (u1) F1 (u1) · · · Fn (u1)...
.... . .
...F0 (un) F1 (un) · · · Fn (un)
b0
b1
...bn
=
p0
p1
...pn
(17.2)
for the unknown control points bj . It has a unique solution if the determinant D of the system’s matrix Adoes not vanish. This is the only condition for the functions Fj (u). In this case the system can be solved bymeans of Cramer’s rule and we obtain control points
bj =
∑ni=0Aijpi
D, (j = 0, 1, . . . , n) (17.3)
where Aij is the signed minor of the element aij in A.If we fix the parameter u and let the value ui vary we obtain the path
g (u, ui) , ui ∈ (ui−1, ui+1)
of the fixed point. Obviously, this path passes through the point pi if u ∈ (ui−1, ui+1).If the functions Fj (u) form a basis of the space of polynomials of degree bounded by n, then the system of
equations has a unique solution. The parameter value uk, (k = 0, 1, . . . , n) appears only in the kth row of A,therefore polynomials of uk are not multiplied when calculating the determinant. Thus, any bj is a rationalfunction of degree at most n in uk, i.e. varying the parameter value uk the control point moves along a pathwhich is a rational curve of degree n. The same holds for the paths of points of the interpolating curve, sincethey are linear combinations of paths of control points.
The paths (both of control points and of points of the interpolating curve) share the same points atinfinity. These paths have a point at infinity where the determinant D vanishes. When modifying the breakpoint uk, D = 0 occurs if uk = uk−1, or uk = uk+1, i.e. when two rows of A are identical. Consequently, thereare only two non-vanishing signed minors and these differ only in their sign.
Therefore, the path of control point bi, (i = 1, 2, . . . , n− 1) has two points at infinity, the directions ofwhich are pk − pk−1 and pk − pk+1. The same holds for the path of points of the interpolating curve.
It can also be proved that varying a parameter value ui every joining line of control points bj and bj+z
has a fix point the barycentric coordinates of which depend on ui. Thus altering the parameter value thecontrol polygon has a kind of constrained movement.
17.3 Quadratic Interpolation
There are given data points p0,p1,p2 and corresponding parameter values u0 < u1 < u2. Hereafter weassume that u0 = 0, u2 = 1. We examine the effect of the modification of u1 on the shape of the interpolatingparabolic arc g (u), and we also study paths of points of g (u), i.e. those curves that are obtained when a uvalue is fixed and u1 varies between u0 and u2.
We consider the Bezier representation of the interpolating parabola. Control points of this quadraticBezier curve are
b0 = p0,
b1 =p1 − p0B
20 (u1)− p2B
22 (u1)
B21 (u1)
, (17.4)
b2 = p2
by the assumption g (ui) = pi, (i = 0, 1, 2). Applying the identity∑2
i=0B2i (u) ≡ 1 the path of the control
point b1 subject to the alteration of u1 is
b1 (u1) =(p1 − p0)B
20 (u1) + p1B
21 (u1) + (p1 − p2)B
22 (u1)
B21 (u1)
(17.5)
= p1 + (p1 − p0)1− u1
2u1+ (p1 − p2)
u1
2 (1− u1). (17.6)
17 Geometric Aspects of Parametrization of Interpolating Bezier Curves 71
This is a quadratic rational curve with two points at infinity (u1 = 0, u1 = 1), therefore it is a hyperbolic arcwith centre p1 and asymptotic directions (p1 − p0) and (p1 − p2) (cf. Figure 17.3).
Equation (17.5) can also be considered as a quadratic rational Bezier curve determined by the controlpoint p1 with weight 1 and by two control vectors (p1 − p0) and (p1 − p2) cf. [1]. (Control vectors are alsocalled infinite control points, cf. [4].)
Fig. 17.1. Paths of points of the interpolating quadratic Bezier curve are hyperbolas the center of which are on theparabola whose control points are the points to be interpolated (u1 = 0.3, u = 0.5)
17.3.1 Well-known parametrizations
The so-called exponential parametrization includes several well-known parametrization scheme. Its generalform is
u0 = 0, ui = ui−1 +|pi − pi−1|e∑n
j=1 |pj − pj−1|e, (i = 1, 2, . . . , n) , e ∈ [0, 1] . (17.7)
The e = 0 case is the uniform parametrization, the e = 1 case is the chord length parametrization whilein case of e = 1/2 we obtain the centripetal parametrization, introduced in [3] . Here we give geometricalinterpretation of these parametrizations for quadratic Bezier curve.
By applying a modified version of Theorem 4 in [2] one can prove the following extremal property.
Theorem 1. The area of the parabolic arc that interpolates data points p0,p1,p2 is minimal if the parame-trization is uniform.
The area of interpolating quadratic Bezier curves varies in the range [|Tmin| ,∞). For an arbitrarilychosen area T ∈ [|Tmin| ,∞) one can compute the corresponding u1. There will always be two solutions, sinceB2
1 (1/2− δ) = B21 (1/2 + δ), ∀δ ∈ R.
For centripetal parametrization the following can be proved.
Theorem 2. In case of centripetal parametrization, the control point b1 is on the axis of the hyperbola (17.6).
17.4 Cubic Bezier Interpolation
Supposing u0 = 0, u3 = 1 and endpoint-interpolation, i.e. b0 = p0 and b3 = p3, the system of equations isas follows:
72 Imre Juhasz and Miklos Hoffmann
1 0 0 0B0 (u1) B1 (u1) B2 (u1) B3 (u1)B0 (u2) B1 (u2) B2 (u2) B3 (u2)0 0 0 1
b0
b1
b2
b3
=
p0
p1
p2
p3
. (17.8)
If the determinant of the system is D, then control points can be expressed as function of the two varyingparameter values:
b1 (u1, u2) = α0p0 + α1p1 + α2p2 + α3p3,
where
α0 =B3
2 (u1)B30 (u2)−B3
0 (u1)B32 (u2)
D, α1=
B32 (u2)
D
α2 = −B32 (u1)
D, α3 =
B32 (u1)B
33 (u2)−B3
3 (u1)B32 (u2)
D.
Since α0 +α1 +α2 +α3 = 1 holds, control point b1 (u1, u2) is barycentric combination of the points pi to beinterpolated. Altering u1 path of b1 is a cubic curve with two asymptotes which are of the form:
limu1→0
d1 (u1, u2) = p1 + (p0 − p1)∞
limu1→u2
d1 (u1, u2) = p1 +B3
3 (u2) (p3 − p1)− 2B30 (u2) (p0 − p1)
B31 (u2)
+ (p2 − p1)∞
Other paths and their asymptotes can be described analogously. It is an interesting fact that modifyingu1 the fix point of the leg b1b2 is as follows
(1− u2)b1 + u2b2 = p2 +(1− u2)
2
3u2(p2 − p0) +
u22
3(1− u2)(p2 − p3).
17.5 Interactive Interpolation
Varying u1 and u2 we obtain two families of paths of b1 and b2, respectively. These paths form parameterlines of the surfaces b1(u1, u2) and b2(u1, u2). These surfaces contain all the possible positions of the controlpoints b1 and b2 preserving the interpolation property. For planar curves these surfaces degenerate to planarregions. This allows us to give an interactive tool for shape control of interpolation curves in a way, thatspecifying the points pi (i = 0, 1, 2, 3), the system provides the planar region where the user can freely pickthe control point b1. Fixing this point the system can calculate the position of b2 (c.f. Fig.17.5). The areacan cover the whole plane in some cases, depending on the relative positions of the asymptotes of the paths.(Certainly, the role of b1 and b2 is interchangeable in this process.)
References
1. Farin, G. Curves and Surface for Computer-Aided Geometric Design, 4th edition, Academic Press, New York, 1997.2. Juhasz, I. (1998), Cubic parametric curves of given tangent and curvature, Computer-Aided Design, 30(1), 1-9.3. Lee, E. T. (1989) Choosing nodes in parametric curve interpolation, Computer-Aided Design, 21(6), 363-370.4. Piegl, L. (1987), On the use of infinite control points in CAGD, Computer Aided Geometric Design, 4(1-2), 155-166.
17 Geometric Aspects of Parametrization of Interpolating Bezier Curves 73
Fig. 17.2. Possible positions of the control points (shaded area) and the parameter lines by modifying purely u1
(green) or purely u2 (red).
18
Rational Parametrized Curves and Surfaces with RationalConvolutions
Miroslav Lavicka and Bohumır Bastl
Dept. of Mathematics, University of West Bohemia in Pilsen, Czech Republic.
Summary. The aim of this article is to focus on the investigation of such parametrized curves and surfaces whichadmit rational convolution generally, or in some special cases. Examples of such curves and surfaces are presented. Wealso aim to examine links between well-known classes of curves and surfaces (e.g. PH/PN or LN) and general objectswhich are defined in this paper.
18.1 Introduction
The process of finding an exact description of convolution is one of the challenging problems in CAGD. We aremainly interested in such curves and surfaces of special classes which yield rational parametric representationssince these descriptions can be easily included into standard CAD systems and then used in the technicalpraxis. Convolutions with spheres (classical offsets) have been studied for many years and most often becausethey play a very important role in the milling theory with the help of sphere cutter. Of course, this is closelyrelated to the theory of PH curves and PN surfaces. Recently, Petternel and Manhart have studied in [3]the convolution between paraboloids and arbitrary rational surfaces. Later, Sampoli, Peternell and Juttlergeneralized in [5] this approach and introduced a very interesting concept of LN surfaces which admit rationalconvolution with an arbitrary surface. From a practical point of view, main difference between PN and LNsurfaces is as follows — while PN surfaces as workpieces allow computation of the rational path just of thesphere cutter, LN surfaces can be in addition taken as universal tool with the guarantee of rationality of thepath and thus they easily admit the concept of generalized offsets.
Of course, LN surfaces are not the only ones which yield the rational convolutions with general parame-trized surfaces and therefore the aim of this paper is to investigate other curves and surfaces which admitrational convolution generally, or in some special cases. Examples of such curves and surfaces will be shown. Inparticular, close connections with above mentioned curve and surface classes (e.g. PH/PN or LN curves andsurfaces) are presented and new classes of geometric objets will be established. Thus, the main contributionof this paper is to generalize the current classification system.
18.2 Preliminaries
Let A and B be smooth hypersurfaces in Rn. The convolution hypersurface C = A ? B is defined
C = A ? B = a + b | a ∈ A,b ∈ B and α(a) ‖ β(b), (18.1)
where α(a) and β(b) are the tangent hyperplanes of A and B at a ∈ A and b ∈ B which are calledcorresponding points. Of course, the convolution hypersurface is invariant under affine transformations becausethe parallelism is generally preserved.
Let A be parametrized by a(u1, . . . , un−1) and B by b(t1, . . . , tn−1) and we assume in further text thatboth of these parametrizations are rational. To find corresponding points at A and B, we have to constructa reparametrization
18 Rational Parametrized Curves and Surfaces with Rational Convolutions 75
φ : (t1, . . . , tn−1)→ (u1(t1, . . . , tn−1), . . . , un−1(t1, . . . , tn−1)) (18.2)
which is defined for certain parts of A and B with the property that tangent hyperplanes α(a) and β(b) ata ∈ A and b ∈ B are parallel. Then, the parametric representation of the convolution hypersurface C = A?Bis given by
c(t1, . . . , tn−1) = a (u1(t1, . . . , tn−1), . . . , un−1(t1, . . . , tn−1)) + b (t1, . . . , tn−1) . (18.3)
Using coordinates of tangent hyperplanes α(a) = (α0, α1, . . . , αn)(ui) and β(b) = (β0, β1, . . . , βn)(ti), i =1, . . . , n− 1, the condition for corresponding points reads
αj(u1, . . . , un−1) = λ · βj(t1, . . . , tn−1), λ 6= 0, j = 1, . . . , n. (18.4)
A and B are rational — thus, without loss of generality, we can assume that all coordinates of tangenthyperplanes are polynomial. Unfortunately, reparametrization φ still doesn’t have to be expressed in explicitform. Therefore, the paper is devoted to a special class of rational parametrized hypersurfaces which yieldrational convolution hypersurfaces — generally (GRC), or just in some special cases (SRC). Moreover, it isalso necessary in the case of SRC hypersurfaces to identify affine classes of geometric objets with which thesehypersurfaces yield rational convolutions.
18.3 Hypersurfaces with Rational Convolution
Generally, there is no one-to-one correspondence between points a ∈ A and b ∈ B such that appropriatetangent hyperplanes α(a) and β(b) are parallel. This can be easily seen from the algebraic point of view.Consider the ideal
I=〈αj(ui)− λβj(ti), 1− wλ〉 ⊂ k(ti)[w, ui, λ], j = 1, . . . , n; i = 1, . . . , n− 1, (18.5)
see the condition (18.4), the last polynomial guarantees λ 6= 0. Using the reduced Grobner basis computationfor (18.5) with respect to the lexicographic order for w > u1 > . . . > un−1 > λ and the Elimination theorem(see e.g. [1]) we can obtain a polynomial P (λ) as the generator of the nth elimination ideal In = I ∩ k(ti)[λ].
Definition 1. A construction of convolution hypersurface A ? B of hypersurface A parametrized by a(ui),i = 1, . . . , n − 1, and hypersurface B parametrized by b(ti), i = 1, . . . , n − 1, (in this order) is said to be ofdegree r if the equation P (λ) = 0 is of degree r.
The degree of the construction shows the number of points a ∈ A corresponding to chosen point b ∈ B.
Lemma 2. The degree of construction of convolution hypersurface A ? B does not depend on the choice ofrepresentative of coordinate vector of tangent hyperplanes of both rational hypersurfaces A and B at thecorresponding points a ∈ A and b ∈ B.
Example 3. Despite the fact that A ? B = B ? A, both constructions may be generally of different degrees.We can illustrate this remark on computation of A?B and B ?A where A is the paraboloid z = x2 + y2 andB is the sphere x2 + y2 + z2 − 1 = 0. Let A be parametrized by
a(u, v) = (u, v, u2 + v2)
and B by
b(s, t) =
(2s
1 + s2 + t2,
2t
1 + s2 + t2,1− s2 − t21 + s2 + t2
).
Two corresponding points are tied to the condition (18.4) where (α1, α2, α3)(u, v) = (−2u,−2v, 1) and(β1, β2, β3)(s, t) =
(2s, 2t, 1− s2 − t2
). The Grobner basis1 G of the ideal I = 〈α1 − λβ1, α2 − λβ2, α3 −
λβ3, 1− wλ〉 ⊂ Q(s, t)[w, u, v, λ] is given by
1We always mean by Grobner basis the reduced Grobner basis with respect to the lexicographic order and forordering of variables given by the definition of the ring.
76 Miroslav Lavicka and Bohumır Bastl
G =
w − β3, u+
β1
2β3, v +
β2
2β3, λ− 1
β3︸ ︷︷ ︸P (λ)
from which it is seen that the construction of A ?B is of degree 1 (which is not surprising because the givenparametrization of paraboloid is of LN type). On the other hand, if we compute the Grobner basis G of theideal J = 〈α1 − λβ1, α2 − λβ2, α3 − λβ3, 1− wλ〉 ⊂ Q(u, v)[w, s, t, λ] we get
G =
w −
4
α21 + α2
2
λ+4α3
α21 + α2
2
, s−2α1
α21 + α2
2
λ+2α1α3
α21 + α2
2
, t−2α2
α21 + α2
2
λ+2α2α3
α21 + α2
2
, λ2 − α3λ−α2
1 + α22
4| z P (λ)
ff
and thus the construction of convolution starting from B is of degree 2. ¥
Remark 4. Example 3 shows us another interesting fact. Starting from the paraboloid a(u, v) = (u, v, u2+v2),the convolution surface of paraboloid and another rational parametrized surface has generally the explicit ra-tional parametric equation. Conversely, starting from the sphere b(s, t) =
(2s/(1 + s2 + t2), 2t/(1 + s2 + t2) ,
(1− s2 − t2)/(1 + s2 + t2)), the convolution surface is rational just in case that the discriminant of the equa-
tion P (λ) = 0 is square of some polynomial σ, i.e. if the other surface fulfills the condition α21(u, v)+α
22(u, v)+
α23(u, v) = σ2(u, v). Of course, the paraboloid is not this type of surface but there exist many surfaces fulfilling
this condition — they are known as PN surfaces in Euclidean space R3.
The above text motivates us to introduce the following concept.
Definition 5. Let A be a rational hypersurface in Rn, parametrized by a(u1, . . . , un−1). This parametrizationis called a GRC parametrization, if and only if the convolution hypersurface A ? B has an explicit rationalparametrization for an arbitrary rational parametrized hypersurface B. Further, A is called a GRC hypersur-face, if and only if it possesses a GRC parametrization.
Definition 6. Let A be a rational hypersurface in Rn, parametrized by a(u1, . . . , un−1). This parametrizationis called a SRC parametrization, if and only if there exists a rational parametrized hypersurface B suchthat the convolution hypersurface A ? B has an explicit rational parametrization. Further, A is called a SRChypersurface, if and only if it possesses a SRC parametrization.
In R2 we speak about GRC curves, or SRC curves and in R3 about GRC surfaces, or SRC surfaces. Ofcourse, the definitions introduced above are meaningful because there are many examples of geometric objectssatisfying these definitions. LN surfaces (including e.g. paraboloids) are typical examples of GRC surfaces; PNsurfaces (generally yielding rational convolution e.g. with spheres) or PH curves (generally yielding rationalconvolution e.g. with circles) are typical examples of SRC objects — and there exist further examples.
Lemma 7. Every LN surface is a GRC surface.
Lemma 8. Every PH curve is a SRC curve and every PN surface is a SRC surface with convolution con-struction of degree 2.
Theorem 9. Let A be a rational hypersurface in Rn, parametrized by a(u1, . . . , un−1). Parametrizationa(u1, . . . , un−1) is a GRC parametrization if and only if for ideal I defined by (18.5) and for its reducedGrobner basis G with respect to the lexicographic order holds
〈LT (I)〉 = 〈LT (G)〉 = 〈w, u, v, λ〉. (18.6)
Theorem 10. Let A be a rational hypersurface in Rn, parametrized by a(u1, . . . , un−1). Parametrizationa(u1, . . . , un−1) is a SRC parametrization if and only if for ideal I defined by (18.5) and for its reducedGrobner basis G with respect to the lexicographic order holds
〈LT (I)〉 = 〈LT (G)〉 = 〈w, u, v, λr〉, where r ≥ 1, r ∈ N. (18.7)
18 Rational Parametrized Curves and Surfaces with Rational Convolutions 77
18.4 RC Properties of Cubic Curves with Cubic Parametrization
We start the investigation of RC properties of parametrized curves with cubics. With an (2 × 4)-matrix B
(where B1:2,1:1 6= 0) and a vector t = (t3, t2, t, 1)T the general form of a polynomial cubic parametrizationR 7→ R2 is
x(t) = Bt . (18.8)
We say that two parametrizations are equivalent if one can be obtained from the other by an affine change ofcoordinates and affine change of parameters. And since convolutions are invariant under affine transforma-tions, these can be used to explore RC properties of different affine types of parametrized cubics. An affinechange of coordinates in R2 is expressed in the form
x′ = Ax + a, (18.9)
where A is a regular (2× 2)-matrix and a is an 2-vector. An affine transformation of parameter is
t = ct+ d, (18.10)
where c 6= 0. The induced change of t is then
t =
c3 3c2d 3cd2 d3
0 c2 2cd d2
0 0 c d0 0 0 1
t . (18.11)
After application of appropriate transformations (of parameter and coordinates) we get four affine classesrepresented by parametrizations mentioned in Table 18.1 — more details in [2]. Our goal is to investigatethe RC properties of these classes of cubic parametrizations. As in Example 3, we construct in all cases theideal I = 〈α1 − λβ1, α2 − λβ2, 1 − wλ〉 ⊂ Q(u)[w, t, λ], where α = (α0, α1, α2)(t) is a tangent of a cubicallyparametrized curve and β = (β0, β1, β2)(u) denotes a tangent of an arbitrary rational curve. For all cases wecompute Grobner basis and thus we can conclude if the appropriate parametrization is GRC, or SRC, or ifit does not lead to any rational convolution.
Type x(u, v) Grobner basis of I RC property
(i) (t2, t3)nw −
3β2
2
4β1, t+ 2β1
3β2, λ− 4β1
3β2
2
oGRC
(ii) (t2, t3 − t)nw −
3β2
2
4λ+ β1, t+ β2
2λ, λ2 − 4β1
3β2
2
λ− 43β2
2
oSRC
(iii) (t2, t3 + t)nw +
3β2
2
4λ− β1, t+ β2
2λ, λ2 − 4β1
3β2
2
λ+ 43β2
2
oSRC
(iv) (t, t3)nw + β2, t
2 + β1
3β2, λ+ 1
β2
o ×Table 18.1. Cubic curves with cubic parametrization
It is clearly seen from the Table 18.1 that among curves with cubic parametrization there is just one classof GRC curves and two classes of SRC curves with convolution construction of degree 2. We can also noteanother interesting thing — the convolution of some curve of affine type (ii) is rational just in case that theother curve fulfills the condition β2
1(u) + 3β22(u) = σ2(u) for some polynomial σ because
λ1,2 =2(β1 ±
√β2
1 + 3β22
)
3β22
. (18.12)
Thus, curves in this class yield generally the rational convolution with ellipses. The affine type (ii) alsoincludes famous Tschirhausen cubic (which is the only example of cubic PH curve in the plane). Similarly,the affine type (iii) consists of curves which have rational convolution with hyperbolas.
78 Miroslav Lavicka and Bohumır Bastl
18.5 RC Properties of Quadratically Parametrized Surfaces
With an (3 × 6)-matrix B (where B1:3,1:3 6= 0) and a vector u = (u2, uv, v2, u, v, 1)T the general form of apolynomial quadratic parametrization R2 7→ R3 is
x(u, v) = Bu . (18.13)
Again, we say that two parametrizations are equivalent if one can be obtained from the other by an affinechange of coordinates and affine change of parameters. An affine change of coordinates in R3 is expressed inthe form
x′ = Ax + a, (18.14)
where A is a regular (3 × 3)-matrix and a is an 3-vector. An affine transformation of parameter is given byan (2× 3)-matrix C
(uv
)= C
uv1
(18.15)
where C = (a, b, c | d, e, f) and C1:2,1:2 6= 0. The induced change of u is then
u =
a2 b2 2ab 2ac 2bc c2
d2 e2 2de 2df 2ef f2
ad be ae+ bd af + cd bf + ce cf0 0 0 a b c0 0 0 d e f0 0 0 0 0 1
u . (18.16)
After application of appropriate transformations (of parameters and coordinates) we get 15 affine classesof quadratically parametrized surfaces — more details in [4]. Table 2 shows all non-quadric ones (i.e. surfacesof degree three or four) — 6 other are eliptic and hyperbolic paraboloids (LN surfaces), cone and paraboliccylinders (developable surface).
Type x(u, v) λ v u RC property
(i) (u+ v, u2, v2) β1
β2β3
β1
2β3
β1
2β2GRC
(ii) (u, u2 + v, v2) 1β3
- β2
2β3- β1
2β2GRC
(iii) (u, uv, v2) 2β1
β2
2
−β1
β2− 2β1β3
β2
2
GRC
(iv) (u, uv, u2 + v) − 1β2
−β2
2
β1β2−2β2
3
−β3
β2GRC
(v) (u, u2 − v2, uv) 2β1
4β2
2+β2
3
− β1β3
4β2
2+β2
3
− 2β1β2
4β2
2+β2
3
GRC
(vi) (uv + u, u2, v2) −8β2
1β3
(β2
1−4β2β3)2
−β2
1
β2
1−4β2β3
2β1β3
β2
1−4β2β3
GRC
(vii) (uv + u+ v, u2, v2) 4β1(β1−2β3)(β1−2β2)
(β2
1−4β2β3)2
−β1(β1−2β2)
β2
1−4β2β3
−β1(β1−2β3)
β2
1−4β2β3
GRC
(viii) (uv, u+ v2, u2)8β3
2
(β2
1−4β2β3)2
− β1β2
β2
1−4β2β3
2β2
2
β2
1−4β2β3
GRC
(ix) (uv − v, u+ v2, u2)4(β2
1+2β2
2)(β2+2β3)
(β2
1−4β2β3)2
−β1(β2+2β3)
β2
1−4β2β3
β2
1+2β2
2
β2
1−4β2β3
GRC
Table 18.2. Non-quadric quadratically parametrized surfaces
In all cases, we compute the Grobner basis of the ideal I = 〈α1 − λβ1, α2 − λβ2, α3 − λβ3, 1 − wλ〉 ⊂Q(s, t)[w, u, v, λ], where α = (α0, α1, α2, α3)(u, v) is a tangent plane of a quadratically parametrized surfaceof the type (i)–(ix) and β = (β0, β1, β2, β3)(s, t) denotes a tangent plane of an arbitrary rational surface. Wesee from the computation that all mentioned surfaces yield generally a convolution which has an explicitrational parametrization for an arbitrary rational parametrized surface. Thus, all non-quadric quadraticallyparametrized surfaces are GRC surfaces. But these are not LN surfaces what we can easily proof by consideringtheir normals, e.g. the normal vector of surface of type (i) is N = (4uv,−2v,−2u), similarly also normal vectorsof other types have quadratic parametrizations.
18 Rational Parametrized Curves and Surfaces with Rational Convolutions 79
18.6 Acknowledgements
Authors have been supported by the MSM 4977751301 Research Plan.
References
1. Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Springer-Verlag, 1992.2. Karger, A.: Bezier Plane Cubics. In Proc. Geometry and Computer Graphics 2004, Praded, Jesenıky, pp. 64–68.3. Peternell, M., Manhart, F.: The Convolution of a Paraboloid and a Parametrized Surface. Journal for Geometry
and Graphics, Volume 7, 2003, pp. 157–171.4. Peters, J., Reif, U.: The 42 Equivalence Classes of Quadratic Surfaces in Afinne n-Space. Computer Aided Geo-
metric Design, Volume 15, 1997, pp. 459–473.5. Sampoli, M.L., Peternell, M., Juttler, B.: Rational surfaces with linear normals and their convolutions with rational
surfaces. CAGD, Volume 23, 2006, pp. 179–192.
19
An Alternative Representation of Helicoids and Catenoids
Esmeralda Mainar Maza1 and Juan Manuel Pena2
1 Dpto. de Matematicas, Estadıstica y Computacion, Universidad de Cantabria, [email protected]
2 Dpto. de Matematica Aplicada, Universidad de Zaragoza, [email protected]
Summary. We discuss the tensor product representation of a patch of some relevant surfaces such us the catenoidor the helicoid. We represent and control the patch with a control mesh. For this purpose, we bring in the bases withoptimal shape preserving properties of some space containing algebraic, trigonometric and hyperbolic polynomials.
19.1 Introduction
Rational Bezier and Non Uniform Rational B-splines are handy and verstatile tools to represent a large varietyof curves and tensor product surfaces. They can be used to represent some curves usually parameterized bymeans of trigonometric functions as the complete circle [7, 12] and some curves as the cardioid, deltoid,trifolium and Limacon [9]. However there are curves and surfaces of interest in geometry and engineering,like the helix, the helicoid the catenoid which can only be approximately represented by rational functions.In order to represent exactly these curves and surfaces, new spaces of functions are needed.
In the last years, a growing interest in the design of curves in spaces mixing algebraic, trigonometric andhyperbolic functions has arisen. This interest is confirmed by the large number of publications dealing withthis subject (cf. [1, 2, 3, 6, 8, 11, 13, 14, 15, 16]). There are some reasons explaining this interest.
On one hand, it is desirable to represent motions of objects with its natural velocity, which eliminatesthe freedom in the parameterization. In particular, it is necessary to represent a circle with its arc lengthparameterization in order to obtain uniform circular motions. On the other hand, it is convenient that alltypes of curves and surfaces which have to be used in the design process can be obtained with the samekind of representation. This may imply working simultaneously with algebraic, trigonometric and hyperbolicfunctions. In this paper, unified systems to represent lines, circles, helices, cycloids and tensor product surfacesare described.
The Bernstein basis is optimal among all other shape preserving bases of the space of polynomials ofdegree not greater than n on a given compact interval [4]. Roughly speaking, a curve designed with theoptimal basis imitates the shape of its control polygon more faithfully than using other representations. In [5]it was proved that each space of functions admitting shape preserving representations always has an optimalbasis called the normalized B-basis.
In [10], we considered spaces of the form
Vn := 〈1, t, . . . , tn−2, u1(t), u2(t)〉,
where the functions ui (i = 1, . . . , 2) are algebraic polynomials, trigonometric or hyperbolic functions. Wefound intervals [0, α] where we can guarantee that the spaces possess normalized totally positive bases (and so,shape preserving representations) and constructed their normalized B-bases (with optimal shape preservingand stability properties). Here, we announce the optimal stability of the tensor product of these bases anddiscuss the representation of some relevant surfaces such as the catenoid or the helicoid. We represent andcontrol the patch with a control mesh.
19 An Alternative Representation of Helicoids and Catenoids 81
19.2 Construction of the Normalized B-basis of Vn
It is well known that the solutions of a homogeneous linear differential equation of order n+ 1
u(n+1)(t) + an(t)u(n)(t) + · · ·+ a0(t)u(t) = 0, t ∈ I, (19.1)
are invariant under translations if and only if the coefficients ai(t) are constant. If U is a finite dimensionalspace of C1(R) which is invariant under reflections, then U is the set of solutions of an homogeneous lineardifferential equation of order n+1 (19.1) such that its characteristic polynomial p(x) = xn+1 +anx
n + · · ·+a0
is either an even or odd function.Let I be an interval of the real line. Let us recall that an extended Chebyshev space is a (n+1)-dimensional
subspace U of Cn(I) such that each nonzero function in U has at most n zeroes counting multiplicities.Now we need to introduce some notations. The collocation matrix of a system of functions (u0(t), . . . , un(t))
t ∈ I, at t0 < · · · < tm in I is given by
M
(u0, . . . , un
t0, . . . , tm
):= (uj(ti))i=0,...,m;j=0,...,n. (19.2)
A matrix is totally positive if all its minors are nonnegative and a system of functions is totally positive whenall its collocation matrices (19.2) are totally positive. If the basis functions form a partition of the unity,we say that the basis is normalized. When the space has a normalized totally positive basis, there exist anormalized B-basis, which presents optimal shape preserving properties (see [5, 6]). A totally positive basis(b0(t), . . . , bn(t)), t ∈ I is called a B-basis if
inf
bi(t)
bj(t)| t ∈ I, bj(t) 6= 0
= 0, ∀i 6= j.
Let us first consider the homogeneous linear differential equation
u′′(t) + a0u(t) = 0, a0 ∈ R. (19.3)
The characteristic polynomial of (19.3), p(x) = x2 + a0, is an even function, and so the space U1 of solutionsof (19.3) is invariant under translations and reflections.
Let us now consider the initial value problem
u′′(t) + a0u(t) = 0, a0 ∈ R,u(0) = 0, u′(0) = 1.
(19.4)
Let S be the unique solution of (19.4). In order to start the iterative procedure for obtaining (n + 1)-dimensional generalized Bernstein bases, let us consider 0 < α < zS where zS denotes the first positive zeroof S and define two initial functions:
u0,1(t) := S(α− t)/S(α), u1,1(t) := S(t)/S(α), t ∈ [0, α]. (19.5)
In [10] we proved the following result.
Theorem 1. The space U1 of solutions of (19.3) is an extended Chebyshev space on [0, α] if and only ifα < zS. Moreover the basis (u0,1, u1,1) of (19.5) is a B-basis of U1.
Let us recall that, for a given space of functions U , the space of the derivatives U ′ is defined by
U ′ := u′ | u ∈ U.
For n > 1, the (n+ 1)-dimensional spaces Un such that U ′k = Uk−1 for all k = 2, . . . , n are:
i) If p(x) = x2, then Un = 〈1, . . . , tn〉.ii) If p(x) = x2 − w2, w 6= 0, then Un = 〈1, . . . , tn−2, cosh(wt), sinh(wt)〉.iii) If p(x) = x2 + w2, w 6= 0, then Un = 〈1, . . . , tn−2, cos(wt), sin(wt)〉.
82 Esmeralda Mainar Maza and Juan Manuel Pena
Observe that, except for the case i), 1 /∈ U1. This implies that U1 has no normalized totally positive basisand therefore it does not possess shape preserving representations. In order to obtain spaces with shapepreserving representations including U1, let us study the space of the integrals of the functions of U1.
In [10] we proved that for n > 1 a normalized basis un = (u0,n, . . . , un,n) of Un can be defined by:
u0,n(t) := 1−∫ t
0
δ0,n−1u0,n−1(s)ds,
ui,n(t) :=
∫ t
0
(δi−1,n−1ui−1,n−1(s)− δi,n−1ui,n−1(s)) ds, i = 1, . . . , n− 1,
un,n(t) :=
∫ t
0
δn−1,n−1un−1,n−1(s)ds, (19.6)
for t ∈ [0, α], where δi,n−1 := 1/∫ α
0ui,n−1(s) ds, i = 0, . . . , n− 1. We also proved the following result.
Theorem 2. For all n ≥ 2, Un is an extended Chebyshev space with a normalized B-basis on [0, α] for anyα < zS. The system (u0,n, . . . , un,n) defined in (19.6) is the normalized B-basis of Un.
19.3 Tensor Product Representation of Catenoid Patches
This section is devoted to the representation of catenoid patches as tensor products of normalized B-bases ofT2 := 〈1, cos t, sin t〉, T3 := 〈1, t, cos t, sin t〉 and H3 := 〈1, t, cosh t, sinh t〉, obtained by (19.6).
The catenoid (catenary of revolution) and plane are the only surfaces of revolution which are also minimalsurfaces. A catenoid patch can be given by the parametric equations
c(u, v) = (cosh v cosu, cosh v sinu, v), 0 ≤ u ≤ α, 0 ≤ v ≤M.
Using (19.6) and (19.5) with S(t) = sin t we have obtained (v0,2, v1,2, v2,2) and (v0,3, . . . , v3,3), the nor-malized B-bases of T2 and T3, respectively, with
v2,2(t) =1− cos t
1− cosα, v0,2(t) = v2,2(α− t), v1,2(t) = 1− v0,2(t)− v2,2(t), t ∈ [0, α]
and
v3,3(t) =t− sin t
α− sinα, v0,3(t) = v3,3(α− t),
v2,3(t) =(1 + cosα)t+ sin(α− t)− sin t− sinα
(1 + cosα)α− 2 sinα− v3,3(t), v1,3(t) = v2,3(α− t), t ∈ [0, α].
Analogously, using (19.6) and (19.5) with S(t) = sinh t we have obtained the normalized B-basis(w0,3, . . . , w3,3) of H3 with
w3,3(t) =t− sinh t
M − sinhM, w0,3(t) = w3,3(M − t),
w2,3(t) =(1 + coshM)t+ sinh(M − t)− sinh t− sinhM
(1 + coshM)M − 2 sinhM− w3,3(t), w1,3(t) = w2,3(M − t),
t ∈ [0,M ].Now we need the coefficients of some functions with respect to the normalized B-basis of T2, T3 and H3.
Table 1 contains the coefficients of the functions t, sin t and cos t with respect to the normalized B-basis ofT2.
function c0 c1 c2
sin t 0 sinα/(1 + cosα) sinα
cos t 1 1 cosα
19 An Alternative Representation of Helicoids and Catenoids 83
Table 1. Coefficients of sin t and cos t with respect the normalized B-basis of T2 on [0, α].Table 2 contains the coefficients of the functions t, sin t and cos t with respect to the normalized B-basis
of T3.
function c0 c1 c2 c3
t 0 (α− sinα)/(1− cosα) (sinα− α cosα)/(1− cosα) α
sin t 0 (α− sinα)/(1− cosα) (sinα− α cosα)/(1− cosα) sinα
cos t 1 1 (α(1 + cosα)− sinα)/ sinα cosα
Table 2. Coefficients of t, sin t and cos t with respect to the normalized B-basis of T3 on [0, α].Table 3 contains the coefficients of t and cosh t with respect to the normalized B-basis of H3.
function c0 c1 c2 c3
t 0 (M − sinhM)/(1− coshM) (sinhM −M coshM)/(1− coshM) M
cosh t 1 1 (M(1 + coshM)− sinhM)/ sinhM coshM
Table 3. Coefficients of t and cosh t with respect to the normalized B-basis of H3 on [0,M ].The catenoid patch c(r, s) = (cosh s cos r, cosh s sin r, s), 0 ≤ r ≤ α, 0 ≤ s ≤M can be written as
c(r, s) =
2∑
i=0
3∑
j=0
qi,jvi,2(r)wj,3(s), c(r, s) =
3∑
i=0
3∑
j=0
qi,jvi,3(r)wj,3(s), r ∈ [0, α], s ∈ [0, 1] (19.7)
where the control points qi,j , qi,j can be immediately obtained from the coefficients in Table 1, Table 2 andTable 3 (see Figure 19.1).
Fig. 19.1. Helicoid and Catenoid patches with their corresponding control meshes.
In [3] we analyze the connection between two ideas of apparently different nature. On one hand, theexistence of an extended Chebyshev basis, which means that the Hermite interpolation problem has alwaysa unique solution. On the other hand, the existence of a normalized totally positive basis, which means thatthe space is suitable for design purposes. We prove that the intervals where the existence of a normalizedtotally positive basis is guaranteed are those intervals where the existence of an extended Chebyshev basis ofthe space of derivatives can be ensured. In that paper we proved, for any space of differentiable functions andinvariant under translations, the existence of a critical length for design purposes ` in the sense that thereexist normalized totally positive bases on and only on compact intervals of length less than `. We appliedour results to the space generated by 1, t, . . . , tn−2, cos t, sin t. We proved that the critical lengths for designpurposes of T2 and T3 are π and 2π, respectively. That means that T2 has shape preserving representationon and only on intervals of length less than π. In figure 19.2 we have represented the control meshes of acatenoid patch with α = 3π/2 using T2 and T3, respectively. We can clearly see that since α = 3π/2 isgreater than the critical length for design purposes of T2, the corresponding control mesh does not preservethe geometric properties of the patch. In contrast with the control mesh corresponding to T3 whose criticallength for design purposes is 2π > 3π/2.
84 Esmeralda Mainar Maza and Juan Manuel Pena
10
5 0 10
0
0,5
5
1
0 -5
1,5
-5
2
-10 -10
2,5
3
Fig. 19.2. Control mesh and catenoid patches obtained with T2 and T3.
19.4 Tensor Product Representation of Helicoid Patches
This section is devoted to the representation of helicoid patches as tensor products of the normalized B-bases of P1 := 〈1, t〉, T3 := 〈1, t, cos t, sin t〉, obtained by (19.6), and the normalized B-basis of T :=〈1, t, cos t, sin t, t cos t, t sin t〉 provided in [11].
The (circular) helicoid is the minimal surface having a (circular) helix as its boundary. It is the only ruledminimal surface other than the plane. For many years, the helicoid remained the only known example of acomplete embedded minimal surface of finite topology with infinite curvature. However, in 1992 a secondexample, known as Hoffman’s minimal surface and consisting of a helicoid with a hole, was discovered (Sci.News 1992). A helicoid patch can be given in parametric form by
h(u, v) = (v cosu, v sinu,Cu), 0 ≤ u ≤ α, 0 ≤ v ≤M.
It is well known [4] that the normalized B-basis (b0,1, b1,1) of P1 coincides with the Bernstein basis of degree1 on [0,M ]:
b0,1(t) := (M − t)/M, b1,1(t) := t/M, t ∈ [0,M ].
Then the helicoid patch h(r, s) = (s cos r, s sin r, Cs), 0 ≤ r ≤ α, 0 ≤ s ≤M can be written as
h(r, s) =
1∑
i=0
3∑
j=0
pi,jbi,1(s)vi,3(r), s ∈ [0, 1] r ∈ [0, α]. (19.8)
where the control points pi,j can be immediately obtained from the coefficients in Table 2.Now we are going to generalize the domain of the parameter to be a trapezium. Consider the helicoid
patchh(r, s) = (r cos s, r sin s, s), 0 ≤ s ≤ α, a0 + b0s ≤ r ≤ a1 + b1s.
The two s-boundaries are h(ai + bis, s) = ((ai + bis) cos s, (ai + bis) sin s, s), 0 ≤ s ≤ α, i = 0, 1. By letting
t := (r − (a0 + b0s)) / ((a1 + b1s)− (a0 + b0s)) ,
the patch can be rewritten as
h(t, s) =
1∑
i=0
bi,1(t)
ai
cos ssin s
0
+ bi
s cos ss sin ss
+ (1− bi)
00s
=
1∑
i=0
5∑
j=0
pi,jbi,1(t)vj,5(s) (19.9)
where (v0,5, . . . , v5,5) is the normalized B-bases of the space T (see [11]). Clearly, the control mesh is formedby the control points of the s-boundaries that can be written as
pi,j = aip2j + bip
1j + (1− bi)p0
j , . . . , j = 0, . . . , 5.
where p2j , p1
j and p0j j = 0, . . . , 5 are the control points with respect the normalized B-basis of T of a circular
arc, a conical solenoid and a segment, respectively.
19 An Alternative Representation of Helicoids and Catenoids 85
References
1. Carnicer JM, Mainar E, Pena JM. Representing circles with five control points. Comput. Aided Geom. Design2003; 20:501–511.
2. Carnicer JM, Mainar E, Pena JM. A unified framework for cubics and cycloids. In: Lyche T, Mazure ML, Schu-maker LL, editors. Curve and surface design: Saint-Malo 2002, Brentwood, TN: Nashboro Press; 2003, p. 31–40.
3. Carnicer JM, Mainar E, Pena JM. Critical length for design purposes and extended Chebyshev spaces. Const.Approx. 2004; 20:55–71.
4. Carnicer JM, Pena JM. Shape preserving representations and optimality of the Bernstein basis. Adv. Comput.Math. 1993; 1:173–196.
5. Carnicer JM, Pena JM. Totally positive bases for shape preserving curve design and optimality of B-splines.Comput. Aided Geom. Design 1994; 11:635–656.
6. Chen Q, Wang G. A class of Bezier-like curves. Comput. Aided Geom. Design 2003; 20:29–39.7. Chou JJ. High order Bezier circles. Computer-Aided Design 1995; 27:303–309.8. Costantini P, Lyche T, Manni C. On a class of weak Tchebycheff systems. Numer. Math. 2005; 101:333-354.9. Lawrence JD. A catalog of special planar curves. Dover; 1972.
10. Mainar E., Pena J. M., Optimal Bases for a class of Extended Chebyshev Spaces. Preprint11. Mainar E, Pena JM, Sanchez-Reyes. Shape preserving alternatives to the rational Bezier model. Computer-Aided
Geom. Design 2001; 18:37–60.12. Piegl L, Tiller W. The NURBS book, 2nd ed. Springer; 1997.13. Pottmann H, Wagner MG. Helix splines as an example of affine Tchebycheffian splines. Adv. Comput. Math.
1994; 2:123–142.14. Wang G, Chen Q, Zhou M. NUAT B-spline curves. Comput. Aided Geom. Design 2004; 21:193–205.15. Zhang J. C-curves: an extension of cubic curves. Comput. Aided Geom. Design 1996; 13:199–217.16. Zhang J. Two different forms of C-B-splines. Comput. Aided Geom. Design 1997; 14:31–41.
20
Implicitizing Rational Curves by Using Bernstein-BezoutianMatrices
Ana Marco and Jose-Javier Martınez
Departamento de Matematicas, Univ. de Alcala, Campus Universitario, 28871-Alcala de Henares, madrid, [email protected], [email protected]
Summary. The approach to curve implicitization through Sylvester and Bezout resultant matrices and bivariateinterpolation in the usual power basis is extended to the case of Bernstein-Bezoutian matrices constructed whenthe polynomials are given in the Bernstein basis. The coefficients of the implicit equation are also computed in thebivariate tensor-product Bernstein basis, and their computation involves the bidiagonal factorization of the inversesof certain totally positive matrices.
20.1 Introduction
When studying rational plane algebraic curves, there are two standard ways of representation, the implicitequations and the parametric equations. The interest in one or other representation depends on the oper-ations that one wants to do with the curve, and hence it is very important to be able to change from onerepresentation to another.
We will concentrate on the implicitization problem, that is to say, on finding an implicit representationstarting from a given rational parametrization of the curve. In [4] we have presented an approach to the implic-itization problem based on resultants and on interpolation using the usual power basis for the correspondingspace of bivariate polynomials. However, the recent work [1] has showed the importance of evaluating re-sultants from Bernstein basis resultant matrices directly, avoiding a basis transformation between Bernsteinand power basis.
So, our aim is to use the Bernstein-Bezout matrix and bivariate interpolation for obtaining in the bivari-ate tensor-product Bernstein basis the implicit equation of a plane algebraic curve given by its parametricequations in Bernstein form. Let us observe that this is the usual situation in the case of Bezier curves, akind of curves which are very popular in computer aided geometric design (CAGD) due to the propertiesthey satisfy.
Let us consider P (t) = (x(t), y(t)) a proper parametrization of a rational plane algebraic curve C, where
x(t) = u1(t)v1(t)
and y(t) = u2(t)v2(t)
and gcd(u1, v1) = gcd(u2, v2) = 1. In this situation the following two theorems
hold [7]:
Theorem 1. The polynomial defining C is Rest(u1(t)−xv1(t), u2(t)−yv2(t)) (the resultant with respectto t of the polynomials u1(t)− xv1(t) and u2(t)− yv2(t)).
Theorem 2. The polynomial F (x, y) defining the implicit equation of C satisfies
degy(F ) = maxdegt(u1), degt(v1),degx(F ) = maxdegt(u2), degt(v2).
Therefore, the polynomial defining the implicit equation of the curve C is F (x, y) =Rest(u1(t) − xv1(t), u2(t) − yv2(t)) and belongs to the polynomial space Πn,m(x, y), wheren = maxdegt(u2), degt(v2) andm = maxdegt(u1), degt(v1). The dimension ofΠn,m(x, y) is (n+1)(m+1).
20 Implicitizing Rational Curves by Using Bernstein-Bezoutian Matrices 87
Assuming that the parametric equations of C are given in the Bernstein basis, we compute F (x, y) =Rest(u1(t) − xv1(t), u2(t) − yv2(t)) by using the Bernstein-Bezout matrix of p(t) =u1(t) − xv1(t) and q(t) = u2(t) − yv2(t). Let us observe that if m = n, the resultant is the determinantof the Bernstein-Bezout matrix, while -as a consequence of the corresponding result for the Bezout resultant[6]- if m > n, that determinant is equal to the resultant multiplied by the factor (pm(t))m−n, where pm(t) isthe leading coefficient of p(t) in the power basis.
Therefore, the fact of knowing the leading coefficients of p and q in the power basis has two importantadvantages: it allows one to work with an interpolation space of smallest dimension, and it avoids the needof polynomial factorization.
20.2 The interpolation problem
Since the expansion of the symbolic determinant of a matrix is very time and space consuming, our aim is tocompute F (x, y) by means of Lagrange bivariate interpolation, but using the Bernstein basis instead of thepower basis.
If we consider the interpolation nodes (xi, yj) (i = 0, · · · , n; j = 0, · · · ,m) and the interpolation spaceΠn,m(x, y), the interpolation problem is stated as follows:
Given (n + 1)(m + 1) values fij ∈ K where i = 0, · · · , n; j = 0, · · · ,m (the interpolation data), find apolynomial
F (x, y) =∑
(i,j)∈I
cijβ(n)i (x)β
(m)j (y) ∈ Πn,m(x, y),
where
β(n)i (x) =
(n
i
)(1− x)n−ixi, β
(m)j (y) =
(m
j
)(1− y)m−jyj ,
are univariate Bernstein polynomials and I = (i, j)|i = 0, · · · , n; j = 0, · · · ,m, such that
F (xi, yj) = fij ∀ (i, j) ∈ I.If we consider for the interpolation space Πn,m(x, y) the basis
B(n,m)ij , i = 0, . . . , n; j = 0, . . . ,m = β(n)
i (x)β(m)j (y), i = 0, . . . , n; j = 0, . . . ,m =
B(n,m)00 , B
(n,m)01 , · · · , B(n,m)
0m , B(n,m)10 , B
(n,m)11 , · · · , B(n,m)
1m , · · · , B(n,m)n0 , B
(n,m)n1 , · · · , B(n,m)
nm with that precise ordering, and the interpolation nodes and the interpolation data with the corresponding
ordering, then the (n+ 1)(m+ 1) interpolation conditions F (xi, yj) = fij can be written as a linear systemAc = f, where the coefficient matrix A is given by a Kronecker product Bx ⊗By, with
Bx = ((β(n)j (xi)), i = 0, . . . , n; j = 0, . . . , n,
By = ((β(m)j (yi)), i = 0, . . . ,m; j = 0, . . . ,m,
c = (c00, · · · , c0m, c10, · · · , c1m, · · · , cn0, · · · , cnm)T ,
andf = (f00, · · · , f0m, f10, · · · , f1m, · · · , fn0, · · · , fnm)T .
We compute each interpolation datum by means of the evaluation of p(t) and q(t) at the node (xi, yj)followed by the computation of the determinant of the corresponding numerical Bernstein-Bezout matrix Bmaking use of the Bini-Gemignani algorithm which constructs the Bernstein-Bezout matrix for the evaluatedpolynomials [1]. In addition, if the determinant of the symbolic Bernstein-Bezout matrix is not the resultantbut a polynomial multiple of it, we must divide the value of the determinant by the polynomial factorevaluated at the node (xi, yj).
As for the interpolation nodes, they must be selected in a way that the matrices Bx and By are nonsingularbecause in this case the matrix Bx⊗By is also nonsingular, and the interpolation problem has unique solution.To avoid divisions by zero we must not use the value of xi for which the leading coefficient of p(t) in the powerbasis evaluates to 0, and the value yj for which the leading coefficient of q(t) in the power basis evaluates to0.
88 Ana Marco and Jose-Javier Martınez
20.3 The solution of the linear system
Our aim in this section is to present an efficient and accurate way of solving the linear system (Bx⊗By)c = fcorresponding to the interpolation problem. As it will be shown, the total positivity of the matrices Bx andBy will be essential for reaching it.
Making use of the results of [3], we know that performing the complete Neville elimination on a strictlytotally positive matrix N of order n a bidiagonal factorization of its inverse N−1 can be obtained, that is tosay, we have
N−1 = G1G2 . . . Gn−1D−1Fn−1Fn−2 . . . F1,
where D−1 is a diagonal matrix and Fi and Gi are bidiagonal matrices.So, after having obtained that factorization, all the linear systems Nz = b with the same coefficient matrix
N can be solved by performing the product
G1G2 . . . Gn−1D−1Fn−1Fn−2 . . . F1b.
On the other hand, from [2] we know that the Bernstein basis of the space of polynomials of degree lessthan or equal to n is a strictly totally positive basis on the open interval (0, 1), which implies that choosingin our situation the interpolation nodes (xi, yj) (i = 0, . . . , n; j = 0, . . . ,m) satisfying x0 < x1 < . . . < xn in(0, 1) and y0 < y1 < . . . < ym in (0, 1), the matrices Bx and By are strictly totally positive and the bidiagonalfactorization of the matrices B−1
x and B−1y can be obtained by means of Neville elimination.
Taking into account that solving the linear system (Bx ⊗ By)c = f is equivalent to solving n + 1 linearsystems with the matrix By and then m+1 linear systems with the matrix Bx [5], we proceed by computingthe bidiagonal factorization of the inverse of By by means of Neville elimination and then by solving eachone of the linear systems Byz = b by computing the product of the bidiagonal matrices by b. After that, weproceed analogously for solving the m+ 1 linear systems with the same coefficient matrix Bx.
Finally, we will briefly examine the computational complexity of our algorithm in terms of arithmeticoperations. For the sake of clarity we will consider here the case m = n.
The factorization of the inverse of a matrix of order n+ 1 by means of complete Neville elimination takesO(n3) operations, but that factorization is used for solving all the systems with the same matrix, so eachof the remaining systems can be solved with O(n2) operations. In this way, the interpolation part of thealgorithm has computational complexity O(n3).
As for the generation of the interpolation data, it has a computational complexity of O(n5) becausethe construction of the numerical Bernstein-Bezout matrix requires O(n2) arithmetic operations and thecomplexity of the computation of each determinant is O(n3). Therefore with our approach the whole processhas complexity O(n5), while using Gaussian elimination it would be O(n6).
Remark. Finally, let us observe that all the linear systems with matrix By can be solved simultaneously,and the same can be said of the systems with matrix Bx. Therefore the algorithm exhibits a high degree ofintrinsic parallelism. This parallelism is also present in the computation of the interpolation data since wecan compute simultaneously the determinants involved in this process.
20.4 References
1. D. A. Bini, L. Gemignani, Bernstein-Bezoutian matrices, Theoretical Computer Science 315 (2004) 319–333.
2. J. M. Carnicer, J.M. Pena, Shape preserving representations and optimality of the Bernstein basis,Advances in Computational Mathematics 1 (1993), 173–196.
3. M. Gasca, J. M. Pena, A matricial description of Neville elimination with applications to total positivity,Linear Algebra and Its Applications 202 (1994) 33–45.
4. A. Marco, J. J. Martınez, Using polynomial interpolation for implicitizing algebraic curves, ComputerAided Geometric Design 18 (2001) 309–319.
5. J. J. Martınez, A generalized Kronecker product and linear systems, Int. J. Math. Educ. Sci. Technol.30(1) (1999), 137–141.
20 Implicitizing Rational Curves by Using Bernstein-Bezoutian Matrices 89
6. T. W. Sederberg, J. Zheng, Chapter 15: Algebraic methods for Computer Aided Geometric Design, in:G. Farin, J. Hoscheck, M. S. Kim (Eds.), Handbook of Computer Aided Geometric Design, Elsevier 2002.
7. J. R. Sendra, F. Winkler, Tracing index of rational curve parametrizations, Computer Aided GeometricDesign 18 (2001) 771–795.
21
Approximating Clothoids by Bezier Curves
Nicolas Montes, Josep Tornero
Dpto. Ingenierıa de Sistemas y Automatica, Technical University of Valencia, 46022 Valencia, [email protected], [email protected]
Summary. Clothoids have been mainly used in road design and recently in defining trajectory references for mobilerobots and vehicles. This paper presents a strategy for approximating clothoids by Bezier curves in order to be used inCAD/CAM systems or any other application requiring parametric curves. The approximation is based on the fact thateach particular point of the clothoid can be described as a linear combination of the Bezier control points. Therefore,these control points can be obtained based on a set of clothoid points using least square identification methods. Inaddition, the Bezier control point obtained for one clothoid can describe any other clothoid just only rescaling thesepoints according to the parameter of the clothoid. The rescaling is easy because each control point moves along astraight line. At the end of the paper, the approximation has been validated for the conditions used in road design.
21.1 Introduction
A clothoid curve or Cornu spiral is a planar curve whose curvature varies linearly with the arc length.Because of this important property, it has been applied in road design for many years [1]. This propertyand his application to the road design have considered by researches in automated guided vehicles [2], [3].Unfortunately, the clothoid curve is defined in terms of Fresnel integrals which are difficult to compute.However, in [8], Fresnel integrals are obtained with an accuracy of 1x10−9.
In addition, clothoids are difficult to be used in CAD/CAM systems as well as any other applicationrequiring parametric curves, such as in automated guided vehicles when defining references in discrete time.
Bezier, B-spline and non-uniform rational B-spline (NURBS) curves are commonly used in CAD andCADG fields. However, NURBS curves are, now, the basic representation form, becoming the standard fordata exchanges among CAD/CAM systems when representing 3D objects.
Some authors proposed approximations to the clothoid into hereinbefore commented standards. That is,in [4] the clothoid is approximated by a Bezier form using Taylor expansion. This approximation is limitedin the interval of the tangent angle [0,π/2], and the well-known clothoid constant parameter set to [30, 3000],used in practical road design. However, the order of the resulting Bezier curve is 23 with an error order of1x10−6.
In [5] the clothoid is approximated by an s-power series, which is a polynomial form. The coefficients canbe translated into the standard Bezier with a transformation matrix [6], but the calculus of the coefficients iscomplicated. In [7] the clothoid is approximated by an arc spline, but the disadvantage is that it is not withcontinuous curvature.
This paper presents a strategy for approximating clothoids by Bezier curves in order to be used inCAD/CAM systems or any other application requiring parametric curves. The approximation is based onthe fact that each particular point of the clothoid can be described as a linear combination of the Beziercontrol points. Therefore, these control points can be obtained based on a set of clothoid points using leastsquare identification methods. Of course, the error committed in this approximation decreases when thenumber of points increases. This fact permits to select a desirable error.
This paper is organized as follow: in Section 2 a brief review of the properties of clothoid curves ispresented. Section 3 explaines how to use least squares identification in order to obtain Bezier control pointsfor a desired piecewise clothoid reference. In section 4, the demonstration of the lineal variability of theBezier control points with different clothoid parameters is presented. Section 5 shows an example for the
21 Approximating Clothoids by Bezier Curves 91
same interval used in [4], appropriate for road design. Conclusions and bibliography are in section 6 and 7respectively.
21.2 Properties of the Clothoid Curve
The Cornu spiral or clothoid curve is defined parametrically in terms of Fresnel integrals as follows:
Q (t) =
(x (t)y (t)
)= B ·
(C(t)S(t)
)=
t∫0
cos π·ξ2
2 dξ
t∫0
sin π·ξ2
2 dξ
where B is a positive real number, parameter t is a non-negative real number. Clothoid curves have thefollowing properties:
1. Angle of tangent:τ = π · t2/2
2. Curvature: k = π · t/B3. Arc length L: L/B · t =
√π ·A · t, where A = B/
√π is the well-known clothoid constant parameter.
The most attractive property of the clothoid curve is that:
1
k= R =
A2
L
where R is the radius of the curvature.
21.3 Bezier Control Points of the Clothoid with Least Squares
A Bezier curve, originally developed by Pierre Bezier in the 1970’s, is the most common form to representplanar curves for CAD/CAM operations. Bezier curves have the formulation:
P (u) =
N∑
k=0
Ck ·N !
k! · (N − k)! · uk · (1− u)n−k
where:Ck : Bezier control pointsu : Intrinsic parameter. [0...1]N : Order of the Bezier equationBezier equation must be rewired to represent a piecewise clothoid in the intervalτi ≤ τ ≤ τf , initial, actual
and end angle of tangent respectively. That is:
P (τ) =
N∑
k=0
Ck ·N !
k! · (N − k)! ·(τf − ττf − τi
)k
·(
1− τf − ττf − τi
)N−k
Bezier equation can be expressed as a lineal equation:
Pτ = C0 ·B0τ + C1 ·B1
τ + ...+ CN ·BNτ
Where Bkτ is the kth Bernstein basis function, which is:
Bkτ =
N !
k! · (N − k)! ·(τf − ττf − τi
)k
·(
1−(τf − ττf − τi
))N−k
For a discrete intervalτi ≤ τ ≤ τf , the resulting equations can be expressed in the following matrix form:
92 Nicolas Montes, Josep Tornero
pτi
.
.pτf
=
B0τi. . BN
τi
. .
. .B0
τf. . BN
τf
·
Ck
.
.CN
P = B · CThis representation permits the use of least squares in order to compute the Bezier control points that betteradjust the desired points of the discrete curveP . The estimation of the Bezier control points can be obtainedby the use of pseudoinverse, that is:
C =(BT ·B
)−1 ·BT · PValue of the variance is obtained by the use of the next equation:
σ2 =∑τf
τi
(Pτ −
N∑
k=0
CkBkτ
)2
Also a percentage of the approximation in the point of maximum variance can be obtained as:
ε =
1−
N∑k=0
CkBkτ
Pτ
· 100
In order to know the desired points of the piecewise clothoid, an approximation of the Fresnel equationsis used [8]. It permits to know the desired points of the piecewise clothoid with an error of 1x10−9. Thedistribution of the tangent angle between the desired points obtained by [8] can be calculated as:
τ =τf − τi
length(P )
where length(P) is the number of points of the piecewise clothoid. This equation is possible because thetangent angle of the clothoid is linearly distributed with the arc length. Next figure shows a Bezier curveapproximation of order 5th and 7th obtained for a clothoid in the interval [0,π/2]. Constant parameter A ofthe clothoid is selected to 300.
Fig. 21.1. Bezier approximation of 5th(left) and 7th (right) order
Table 1 shows the variance, maximum variance and errors for the hereinbefore Bezier approximation withdifferent orders of complexity.
The precision of the approximation has a relationship with the interval of the desired piecewise clothoidand the order of the Bezier curve. Next table show the paremeters of the approximation in the interval [0, 3·π
2 ]and constant parameter A equal to 300.
21 Approximating Clothoids by Bezier Curves 93
N σ2x σ2
y Max(σ2x) Max(σ2
y) |εx| (%) |εy| (%)
5 0.2245 0.0751 0.0103 0.0046 0.03 0.46
6 0.0057 0.0011 2.9·10−4 6.2·10−5 0.029 0.0044
7 1.8·10−5 1.8·10−4 1.8·10−6 8.8·10−6 1.5·10−4 8.8·10−4
8 1.3·10−6 2.5·10−6 5.1·10−8 1.5·10−7 5.1·10−6 1.5·10−5
9 8.4·10−8 1.4·10−9 4.5·10−9 1.4·10−10 4.5·10−7 1.4·10−8
Table 21.1. Parameters of the approximation for different Bezier curve orders
N σ2x σ2
y Max(σ2x) Max(σ2
y) |εx| (%) |εy| (%)
8 0.0109 0.72 0.0089 0.0276 0.88 2.75
9 0.0231 0.0367 8.2·10−4 0.0021 0.02 0.2144
10 0.0038 1.6·10−4 1.7·10−4 6.7·10−6 0.01 0.001
11 7.4·10−5 1.6·10−4 4.6·10−6 5.6·10−6 4.8·10−4 9.8·10−4
12 1.9·10−6 1.1·10−5 1·10−7 5.3·10−7 2.3·10−4 5.3·10−5
Table 21.2. Parameters of the approximation for different Bezier curve orders
21.4 Variability of the Bezier Control Points for Clothoid Parameter
As explained in section 3, the tangent angle of the piecewise clothoid has an influence in the precision (errorand variance) and in the order of the Bezier curve. On contrary, the variation of the clothoid constantparameter (A) along the selected interval has minimum influence in the error of the approximation. Nextfigure (left) shows an approximation obtained for six piecewise clothoids in the interval [0,π/2] using 7thorderBezier curves for different constant parameters.
Fig. 21.2. Bezier approximations with constant angle
Table 3 shows the variance of the approximation for different clothoids constant parameters according toSection 3.
Figure 2(right) shows that the Bezier control points of the approximation are located in a straight linebetween selected ranges of the clothoids constant parameter (A). These straight lines can also be approximatedwith Bezier curves of 1storder. This implies that only two Bezier control points located at the start and theend of the straight line are required.
Least squares can also be used in order to demonstrate that Bezier control points are allocated in astraight line. Table 4 shows the parameters of the approximation for a Bezier curve of 1st order.
The invaluable error committed for the approximation of 1storder Bezier Curve demonstrates that theBezier control points are allocated in a straight line.
94 Nicolas Montes, Josep Tornero
A σ2x σ2
y Max(σ2x) Max(σ2
y) |εx| (%) |εy| (%)
200 8·10−6 8·10−5 6.75·10−7 3.9·10−6 6.7·10−5 3.9·10−4
400 3.2·10−5 3.2·10−4 2.7·10−6 1.5·10−5 2.7·10−4 1.5·10−3
800 1.2·10−4 1.2·10−3 1·10−5 6.28·10−5 1·10−3 6.2·10−3
1500 4.5·10−4 4.5·10−3 3.8·10−5 2.2·10−4 3.8·10−3 0.022
3000 1.8·10−3 0.018 1.5·10−4 8.8·10−4 0.015 0.088
Table 21.3. Parameters of the approximation for different constant parameters (A)
C σ2x σ2
y Max(σ2x) Max(σ2
y) |εx| (%) |εy| (%)
1 5·10−23 8·10−24 1·10−23 2·10−24 4·10−8 7·10−9
2 4·10−25 1·10−22 1·10−26 2·10−23 6·10−14 2.1·10−9
3 8·10−24 9·10−22 5·10−25 1·10−22 3 ·10−13 1·10−9
4 3·10−24 1·10−21 4·10−25 2·10−22 4·10−14 2.8·10−11
5 1·10−23 9·10−22 4·10−24 1·10−22 1·10−13 7·10−12
6 1·10−23 2·10−22 1·10−24 5·10−23 6·10−14 1.4·10−12
7 3·10−23 4·10−23 1·10−24 6·10−24 2·10−13 3.6·10−13
8 2·10−23 2·10−23 3·10−24 5·10−24 7·10−14 1.5·10−13
Table 21.4. Parameter approximation of the straight lines
In conclusion, just only it is neccesary to approximate first and last selected piecewise clothoid in the rangeof Ai ≤ A ≤ Af and for the selected range of τi ≤ τ ≤ τf with least squares using the same tangent angle inboth approximations. The resulting points are the start and the end Bezier control points of a Bezier curveof 1st order that approximates the variation of the points in the range of A. With this, an approximation ofa desired work range is obtained. Equation of the Bezier curve of the desired piecewise clothoid is:
P (A, τ) =
N∑
kτ=0
[∑1
KA=0Ckτ
A · βkA
A
]· βkτ
τ
where Ckτ
A are the Bezier control points of the straight lines and the βkA
A ,βkττ are the kth Bernstein basis
functions as described below,
βkAA =
1!
k! · (1 − k)!·
„Af −A
Af −Ai
«k·
„1 −
„Af −A
Af −Ai
««N−k
, βkττ =
N !
k! · (N − k)!·
„τf − τ
τf − τi
«k·
„1 −
„τf − τ
τf − τi
««N−k
21.5 Example in the Range of Road Design
In this section, an example of the approximation selected for the interval of the tangent angle τ ∈ [0, π/2],and the clothoid constant parameter A ∈ [30, 3000] is described.
It is interesting to remark that the range is used in practical road design [4]. The location of the Beziercontrol points are showing in the hereinafter table.
Next table shows the error committed for two values of parameter A.
21.6 Conclusions
In this paper, a strategy to obtain the approximation of a selected piecewise clothoid into Bezier curve ispresented. It is possible for two main reasons. First is that Bezier curves can be expressed as a linear equationpermitting the use of least square identification. Second is that for different constant parameters A, but thesame tangent angle interval, the resulting Bezier control points are allocated in straight lines. It permits theuse of Bezier curves of 1storder in order to approximate the location of the points.
21 Approximating Clothoids by Bezier Curves 95
CkτA Start Point End Point
kτ X Y X Y
1 -8.5·10−5 -2.1·10−4 -8.5·10−3 -2.1·10−2
2 5.967 2.7·10−3 596.7 0.2703
3 11.926 -1.2·10−2 1192.6 -1.21
4 17.916 0.636 1791.6 63.699
5 23.82 2.362 2382 236.29
6 29.46 6.141 2946.5 614.1
7 32.781 11.974 3278.1 1197.4
8 33.072 17.93 3307.2 1793
Table 21.5. Bezier control points
A σ2x σ2
y Max(σ2x) Max(σ2
y) |εx| (%) |εy| (%)
30 6.1·10−8 6·10−7 7.33·10−9 4.54·10−8 7.3·10−7 4.5·10−6
3000 6.1·10−4 6·10−3 7.33·10−5 4.54·10−4 0.0073 0.0454
Table 21.6. Variances and errors
The resulting approximation is a low order Bezier equation for an accurate approximation. For instance,in the interval of τ= [0,π/2], A= [30, 3000], and the selected 7th order Bezier equation, the maximumvariance in the worst case is 4.54·10−4, see Table 6. The error committed in the approximation of straightlines is invaluable as seeing in Table 4. This offline approximation permits to obtain the representation of thepiecewise clothoid in the selected work range. This representation can be easily introduced in CAD/CAMfields because it is expressed in Bezier, one of the standard forms. In addition, these Bezier curves can alsobe used other application requiring parametric curves such as robotics and control.
References
1. Ignacio de Corral Manuel de Villena. Topografıa de obras. SPUPC 1999.2. Scheuer and Fraichard .Planning Continuos-Curvature Paths for Car-Like Robots. IEEE Int Conf. on Intelligent
Robots and Systems. November 1996.3. Scheuer and Fraichard .Collision-Free and Continuous-Curvature Path Planning for Car-Like Robots. IEEE Int.
Conf. on Robotics and Automation, April 1997.4. L.Z. Wang, K.T.Miura, E.Nakamae, T.Yamamoto, T.J.Wang. An approximation approach of the clothoid curve
defined in the interval [0,π/2] and its offset by free-form curves. Computer Aided Design 33, pp. 1049-1058, 2001.5. J.Sanchez-Reyes, J.M.Chacon. Polinomial approximation to clothoids via s-power series. Computer –Aided Design.
December 2003.6. J.Sanchez-Reyes. The symmetric Analogue of the Polinomial Power Basis. ACM Transactions on graphics, vol 16,
No 3, pp. 319-357. July 1997.7. D.S.Meek, D.J.Walton. An arc spline approximation to a clothoid.. Journal of computational and applied math-
ematics. 170 pp. 59-77, 2004.8. Klaus D. Mielenz. Computation of Fresnel Integrals II. Journal of Research of the National Institute of Standard
and Technology. Vol 105, No4, 2000.
22
Parallelization of Triangular Decompositions
Marc Moreno Maza and Yuzhen Xie
ORCCA, University of Western Ontario, London, Ontario, Canada.
Summary. We discuss the parallelization of algorithms for solving polynomial systems symbolically. We address thefollowing questions: How to discover geometrical information, at an early stage of the solving process, that would befavorable to parallel execution? How to ensure load balancing among the processors? We answer these questions in thecontext of triangular decompositions. We show that rich opportunities for parallel execution are obtained by combiningmodular techniques together with a solving process producing components by decreasing order of dimension.
22.1 Introduction
Since the discovery of Grobner bases, the algorithmic advances in Commutative Algebra have made possibleto tackle many classical problems in Algebraic Geometry that were previously out of reach. However, algo-rithmic progress is still desirable, for instance when solving symbolically a large system of algebraic non-linearequations. For such a system, in particular if its solution set consists of geometric components of differentdimension (points, curves, surfaces, etc) it is necessary to combine Grobner bases with decomposition tech-niques, such as triangular decompositions. Ideally, one would like each of the different components to beproduced by an independent processor, or set of processors. In practice, the input polynomial system, whichis hiding those components, requires some transformations in order to split the computations into sub-systemsand, then, lead to the desired components. The efficiency of this approach depends on its ability to detectand exploit geometrical information during the solving process. Its implementation, which naturally mustinvolve parallel symbolic computations, is yet another challenge.
Our work addresses two questions: How to discover geometrical information, at an early stage of the solvingprocess, that would be favorable to parallel execution? How to ensure load balancing among the processors?We answer these questions in the context of triangular decompositions [3] which are a popular way of solvingpolynomial systems symbolically. They are used in geometric computations [5] such as implicitization andclassification problems [8, 9]. They have interesting properties [6] for developing modular methods and stablenumerical techniques.
Algorithms computing triangular decompositions tend to split the input polynomial system into subsys-tems and, therefore, are natural candidate for parallel implementation. However, the only such method whichhas been parallelized so far is the Characteristic Set Method of Wu, as reported in [1, 11]. This approach suf-fers from several limitations. For instance, the computation of the second component cannot start before thatof the first one is completed; this is a limitation in view of coarse-grain parallelization. Actually, the operationwhich is parallelized in [1, 11] is the pseudo-reduction of several polynomials w.r.t. a triangular set. This issimilar to most approaches in parallelizing Buchberger’s algorithm [4, 2]; we view them as medium-grainparallelization.
The algorithms in [10], that we call Triade, for TRIAngular DEcompositions, follows a different approach,summarized in Section 22.2. It appears to be a natural candidate for coarse-grain parallel implementation,based on geometrical considerations.
However, several challenges remain to be considered. First, load balancing is very difficult to control dueto irregular tasks. Even worse: for some input polynomial systems, resource consuming tasks may not beexecuted concurrently. (In characteristic zero, most examples from www.SymbolicData.org have a triangular
22 Parallelization of Triangular Decompositions 97
decomposition consisting of a single component.) Second, data communication can be very heavy due to largeintermediate results.
In order to achieve load balancing we rely on the following facts. For an input polynomial system, theTriade algorithm can generate the (intermediate or output) components by decreasing order of dimension,as explained in Section 22.2. As a consequence, expensive tasks (those in lower dimension) can be processedconcurrently. In addition, when solving a (non-trivial) polynomial system modulo a prime integer, the numberof these tasks may be sufficient for expecting a good speed-up in a parallel execution. The case of polyno-mial systems with integer coefficients can also benefit from these features by using the modular techniquesintroduced in [6].
We have developed a parallel scheme for the Triade algorithm, presented in Section 22.3, aiming atminimizing data communication overhead. Tasks are scheduled and updated by a process manager. Individualtasks are solved “lazily” by process workers. However, each process worker keeps track of enough informationsuch that it can continue the solving of some of these tasks, when needed. We have realized a preliminaryimplementation on a shared memory multiprocessor. In Section 22.4, we report on our experimental results.
22.2 Intersecting Varieties with Quasi-Components
This section is an overview of the Triade algorithm [10] with an emphasis on its properties that are favorableto its parallelization. The notions of a regular chains and a quasi-component can be found in [3]. Definitions 1and 2 introduce notions specific to the algorithm, after some notations.
Let K be a field and X = x1 < · · · < xn be ordered variables. For a subset F ⊂ K[X], we denote byV (F ) the zero set of F in the affine space K
nwhere K is an algebraic closure of K. For a subset W ⊂ K
n,
we denote by W the Zariski closure of W w.r.t. K. For a regular chain T ⊂ K[X], we denote by W (T ) itsquasi-component, by Sat(T ) its saturated ideal, and, for F ⊂ K[X], we denote by Z(F, T ) the intersectionV (F )∩W (T ).
Definition 1. We call a task any couple [F, T ] where F ⊂ K[X] is a polynomial set and T ⊂ K[X] is aregular chain. The task [F, T ] is solved if F is empty, otherwise it is unsolved. By solving a task, we meancomputing regular chains T1, . . . , T` such that we have:
V (F ) ∩ W (T ) ⊆ ∪ei=1W (Ti) ⊆ V (F ) ∩ W (T ).
The goal of the Triade algorithm is to solve tasks in the above sense. Moreover, a motivation in thealgorithm design is to generate all regular chains (final or intermediate) by increasing order of their cardinality,and, thus by decreasing order of the dimension of their saturated ideals. However, the “basic routine” is thecomputation of intersections of the form Z(p, T ) (for a polynomial p) which may consist of components ofdifferent dimensions. Resolving this conflict of interest is achieved by a form of lazy evaluation, formalizedbelow.
Definition 2. The tasks [F1, T1], . . . , [Fd, Td] form a delayed split of the task [F, T ] and we write [F, T ] 7−→D
[F1, T1], . . . , [Fd, Td] if the following five properties together hold: (i) Z(Fi, Ti)≺Z(F, T ), (ii) Z(F, T ) ⊆Z(F1, T1) ∪ · · · ∪ Z(Fd, Td), (iii) Sat(T ) ⊆ Sat(Ti), (iv) Fi 6= ∅ =⇒ F ⊆ Fi, (v) Fi = ∅ =⇒ W (Ti) ⊆V (F ).
Property (i) means that each “output” task [Fi, Ti] is more solved than the “input” one [F, T ] (in a sensethat we do not precise here and which is based on Ritt-Wu ordering for characteristic sets [10]). Properties(ii) to (v) imply:
V (F ) ∩ W (T ) ⊆ ∪di=1Z(Fi, Ti) ⊆ V (F ) ∩ W (T ).
The properties of delayed splits show that solving the task [F, T ] reduces to solving tasks of the form [p, T ]where T∪t is a triangular set, but not necessarily a regular chain, or of the form [p, T∪t] where p, t arenon-constant polynomials with the same main variable v and such that both T∪t and T∪t are regularchains. By means of polynomial GCDs and resultants, in some general sense defined in [10], one can design anoperation extend(p, T ) producing a delayed split of the above task [p, T ] and, an operation decompose(p, T ∪ t)producing a delayed split of the above task [p, T∪t]. Each of these operations satisfies the following key
98 Marc Moreno Maza and Yuzhen Xie
property: for every output task [Fi, Ti] we have Fi = ∅ ⇐⇒ |Ti| = |T | + 1. Hence, these operations solve“lazily” in each branch where the dimension drops and solve “completely” in the others.
We sketch now a procedure solving an input task [F, T ] and generating all computed regular chains (finalor intermediate) by decreasing order of the dimension of their saturated ideals.
This procedure uses a list U consisting of all unsolved tasks, a list S consisting of solved tasks, and aninteger H which is the current size of the regular chains being computed. Initially U = [[F, T ]], S is the emptylist and H = |T |. Our procedure can be sketched as follows
(1) Let V be the list of all tasks [F ′, T ′] in U with |T ′| = H.(2) Let V ′ be the list of all tasks [F ′, T ′] in V to which the operation decompose applies.(3) If V ′ 6= ∅, then apply decompose to its elements, update the lists U and S, and go to (1).(4) If V ′ = ∅, then apply extend to each element in V , update the lists U and S, replace H by H + 1 and go
to (1).
All steps (1) to (4) lead naturally to parallel execution. This procedure is also different from Algorithm 2 in[10], which was meant to be executed sequentially.
Generating regular chains by decreasing size has at least two benefits. First, it allows to detect redundantcomponents (by means of an inclusion test) at an early stage of the solving process. Indeed, if W (T1) containsW (T2) then we must have |T1| ≤ |T2|. Second, it forces the algorithm to delay the computations in lowerdimension toward the end of the solving process, which increases the opportunities for parallelization. Indeed,when computing decompose(p, T ∪ t) the larger is |T |, the more expensive are calculations modulo Sat(T ).See for instance the complexity results in [7].
22.3 Parallel Implementation Scheme
Based on the procedure of Section 22.2, we propose now an implementation scheme for parallel triangulardecompositions. We denote by P0 the Process Manager. Any other process worker will be denoted by somePi for i > 0. Each task [Fi, Ti] has a unique task identifier (TID), and each worker Pi owns a unique workeridentifier (WID). The manager P0 maintains a taskTable, which contains the tasks that are already solved,and those which are waiting (to be selected). For each task, we also record its TID and WID (the worker thathas produced it). Every process worker Pi records all the tasks that Pi has computed in a localTaskTable (Pi
stores their actual values and not only their identifiers). Recall that the key idea of the algorithm is to solveby decreasing order of dimension. To this effect, we have a global variable (visible by all workers) H takingsuccessively the values |T |, |T |+1, . . .. WhenH = k, then all the regular chains in the triangular decompositionof V (F ) which have less than k elements have already been computed. In addition, the algorithm is currentlycomputing those regular chains which have k elements. We describe below the algorithm of P0 with the inputtask [F, T ]. Initially H := |T |.(1) Select all the tasks which can lead to components of dimension n−H, denoted as R1, . . . , Rs.(2) Find s workers to solve these tasks. This may result in launching new workers, but try to solve (as much
as possible) each task on the worker that created it (when this is possible, only the TID needs to be sentto the worker).
(3) Collect all the tasks (solved or unsolved) produced by the workers and update taskTable.(4) If all the tasks are solved, send a stop message to each worker and halt, otherwise H := H + 1 and go
to (1).
We describe now the algorithm of process worker Pi. Initially localTaskTable := .(1) Receive a task, or a message from manager P0. In case of a “stop” message, then halt.(2) In case of a TID, look up the localTaskTable for this task, say [Fj , Tj ]. Compute a delayed split of
[Fj , Tj ] such that all unsolved tasks are contained in components of dimension (strictly) less than n−H.Update the output tasks and localTaskTable by removing redundant ones.
(3) Send the output tasks to manager P0 and go to (1).
As much as possible we let process workers continue solving the tasks they have produced, so as to reducethe data communication overhead. Besides, new worker processes are activated only when the number ofselected tasks exceeds the number of running workers. This strategy helps minimizing the cost of dynamicprocess creation and management.
22 Parallelization of Triangular Decompositions 99
22.4 Experimentation
We have realized a preliminary implementation of the algorithm in Section 22.2 on a shared memory multi-processor, AuthenticAMD with four CPUs (2390MHz) and 8 GB total memory. Implementation is based onthe existing BasicMath library written in Aldor, see www.aldor.org, which is a sequential implementationof the Triade algorithm. We have created two binary modules: Process Manager and Process Worker. Thelatter one is activated and terminated dynamically by the former one as soon as a delayed split computa-tion is required. Between theses binaries, data communication is done through shared memory segments,which are efficient ways for System V inter process communication. To this end, tasks, and thus multivariatepolynomials, are tuned into machine integer arrays through Kronecker substitution.
The polynomial systems that are used in this experimentation are taken from www.SymbolicData.org.For each system, n denotes the number of variables, d is the total degree of the polynomial system and p isthe prime number used in both the sequential and the parallel solving. We report in the following figures twoexamples, namely: gametwo5 and Uteshev-Bikker. For each of them: (1) we plot the number of processes actingat a time during the whole solving procedure, and (2) we show the average number of processes. Although thespeed-up ratio we gained is between 1.5 and 2 by comparing with the sequential implementation in Aldor,this is very promising for such a coarse-grain parallel implementation, where truly expensive tasks can beprocessed in parallel. It is noticed that the average number of processes in the entire solving procedure isabout 3.
The implementation of the scheme in Section 22.3 is work in progress. In our current implementation,branching one expensive task into two expensive tasks relies only on the D5 Principle. Applying factorizationinto irreducible polynomials should improve the speed-up ratio. Further more, we need to integrate fine-grainparallelization, for GCD/resultant computations, which should improve the time and space efficiency of ourparallel triangular decompositions.
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References
1. I. A. Ajwa. Parallel Algorithms and Implementations for the Grobner Bases Algorithm and the Characteristic SetMethod. PhD thesis, Kent State University, Kent, Ohio, 1998.
2. G. Attardi and C. Traverso. Strategy-accurate parallel Buchberger algorithms. Journal of Symbolic Computation,21(4):411–425, 1996.
3. P. Aubry, D. Lazard, and M. Moreno Maza. On the theories of triangular sets. J. Symb. Comp., 28(1-2):105–124,1999.
4. S. Chakrabarti and K. Yelick. Distributed data structures and algorithms for Grobner basis computation. Lispand Symbolic Computation: An International Journal, 7:147–172, 1994.
5. F. Chen and D. Wang, editors. Geometric Computation. Number 11 in Lecture Notes Series on Computing.World Scientific Publishing Co., Singapore, New Jersey, 2004.
6. X. Dahan, M. Moreno Maza, E. Schost, W. Wu, and Y. Xie. Lifting techniques for triangular decompositions. InISSAC’05, ACM Press, 2005.
7. X. Dahan, M. Moreno Maza, E. Schost, and Y. Xie. On the complexity of the D5 principle. In Proc. of TransgressiveComputing 2006, Granada, Spain, 2006.
100 Marc Moreno Maza and Yuzhen Xie
8. M.V. Foursov and M. Moreno Maza. On computer-assisted classification of coupled integrable equations. Inproceedings of ISSAC 2001, pages 129–136, London, Ontario, 2001. ACM Press.
9. I. A. Kogan and M. Moreno Maza. Computation of canonical forms for ternary cubics. In Teo Mora, editor, Proc.ISSAC 2002, pages 151–160. ACM Press, July 2002.
10. M. Moreno Maza. On triangular decompositions of algebraic varieties. Technical Report TR 4/99, NAG Ltd,Oxford, UK, 1999. Presented at the MEGA-2000 Conference, Bath, England.
11. Y.W. Wu, G.W. Yang, H. Yang, W.M. Zheng, and D.D. Lin. A distributed computing model for wu’s method.Journal of Software (in Chinese), 16(3), 2005.
23
The Geometry of Flags
Boris Odehnal
Vienna Univ. of Tech., Inst. f. Diskrete Math. u. Geometrie, Wiedner Hauptstrasse 8-10/104/3, 1040 Wien, Austria.
Summary. We present a low dimensional point model for the set of flags in R3 which is contained in a six-dimensionalalgebraic variety M6
8 of degree eight embedded in a nine-dimensional projective space. It turns out that M 68 can be
parameterized rationally. We study the geometry of the manifold and show the relations between the geometry offlags and kinematics in Euclidean R3 and Non-Euclidean geometries.
23.1 Introduction
The study of higher dimensional algebraic (and other) manifolds has a long history. Grassmann, Segre,and Veronese varieties as well as Schubert manifolds have been a field of intensive research [3, 4, 5, 10].The major benefits of geometries of this type are the following: (1) They allow to treat objects consisting of acollection of elements of projective spaces as points in some (of course higher dimensional) projective space.(2) The transformations of these objects can be represented by collineations (i.e. linear transformations) inthe model space. The obvious disadvantage (in view of applications) of these models is their relatively highdimensional model space. Computations can become long winded and costly.
Nowadays point models for some special manifolds appear frequently in application areas: The Klein
model of line space as well as other models of line space can be used for approximation and interpolationproblems in line space [13, 14]. Even a new model for the set of line elements in Euclidean three-space wasdeveloped in [12] for the recognition and reconstruction of spiral surfaces, see [6].
In the following we study the manifold of flags in Euclidean three-space R3. The investigations are not donefor their own sake. We show a tricky way to parameterize this manifold by means of rational functions. Thisparameterization method also applies to other parameterization problems. Further we discuss the geometryof this six-dimensional manifold in order to get insight and understand it. Applications of this are not wellstudied until now. Though motion planning by means of subdivision motions on this manifold could be offuture interest.
23.2 Equation of the Manifold M 6
8
23.2.1 Lines in space
Since we are dealing with Euclidean three-space we use Cartesian coordinates p = (p1, p2, p3) in order torepresent points P . A line L in Euclidean three space will be described by normalized Plucker coordinates.Assume that L is parallel to the unit vector l 6= 0 (i.e. its Euclidean length ‖l‖ equals 1) and passes throughthe point P with coordinate vector p. Then we define the normalized Plucker of L coordinates as
L = (l, l) = (l1, l2, l3; l4, l5, l6), (23.1)
where l = p× l is the momentum vector of L. Here and in the following the cross product of vectors in R3 isdenoted by ×. The coordinates of L do not depend on the choice of P on L.
With 〈., .〉 we denote the standard scalar product of vectors in R3. It is obvious that
102 Boris Odehnal
M42 : 〈l, l〉 = l1l4 + l2l5 + l3l6 = 0 (23.2)
holds.If we drop the normalization of l the coordinates li can be considered as homogeneous coordinates of
points in a projective five-space P5. M42 is the Klein quadric or Plucker quadric (i.e. the Grassmannian
G31). It is a point model for the set of lines in three-space, see [3, 7, 13]. M 4
2 is a regular quadric carrying twothree-parameter families of planes corresponding to bundles and fields of lines.
23.2.2 Line elements in space
The pair (P,L) consisting of a line L and a point P on it will be called line element. In order to describeline elements we assign coordinates to them in the following way (cf. [6, 12]): The line L is described byits normalized Plucker coordinates (l, l). In order to fix the point P on L we add a seventh coordinateλ := 〈p, l〉 to the Plucker coordinates of L and let
(P,L) = (l, l, λ) = (l1, l2, l3; l4, l5, l6; l7) (23.3)
be the coordinates of the line element (P,L).The point P can be recovered from the line element coordinates by p = l × l + λl. Note that this is true
because ‖l‖ = 1, which is yet assumed. The line element coordinates of (P,L) thus satisfy (23.2).Again we can drop the normalization condition of l. Then we can consider the seven coordinates of line
elements as homogeneous coordinates of points in a projective six-space P6. Since l7 does not appear inEq. (23.2), we see that Eq. (23.2) (interpreted in P6) is the Equation of a quadratic cone M 5
2 whose pointscorrespond to line elements in Euclidean three-space. M 5
2 has a point for its vertex and two three-parameterfamilies of three-dimensional generators. The points contained in the generator l1 = l2 = l3 = 0 do notcorrespond to line elements in Euclidean R3.
23.2.3 Flags in space
We follow [10] and define:
Definition 1. A flag F in Euclidean three-space R3 is a triplet (P,L,E) where P is a point, L is a line, andE is a plane and P ∈ L ⊂ E.
Obviously a flag F = (P,L,E) consists of a line element (P,L) and a plane carrying it. Assume that the
unit vector l is perpendicular to the plane E. In order to assign coordinates to the flag F we add the vectorl to the coordinates of the line element (P,L). So we define
(P,L,E) = (l, l, l, λ) = (l1, l2, l3; l4, l5, l6; l7, l8, l9; l10) (23.4)
as coordinates of the flag F = (P,L,E).The coordinates of F satisfy the following conditions:
〈l, l〉= 0 =l1l4 + l2l5 + l3l6, (23.5)
〈l, l〉= 0 =l1l7 + l2l8 + l3l9, (23.6)
〈l, l〉 − 〈l, l〉= 0 =l21 + l22 + l23 − l27 − l28 − l29. (23.7)
Any vector (l1, . . . , l10) satisfying Eqs. (23.5), (23.6), and (23.7) defines (up to orientations) a unique flagF in R3. The point P , the line L, and the plane E can be recovered from the coordinates of F by p = l×l+λl,L = (l, l), and E : 〈l, x〉 = det(l, l, l).
23 The Geometry of Flags 103
23.3 Some Facts on M6
8
Now we can drop the norming conditions on l and l, respectively. We only assume that they are of equallength, i.e. ‖l‖ = ‖l‖. We observe that the coordinates of F are homogeneous in the following sense: If we
scale l and l by a real non-vanishing factor, say c, we find the momentum vector changes to cl and λ changesto cλ. This makes it possible to interpret the coordinates li of F as homogeneous coordinates of points in aprojective space P9 of dimension nine. Eqs. (23.5), (23.6), and (23.7) define an algebraic variety M 6
8 in P9.We have:
Theorem 2. The algebraic degree and the dimension of M 68 equal eight and six, respectively.
Proof. The dimension d of M 68 equals six, which is clear since we need six parameters in order to fix a flag
in R3.The algebraic degree follows from the Hilbert-Polynomial (for definition and properties see [15])
H(t) =1
90t6 +
2
15t5 +
25
36t4 + 2t3 +
593
180t2 +
43
15t+ 1
since degM68 = d!c6 = 8, where c6 is the coefficient of the leading monomial.
Suprisingly we find the following result:
Theorem 3. The manifold M 68 allows a rational parameterization.
Proof. We construct a rational parameterization Φ of M 68 . We let P be represented by p = (u3, u4, u5) ∈ R3.
Further we assume that l = (2u1, 2u2, 1 − u21 − u2
2)/M , where M = 1 + u21 + u2
2. The momentum vector isgiven by p× l according to its definition.
Since l is a rational isothermal parameterization of the Euclidean unit sphere S2 we find that its partialderivatives (with respect to the parameters ui) are of equal length and orthogonal to each other, i.e. we have‖l,1‖ = ‖l,2‖ = 2M−1 and 〈l,1, l,2〉 = 0. Note that l,1, l,2, l is an orthogonal frame.
Eqs. (23.5), (23.6), and (23.7) tell us that l is orthogonal to l and thus we can write l = l,1/‖l,1‖c3 +
l,2/‖l,2‖s3. We substitute c3 = (1−u23)/N and s3 = 2u3/N with N = 1+u2
3 which guarantees that ‖l‖ = ‖l‖and find
MNΦ(u1, u2, u3, u4, u5, u6) = [2u1N, 2u2N,NS;
N(u5S − 2u2u6), N(2u1u6 − u4S),
2N(u2u4 − u1u5);
(1− u23)(M − 2u2
1)− 4u1u2u3,
2u3(M − 2u22)− 2(1− u2
3)u1u2,
− 2u1(1− u23)− 4u2u3;
N(2u1u4 + 2u2u5 + u6S)],
(23.8)
where S = 1− u21 − u2
2.We note that (u1, . . . , u6) is an affine parameter and (23.8) does not reach the entire surface M 6
8 .
23.4 Geometric Properties of M 6
8
In Eqs. (23.5), (23.6), and (23.7) the coordinate l10 does not show up. Thus we have:
Theorem 4. The manifold M 68 is a cone with the thenth base point of the standard projective frame for its
vertex.
104 Boris Odehnal
From Eqs. (23.5), (23.6), and (23.7) we conclude that M 68 is the intersection of three quadratic cones
∆i. At least from the viewpoint of projective geometry (over the real number field) these three cones do notdiffer: One can find collineations of P9 (i.e. linear mappings in R10) transforming one cone ∆i into anothercone ∆j .
The cone ∆1 has the Klein quadric M42 contained in the subspace B1 : x7 = x8 = x9 = x10 = 0 for a
director quadric. Its vertex is the three-dimensional subspace V1 : x1 = x2 = x3 = x4 = x5 = x6 = 0. SinceM4
2 carries two three-parameter families of planes ∆1 has two three-parameter families of six-dimensionalprojective subspaces for its generators. The cones ∆2 and ∆3 have similar properties. The vertex space of∆1 is entirely contained in ∆2 and vice versa.
We use the following definition:
Definition 5. The set of flags sharing two components is called pencil of flags.
We can distiguish between three types of pencils of flags: (1) flags with fixed line and plane component,(2) flags with fixed point and plane component, and (3) flags with fixed point and line component (see Fig.23.1).
Fig. 23.1. Flags sharing two components.
The three types of pencils of flags correspond to certain subspaces in M 68 :
Theorem 6. Pencils of flags correspond to lines in M 68 .
Proof. Parameterizing these pencils and computing their flag coordinates leads to linear parameterizationsof the one-dimensional subspaces in M 6
8 corresponding to the pencils.There are other simple manifolds of flags:
Definition 7. The set of flags sharing one component is called bundle of flags.
We find three different types of bundles of flags (see Fig. 23.2): (1) flags with a common point, (2) flagswith a common plane, and (3) flags with a common line. Unfortunately only one type of these correspondsto projective subspaces in M 6
8 . We have:
Theorem 8. The bundels of flags with a common line component correspond to planes in M 68 .
Proof. The proof is as simple as in the case of pencils.
23.5 Euclidean Kinematics and Non-Euclidean Geometries
In this section we finish the discussion of bundels of flags and corresponding subspaces. For that we look closerto the definition of flag coordinates. We recall that the vectors l and l were unit vectors in the beginning.Consequently the line L (and thus the line element) and the plane E are oriented. Therefore a flag can be
oriented in four different ways depending on the respective choices of the orientations of l and l.As is clear from the definition the triplet l, l, l× l is a Cartesian frame attached to the flag F = (l, l, l, λ).
Assume an orientation is fixed and a certain proto flag F0 (system of reference, referred to as the fixed system)is chosen. Then there exits a unique Euclidean motion transforming F0 into F . Thus we have:
23 The Geometry of Flags 105
Fig. 23.2. Flags sharing only one component.
Theorem 9. There is a bijective correspondence between the set of oriented flags in Euclidean three-spaceand the set of Euclidean motions.
Using the calculus of dual quaternions (see [7, 8]) one finds that Euclidean motions can be mapped topoints on a certain quadric S6
2 ⊂ P7, the so called Study quadric, see [2, 7, 16]. By identifying Euclideanmotions and oriented flags we find:
Theorem 10. There is a one-to-one correspondence between the set of oriented flags in Euclidean three-spaceand the points of the Study quadric S6
2 .
The Study quadric is a hyper surface in P7 and it is not possible to find a lower dimensional point modelfor the set of (oriented) flags in Euclidean three-space.
Note that the geometric object F = (P,L,E) carries four different orientations. These correspond to fourEuclidean motions transforming the proto flag into F . Thus the manifold of Euclidean motions is coveredfour times by the set of flags (without orientations) in Euclidean three space.
The bundles of flags with fixed plane or point component are closely related to Non-Euclidean geometries:
Theorem 11.(1) The set of oriented flags through a fixed point form an elliptic three-space.(2) The set of flags with a fixed plane component form a quasi elliptic three-space.
Proof. (1) Fix one oriented flag in the bundle, it then serves as proto flag F0. The Euclidean motionstransforming F0 into any other flag F in the bundle are rotations about the common point. It is well knownthat the rotations about one fixed point form an elliptic three space, see [1, 4, 9].(2) Fix an oriented line element (P0, L0) in the plane E as proto element. Now there exists a unique planarEuclidean motion transforming the proto element in a certain oriented line element (P,L). So any orientedline element in E can be identyfied with a certain planar Euclidean motion, which can be mapped to exactlyone point of a quasi-elliptic three-space (via the Blaschke-Grunwald mapping), see [4, 7, 13].
Note that the elliptic three-space as well as the quasi-elliptic three-space appearing in the above theoremare covered more than once.
23.6 Final Remarks
The manifold of oriented flags in Euclidean R3 which can also be seen as the Euclidean motion group can beused for motion planning. Several algorithms for that are developed. Most of them use orthogonal projectiononto the group of motions. Subdivision schemes are not restricted to polygons/polyhedra in Euclidean spacesthey can also be applied in any Riemannian manifold such as the Euclidean motion group, see [17]. A manifoldwith explicitly known rational parameterization of relatively low degree may support subdivion algorithms.
106 Boris Odehnal
The construction of the rational parameterization of M 68 given in (23.8) does not use an algorithm and
is mainly discovered by close inspection. A similar approach works for another algebraic manifold M 55 which
serves as a point model for the set of line elements in projective space P3 [11].It would be of interest to find techniques or algorithms for the construction of low degree parameterizations
of such manifolds. Maybe the study of geometric properties and a deeper insight in projective generations ofsuch manifolds can help to find appropriate algorithms.
References
1. W. Blaschke: Kinematik und Quaternionen. VEB Dt. Verlag der Wissenschaften, Berlin, 1960.2. O. Bottema, B. Roth.: Theoretical kinematics. Dover Publ., New York, 1990.3. W. Burau: Mehrdimensionale projektive und hohere Geometrie. VEB Dt. Verlag der Wissenschaften, Berlin,
1961.4. O. Giering: Vorlesungen uber hohere Geometrie. Vieweg, Braunschweig-Wiesbaden, 1982.5. H. Havlicek, K. List, C. Zanella: On automorphisms of flag spaces. Linear a. Multilinear Algebra, Vol. 50:3,
2002, 241–251.6. M. Hofer, B. Odehnal, H. Pottmann, T. Steiner, J. Wallner: 3D shape recognition and reconstruction
based on line element geometry. In: Thenth IEEE International Confernence on Computer Vision, volume 2, pages1532–1538. IEEE Computer Society, 2005, ISBN 0-7695-2334-X.
7. M. Husty, A. Karger, H. Sachs, W. Steinhilper: Kinematik und Robotik. Springer, Berlin, 1997.8. A. Karger: Space Kinematics and Lie Groups. Gordon & Breach Science Publishers, New York, 1985.9. H. R. Muller: Spharische Kinematik. VEB Dt. Verlag der Wissenschaften, Berlin, 1962.
10. B. Odehnal: Flags in Euclidean three-space. Mathematica Pannonica 17/1 (2006), 29–48.11. B. Odehnal: Die Linienelemente des P3. Sitzungsber. Akad. Wiss. Wien, (2006) to appear.12. B. Odehnal, H. Pottmann, J. Wallner: Equiform kinematics and the geometry of line elements. Geometry
Preprint No. 129, Vienna University of Technology, 2004.13. H. Pottmann, J. Wallner: Computational line geometry. Springer, Berlin, 2001.14. H. Pottmann, M. Hofer, B. Odehnal, and J. Wallner: Line geometry for 3D shape understanding and
reconstruction. In: T. Pajdla and J. Matas, editors, Computer Vision - ECCV 2004, Part I, volume 3021 ofLecture Notes in Computer Science, pages 297–309. Springer, 2004, ISBN 3-540-21984-6.
15. I. R. Shafarevich: Basic Algebraic Geometry I. Springer, New York, 1994.16. E. Study: Geometrie der Dynamen. B. G. Teubner, Leipzig, 1903.17. J. Wallner & H. Pottmann: Intrinsic subdivision with smooth limits for graphics and animation. ACM Trans.
Graphics 25/2 (2006),356–374.
24
Sphere-Geometric Aspects of Bisector Surfaces
Martin Peternell
Inst. of Discrete Math. and Geometry, Univ. of Techn. Vienna, Wiedner Hauptstrasse 8-10, 1040 Vienna, [email protected]
Summary. The bisector surface B of two smooth input objects P and Q is the locus of centers of spheres which aretangent to P and Q, respectively. This definition already indicates that methods from sphere geometry, in particularLie-sphere geometry apply nicely to the construction of these surfaces. The computation of bisector surfaces of generalinput surfaces results in the solution of a system of nonlinear equations. We show that if both surfaces are canal surfacesor if one surface is a Lie-sphere, the construction is elementary.
24.1 Introduction
Given two geometric objects P and Q in Euclidean 3–space R3, their bisector B is defined as locus ofequidistant points from P and Q. Since distances are measured orthogonal to both objects, the bisector Bis the set of centers of spheres touching both P and Q. We do not require that the distance between B andP (or Q) is minimal and discuss only the untrimmed bisector, see [8]. The objects P and Q shall be points,smooth curves and surfaces.
A possible method to study bisectors in R3 from a sphere-geometric point of view uses concepts fromLaguerre geometry. A detailed description of the planar case is given in [11], general monographs on spheregeometry are for instance [2, 4] and a brief introduction and some details are given in [13]. Within Laguerregeometry one still distinguishes between oriented spheres and oriented planes. Lie-sphere geometry providesa unifying concept to deal with all these elements and this will be applied in the following.
A Lie-sphere is defined to be either an oriented sphere or an oriented plane or a point in R3. We will usea quadric model L where Lie-spheres are represented as points and oriented contact between two elementsis determined by the conjugacy relation with respect to the quadric L. Envelopes of one-parameter familyof Lie-spheres are called Lie-canal surfaces. This class of surfaces consists of canal surfaces, together withcurves and developable surfaces, as one parameter families of points and oriented planes. All other smoothsurfaces are denoted by general surfaces, from the sphere geometric point of view.
The computation of the bisector surface of two input surfaces is difficult in general and results in a systemof nonlinear equations. We show that if both input surfaces are Lie-canal surfaces or one input surface isa Lie-sphere, the construction of the bisector is either linear or quadratic and we call this an elementaryconstruction. Several results about geometric properties of bisector surfaces can be found in [5, 6, 8, 12].
24.2 The Quadric Model of Lie-Sphere Geometry
Let R3 be the real Euclidean 3–space. We identify points in R3 by their coordinate vectors x = (x1, x2, x3)>
with respect to a Cartesian coordinate system. The canonical Euclidean dot product is denoted by x> · yand for the squared norm ‖x‖2 of vectors we use also x2.
An oriented sphere S : (x−m)2 − r2 = 0 in R3 is uniquely determined by its center m = (m1,m2,m3)>
and its signed radius r. Thus there is a bijective correspondence between points M = (m1,m2,m3, r)> in R4
and oriented spheres in R3. We make the arrangement that positive radii shall represent spheres with normalvectors pointing outwards. Points x = (x1, x2, x3) in R3 are considered as spheres of zero radius.
108 Martin Peternell
Two spheres S1, S2 with centers m1,m2 and radii r1, r2 respectively, are in oriented contact if
(m1 −m2)2 − (r1 − r2)2 = 0. (24.1)
An oriented plane E in R3 is determined by an equation e0 + x1e1 + x2e2 + x3e3 = 0. We also use thenotation e0 + e> · x = 0 and assume the normalization e> · e = 1, which makes the description unique. Anoriented plane E : e0 + e> ·x = 0 is in oriented contact to an oriented sphere S : (x−m)2− r2 = 0 exactly if
e0 + e> ·m + r = 0, with e> · e = 1. (24.2)
Points and spheres or points and planes are said to be in oriented contact in case of incidence.A bijective mapping is called a Lie-transformation if it maps Lie-spheres to Lie-spheres and preserves
oriented contact. These transformations are not necessarily point-preserving, but they can map points tooriented spheres. While R4 can be considered as model of Laguerre geometry, where points represent orientedspheres, a point model of Lie-sphere geometry is given by a quadric L of index 1 in R5, where the indexdenotes the largest dimension of subspaces contained in the quadric.
Let P 5 be the projective extension of R5. Points in P 5 are denoted by capital bold face letters X and areidentified with their homogeneous coordinate vectors X = (X0, . . . , X5)R. The Lie quadric L is determinedby the quadratic equation
L : 2X0X5 +X21 +X2
2 +X23 −X2
4 = 0. (24.3)
The correspondence between Lie-spheres X in R3 and points X ∈ L ⊂ P 5 is given by the mapping λ,
sphere (m, r) 7→M = (1,m, r,− 12 (m2 − r2))R,
point (p) 7→ P = (1,p, 0,− 12p
2)R,
plane (e0, e) 7→ E = (0, e,−1, e0)R, with ‖e‖ = 1.
(24.4)
Points x ∈ R3 have λ-images X in X4 = 0, while oriented planes E are mapped to points E ∈ X0 = 0. Thepoint Z = (0, 0, 0, 0, 0, 1)R does not occur as λ-image of spheres, planes or points of R3 but is consideredas λ-image of the ’point’ ∞, which is used to compactify R3 in the sense of Mobius geometry (one-pointcompactification). The quadratic cone X0 = 0, X2
1 +X22 +X2
3 −X24 = 0 consists of the λ-images of oriented
planes and is often referred to as Blaschke cone (cylinder). The quadric L∩X4 = 0 is projectively equivalentto S3 and is a point model of the Mobius geometry in R3.
The group of Lie-transformations is represented by the bijective projective transformations κ : P 5 → P 5
which map L onto itself. It is a 15-parametric group and contains the groups of Laguerre transformationsand Mobius transformations as subgroups. Laguerre transformations preserve oriented planes and Mobiustransformations preserve points.
The mapping λ is quadratic. The projection
π : L → P 3 : X4 = X5 = 0, (24.5)
with ’center’ Z∨(0, 0, 0, 0, 1, 0)R onto P 3 maps a finite point X withX0 6= 0 to the center (X1/X0, X2/X0, X3/X0)of the corresponding Lie-sphere X. This fact shall motivate the special choice of the coordinate system, al-though the classical literature [1] often uses a different normal form of L.
The stereographic projection from Z onto X5 = 0 leads to a point model of Laguerre geometry notconsidering the images of oriented planes. We note that R4 is considered as Lorentz space with (+ + +−) asthe signature of the corresponding scalar product.
The oriented contact of Lie spheres is determined by the bilinear form
〈X,Y〉 = X0Y5 +X5Y0 +X1Y1 +X2Y2 +X3Y3 −X4Y4, (24.6)
with respect to the Lie-quadric L (24.3). It is not difficult to show that two Lie-spheres X,Y are in orientedcontact exactly if 〈X,Y〉 = 0. We note that any oriented plane is in oriented contact with ∞ and orientedspheres or points are never in contact with ∞.
24 Sphere-Geometric Aspects of Bisector Surfaces 109
24.3 Bisector Constructions
Let P and Q be two general parametrized oriented surfaces in R3 whose λ-image surfaces are parametrized byP(u, v) and Q(s, t). A Lie-sphere X is tangent to P and Q if its image point X satisfies the linear conjugacyrelations
〈P,X〉 = 0, 〈Pu,X〉 = 0, 〈Pv,X〉 = 0,〈Q,X〉 = 0, 〈Qs,X〉 = 0, 〈Qt,X〉 = 0,
(24.7)
where partial derivatives ∂F/∂t of a function F are denoted by Ft. Additionally 〈X,X〉 = 0 holds, since Xis λ-image of a Lie-sphere. If X0 6= 0 then the center of the Lie-sphere X is contained in the bisector B ofPand Q.
In general the computation of B amounts to the problem of eliminating the parameters u, v and s, t fromthe equations (24.7). This leads to a system of two equations F (X) = 0, G(X) = 0. Together with 〈X,X〉 = 0this defines a two-dimensional manifold B in L. The projection B = π(B) of B, B0 6= 0 onto R3 is the bisectorsurface of the two input surfaces P and Q.
Consider rational input surfaces P and Q, the elimination leads to rather complicated equations F (X) =G(X) = 0 even for simple input surfaces. Moreover it is difficult to understand geometric properties of thesolution.
24.3.1 Offset-invariance of the bisector surface
Let P and Q be two surfaces in R3 and let B be their bisector surface. We consider offset surfaces Pd andQd which are the envelopes of oriented spheres with radii d centered at P and Q, respectively.
B is the locus of centers of spheres S(u, v) which are tangent to P and Q. By reducing the radius functionof S(u, v) by −d and keeping their centers fixed one obtains a family of spheres tangent to the offset surfacesPd and Qd. Thus B is also the bisector surface of Pd and Qd.
The offset surfaces Pd andQd are images of P andQ under a Lie-transformation (Laguerre transformation)δ which increases the radius of oriented spheres by d.
24.3.2 Bisector surfaces of Lie spheres
Let P and Q be two Lie-spheres 6=∞ in R3 which are not in oriented contact. The λ-image B of their bisectorB is the intersection of L with the 3-space 〈P,X〉 = 〈Q,X〉 = 0. In general B is a quadric of index 0, butfor certain configurations B is a plane considered as set of points. This happens if P,Q are points or sphereswith equal radii or if P,Q are oriented planes.
We consider three Lie-spheres P,Q and R which are not touching each other and compute the set of allLie-spheres touching them. The image points P,Q,R define a plane E ∈ P 5 which intersects L in a conic C.The family of spheres C corresponding to C envelop a Dupin cyclide Φ in general. The conjugacy conditions
〈P,X〉 = 〈Q,X〉 = 〈R,X〉 = 0
define a plane F , the polar plane of E with respect to L. The conic D = F ∩ L corresponds to a family ofspheres touching the same Dupin cyclide, since Φ admits two generations as canal surface. If P,Q and R areoriented planes in general position, the oriented planes C envelope a cone of revolution Φ and D is the familyof spheres touching Φ with centers at Φ’s axis.
24.4 Lie Canal Surfaces
One-parameter families of Lie-spheres C(u) correspond to curves C(u) on L. We assume sufficient smoothness.The tangent lines of C are T(u0) = λC(u0) + µC(u0), where C = dC/du.
If 〈C, C〉 ≥ 0 holds, these Lie-spheres C(u) envelope a Lie canal surface. This is either a canal surface, acurve in case of C4(u) = 0 or a developable surface in case of C0(u) = 0. If 〈C, C〉 ≤ 0 holds, no real envelopeexists.
110 Martin Peternell
Fig. 24.1. Dupin cyclides
If 〈C, C〉(u) = 0 holds identically in an interval, C is an asymptotic curve in L. The curve C(u) ∈ Ldefine a one-parameter family of planes E(u) determined by T ∩X0 = 0 and an incident curve, determinedby T ∩X4 = 0. This is called a surface strip.
Lie spheres tangent to C(u) are computed as solutions of
〈C(u),X〉 = 0, 〈C(u),X〉 = 0, 〈X,X〉 = 0. (24.8)
24.4.1 Bisector surfaces of two Lie-canal-surfaces
PB Q P
B
Q
Fig. 24.2. Bisector surfaces of Lie canal surfaces
We are given two Lie-canal-surfaces P and Q with λ-images P(u) and Q(v) in L. The bisector surface Bconsists of all centers of oriented spheres X tangent to P and Q. Any solution B ∈ L of
〈P,X〉 = 0, 〈dP/du,X〉 = 0,〈Q,X〉 = 0, 〈dQ/dv,X〉 = 0,
(24.9)
satisfying 〈X,X〉 = 0 and X0 6= 0 projects onto a point π(B) of the bisector surface B. For any (u, v) thesolution of (24.9) is a line G(u, v) in P 5. Thus we obtain B(u, v) as intersection G(u, v) ∩ L which proves
Theorem 1. The bisector surface of two Lie canal surfaces P,Q ∈ R3 can be constructed in an elementaryway. The parametrization of the bisector B of two Lie canal surfaces is given by square roots. The constructionis linear if P and Q are both curves or developable surfaces.
We note that the bisector of two rational developable surfaces or two rational curves is a rational surfacesince G(u, v) ∩ L is linear in u, v.
24 Sphere-Geometric Aspects of Bisector Surfaces 111
24.5 Two Parametric Families of Lie Spheres
Two parametric families of Lie-spheres correspond to surfaces P(u, v) in L. The tangent space T 2 is spannedby P and the derivative points Pu,Pv. The real intersection L ∩ T 2 can consist of one point, or one or twolines of L. Of particular interest are the two-dimensional sub-manifolds P(u, v) = (1,p(u, v), 0,− 1
2p2(u, v))R
of the Mobius-quadric M = L ∩ y4 = 0. Since M is of index 0, T 2 intersects L in a single point. This alsofollows from intersecting sPu + tPv with L. Since 〈Pu,Pu〉 = p2
u, 〈Pv,Pv〉 = p2v and 〈Pu,Pv〉 = p>
v ·pv, weobtain
s2p2u + 2stp>
u · pv + t2p2v > 0.
24.5.1 Lie-sphere and a parametrized surface
We consider a general parametrized surface P = p(u, v) with λ-image P(u, v) in L and a Lie-sphere Q. Thebisector surface B of P and Q is the projection π(B) of the surface B ∈ L which is a solution of
〈P,X〉 = 0, 〈Pu,X〉 = 0, 〈Pv,X〉 = 0,〈Q,X〉 = 0.
(24.10)
Since (24.10) determines lines G(u, v) in P 5, we obtain B(u, v) = G(u, v) ∩ L. We can state
Theorem 2. The bisector surface of a parametrized surface P ∈ R3 and a Lie-sphere Q can be constructedin an elementary way and the parametrization of B involves square roots. If Q is a point or an oriented planethe construction is linear. If P is a rational offset surface, B is rational too.
The bisector surface B of P and an oriented plane E is an anticaustic of P with respect to parallelillumination perpendicular to the plane E.
24.5.2 Lie canal surface and parametrized surface
Let Q be a Lie canal surface with λ-image L(t) and let P be a general surface whose λ-image has theparametrization P(u, v). We consider a fixed oriented sphere L(t0). The bisector Bt0 of Q and L(t0) isobtained as projection of the solution Bt0 ∈ L of
〈P,X〉 = 0, 〈Pu,X〉 = 0, 〈Pv,X〉 = 0,〈L(t0),X〉 = 0.
(24.11)
Only those points X ∈ Bt0 contribute to B which satisfy the linear relation
R4(t0) : 〈Lt(t0),X〉 = 0.
Performing this for all t leads to a representation of B.
References
1. Blaschke, W.: Vorlesungen uber Differentialgeometrie III2. Cecil, Th.E.: Lie Sphere Geometry, Springer, New York, 1992.3. Choi, J.J., Kim, M.S. and Elber, G.: Computing Planar Bisector Curves Based on Developbable SSI, research
report, Dept. of Comput. Science, Postech, Korea, 1997.4. Coolidge, J.L.: A Treatise on the Circle and the Sphere, Oxford, Clarendon Press, 1916.5. Dutta, D. and Hoffmann, C.: On the Skeleton of Simple CSG objects, ASME J. Mech. Des. 155, 87-94, 1993.6. Elber, G. and Kim, M–S.: The Bisector Surface of Rational Space Curves, ACM Transactions on Graphics, Vol.
17, No. 1, 32–49, 1998.7. Elber, G. and Kim, M–S.: Rational Bisectors of CSG Primitives Proc. of ACM Symp. on Solid Modeling and
Applications, Ann Arbor, Michigan, 1999.8. Farouki, R.T. and Johnstone, J.K.: Computing point/curve and curve/curve bisectors, in The Mathematics of
Surfaces V, (R.B. Fisher, Ed.), Oxford University Press, 327–354, 1994.
112 Martin Peternell
9. Kim,D-S., Cho, Y. and Kim,D.: Edge-tracing algorithm for Euclidean Voronoi diagram of 3D spheres, Proceedingsof the 16th Canadian Conference on Computational Geometry, 2004, 176–179.
10. Krasauskas, R. and Maurer, Ch.: Studying cyclides with Laguerre geometry, submitted to Computer Aided Geo-metric Design 17, 101-126, 1999.
11. Muller, E. and Krames, J.: Vorlesungen uber Darstellende Geometrie II, Deuticke, Leipzig und Wien, 1929.12. Peternell, M.: Geometric Properties of Bisector Surfaces, Graphical Models 62, 202–236, 2000.13. Pottmann, H. and Peternell, M.: Applications of Laguerre Geometry in CAGD, Computer Aided Geometric Design
15, 165–186, 1998.
25
Fast Computation of the Implicit Ideal of a Hypercircle
Tomas Recio1∗, J. Rafael Sendra2†, Luis Felipe Tabera1‡, and Carlos Villarino2
1 Dto. Matematicas, U. Cantabria, Spain.reciot,[email protected]
2 Dto. Matematicas, U. Alcala, Spain.rafael.sendra,[email protected]
Summary. In this extended abstract, we present an effective method to compute the implicit ideal of a parametricallygiven hypercircle. An experimental comparison with classical implicitation methods is also shown.
25.1 Introduction
Let K be a computable field of characteristic zero, not necessarily algebraically closed. Let F be its algebraic closure.Let K ⊆ K(α) be an algebraic extension of degree n. Let u(t) = at+b
ct+d, ad − bc 6= 0, be a unit of K(α)(t), where t is
transcendental over K. Write
u(t) =
n−1X
i=0
φi(t)αi, φi(t) ∈ K(t).
The curve U in Fn given by the parametrization Φ(t) := (φ0, . . . , φn−1) is called hypercircle [1]. These curves havebeen proposed as an auxiliary tool for the algebraically optimal reparametrization problem for curves [1], [5], [6].Namely, let C ' (η1(t), . . . , ηm(t)) be a parametric curve given by a parametrization with coefficients in K(α). Replacet by
Pn−1i=0 tiα
i in ηj(t) and write
ηj
n−1X
i=0
tiαi
!=
n−1X
i=0
qij(t0, . . . , tn−1)
N(t0, . . . , tn−1)αi, 1 ≤ j ≤ m
where t0, . . . , tn−1 are new variables. In this situation C is K−definable if and only if
U = V (qij , 1 ≤ i ≤ n− 1, 1 ≤ j ≤ m) \ V (N)
is of dimension 1. Moreover, C is K−parametrizable if and only if U is a hypercircle and the unit u(t) associated to Ureparametrizes C over K [1]. Thus, in order to reparametrize C over K it suffices to know a unit defining the associatedhypercircle. Here, we focus on specific algorithms for hypercircles. Namely, we present a method for computing theimplicit equations of a hypercircle U from its parametric representation Φ(t). For this purpose, we will use some ofthe main geometric properties of hypercircles; for details on these auxiliary results we refer to [4].
25.2 Description of the Methods
Given a unit u(t) = at+bct+d
, it is straightforward to check that, if c = 0, the hypercircle U is a line. Since the implicitizationproblem is trivial for lines, we assume in the sequel that c = 1. In [4], it is shown that U is a regular curve ofdegree r = [K(d) : K]. Hence, if d /∈ K, U is not a line. Let M(t) be the minimal polynomial of −d over K, let
m(t) = M(t)t+d
∈ K(α)[t], and (at+ b)m(t) =Pn−1i=0 pi(t)α
i. Then, φi(t) = pi(t)M(t)
.A naive method to compute the implicit ideal is to use classical implicitation methods, for example, Grobner
bases method. However these universal methods do not exploit the geometric properties of hypercircles. The followingmethod uses the fact that the hypercircle might be defined as the image of K ⊆ K(α) under the map u(t) : K(α) →K(α).
∗The authors are partially supported by the project MTM2005-08690-CO2 “Ministerio de Educacion y Ciencia”.†Partially supported by CAM-UAH2005/053 “Direccion General de Universidades de la Consejerıa de Educacion
de la CAM y la Universidad de Alcala”.‡L.F.Tabera also supported by a FPU research grant.
114 Tomas Recio, J. Rafael Sendra, Luis Felipe Tabera, and Carlos Villarino
Theorem 1. Let U be a hypercircle associated to the unit u(t). Let v(t) be the inverse of u(t), v(t) = −dt+bt−a
. Write
v
n−1X
i=0
αiXi
!=
n−1X
i=0
ri(X0, . . . , Xn−1)
s(X0, . . . , Xn−1)αi.
Then, a set of generators of the ideal of U is the elimination ideal with respect to Z
I(U) = (r1(X), . . . , rn(X), s(X)Z − 1) ∩ F[X0, . . . , Xn−1]
This method also needs to perform some elimination, but it starts with a non trivial ideal (r1, . . . , rn) which is zeroon U . Also, notice that we need to know in advance a unit u(t) associated to our hypercircle. It may happen that weare only given a parametrization of the hypercircle in standard form. That is, a parametrization Ψ(t) = (ψ0, . . . , ψn−1)with coefficients over K(α) such that
Pn−1i=0 α
iψi(t) = t. In this case, [3] shows how to proceed.The following method does not use any implicitation method. It is essentially based on elementary linear algebra
and on the following result.
Theorem 2. Let U be the hypercircle associated to the unit u(t) = at+bt+d
∈ K(α)(t) and let r = [K(d) : K]. Then, thereexists a regular projective transformation
χ : P(F)n −→ P(F)n
defined over K such that the curve χ(U) is the rational normal curve of degree r. Therefore, it can be parametrized as:
eχ(t : s) = [sr : sr−1t : · · · : str−1 : tr : 0 : · · · : 0].
This means that the homogenization of p0(t), . . . , pn−1(t),M(t) form a generating system of the K-vector spaceof r-degree forms. In other words, p0(t), . . . , pn−1,M(t) is a generating set of K[t]r, the polynomials over K ofdegree at most r. Geometrically, one can see this by requiring the curve to have degree r and not being contained in
any (r − 1)-dimensional linear subspace. It follows that, given“q0(t)N(t)
, . . . ,qn−1(t)
N(t)
”, any proper parametrication of U
with coefficients in F, q0(t), . . . , qn−1(t), N(t) is a generator set of F[t]r. That is, our method applies to any properparametrization.
Theorem 3. Let P (t) = (p0(t), . . . , pn−1(t),M(t)) be as above, where pn(t) = M(t). Suppose w.l.o.g. that p0(t), . . . ,pr(t) is a basis of K[t]r. Let pi(t) =
Pr
j=0 aijtj, 0 ≤ i ≤ r. Write pk(t) =
Pr
i=0 λk−r,ipi(t), r + 1 ≤ k ≤ n. LetA = (aij) ∈ Mr+1×r+1(K) and B = (λki) ∈ Mn−r×r+1(K). Then
Q =
»A−1 Or+1×n−r
−B In−r×n−r
–∈ Mn+1×n+1(K)
is a regular matrix defining a regular projective transformation mapping U onto the rational normal curve χ(U).
Corollary 4. Let Q be the matrix in Theorem 3. Let h1(Y ), . . . , hs(Y ) be a set of homogeneous generators of the
ideal of χ(U). Then fi(X) = hi“Pn
j=0 Q0jXj , . . . ,Pn
j=0 QnjXj”, 1 ≤ i ≤ s is a set of generators of the projective
ideal of U .
Proposition 5. YiYj−1 − Yi−1Yj , Yk | 1 ≤ i < j ≤ r, r + 1 ≤ k ≤ n is a set of generators of the ideal of χ(U).
Proof. Note that eχ(t : s) (see Theorem 2) provides the subspace containing the curve, and that the set of quadraticpolynomials is a well known set of generators (see, e.g., [2]). 2
Thus, this method provides a fast implicitation method for hypercircles. Note that the computation of Q can beperformed, using linear algebra, in O(n3) field computations. Then, we have to compute up to O(n2) products oflinear polynomials in n variables. Hence, the total amount of field operations is dominated by O(n4).
In the next table, we show a comparison of the running times of three implicitation methods. The first onecorresponds to classic elimination techniques using Grobner bases. The second method implicitizes via the inverseunit (see Theorem 1), and the third uses the normal rational curve (see Corollary 4). These methods have been testedon a Mac Xserver with 2 processors 2.3GHz using Maple 10.
25 Fast Computation of the Implicit Ideal of a Hypercircle 115
Algebraic Number Unit G. Basis Inv. Unit Normal Curve
α4 + α2 − 3 (1−α3)t+α2
t+1+2α−3α2 0.403 2.757 0.035
α5 + 2α3 + 2 (α+α4)t−1−α2
t+α0.891 32.363 0.026
α6 + 3 3αt+1t+α+α4 2.010 > 300 0.038
α12 + 2α+ 1 α3t+3−αt−α
> 300 > 300 0.238
References
1. Andradas C., Recio T. and Sendra J.R. (1999). Base Field Restriction Techniques for Parametric Curves. Proc.ISSAC 1999, ACM Press, pp. 17-22.
2. Harris, J. (1992). Algebraic Geometry, a First Course. Springer-Verlag.3. Recio T., Sendra J.R. and Villarino C. (2004). From Hypercircles to Units. Proc. ISSAC 2004, ACM Press,, pp.
258-265.4. Recio T., Sendra J.R., Tabera L.F. and Villarino C. Generalizing Circles Over Algebraic Extensions. Preprint.5. Sendra J.R. and Villarino C. (2001). Optimal Reparametrization of Polynomial Algebraic Curves. International
Journal of Computational Geometry and Applications vol. 11/4, pp. 439–453.6. Sendra J.R. and Villarino C. (2002). Algebraically Optimal Reparametrizations of Quasi-Polynomial Algebraic
Curves. Journal of Algebra and its Applications vol. 1/1, pp. 51–74.
26
Subresultants and Implicit Equations of Rational Curves
Ignacio Fernandez Rua∗ and Laureano Gonzalez–Vega†
Departamento de Matematicas, Estadıstica y Computacion, Universidad de Cantabria, Santander, [email protected], [email protected]
Summary. This paper is devoted to show how the subresultant sequence of the two polynomials defining a ratio-nal curve contains very relevant information concerning the so called implicitation, reparametrization and inversionproblems.
26.1 Introduction
One interesting problem concerning the manipulation of parametric curves and surfaces in computer–aided geometricdesign is the finding of efficient algorithms for computing the implicit equation of curves and surfaces parametrizedby rational functions (see for example chapters 5 and 7 in [5]).
The implicitization problem for hypersurfaces parametrized in a rational way can be stated in the following terms:let V be a hypersurface in Cn parametrized by (i ∈ 1, . . . , n):
xi =fi(t1, . . . , tn−1)
gi(t1, . . . , tn−1)=fi(t)
gi(t)
where fi and gi belong to C[t1, . . . , tn−1] with gcd(fi, gi) = 1. The implicitization problem for V is the finding of anon zero element RV(x1, . . . , xn) in C[x1, . . . , xn], with the smallest posible total degree, such that:
RV(f1(t)/g1(t), . . . , fn(t)/gn(t)) = 0.
More general formulations of the implicitization problem for arbitrary parametric varieties can be found, for example,in [1], chapter 3 of [2] or [3].
In this paper we address the particular case of rational plane curves. Our approach is based on the subresultantsequence of two defining polynomials of the curve. This sequence contains very relevant information that can be usedto solve the implicitation, reparametrization and inversion problems for these curves.
26.2 Preliminaries
Throughout this paper C will be an affine algebraic curve implicitly defined by a non-constant bivariate polynomialh(x, y) ∈ C[x, y], that is, C is the zero zet C = V (h) = (x0, y0) ∈ C2 : h(x0, y0) = 0. Our main interestwill be rational curves in C2, that is, curves C that can be defined by a rational parametrization P(t) =“p11(t)p12(t)
, p21(t)p22(t)
”where pij(t) ∈ C[t], i, j = 1, 2, are polynomials with coefficients in C such that h
“p11(t)p12(t)
, p21(t)p22(t)
”= 0,
for all t ∈ C \ V (p12p22), and, for all but finitely many points (x0, y0) ∈ C, there exists a parameter value t0 ∈ C such
that (x0, y0) =“p11(t0)p12(t0)
, p21(t0)p22(t0)
”. Moreover, we will assume that the parametrization is given in reduced form, i.e.,
gcd(p11, p12) = gcd(p21, p22) = 1. Finally, we will not consider the case where one of the quotients p11p12
, p21p22
is constant.These parametrizations correspond to a horizontal or vertical line and the solution to the problems we are about tostudy is trivial for these curves. Not all algebraic curves have a rational parametrization. Only irreducible curves ofgenus 0 have such a parametrization. This curves are called unirational.
∗Partially supported by the Spanish funded project MTM2004-08115-C04-01 (Ministerio de Educacion y Ciencia).†Partially supported by the Spanish funded project MTM2005-08690-C02-02 (Ministerio de Educacion y Ciencia).
26 Subresultants and Implicit Equations of Rational Curves 117
A rational parametrization P(t) =“p11(t)p12(t)
, p21(t)p22(t)
”of the curve C is called proper provided that the map
P : C −→ C is birational, i.e., for all but a finite number of points (x0, y0) in the curve C, there exists a unique t0 ∈ C
such that (x0, y0) =“p11(t0)p12(t0)
, p21(t0)p22(t0)
”.
Theorem 1. [6] Let Q(t) be a proper parametrization of a rational curve C, and let P(t) be any other parametrizationof C. Then there exists a non-constant rational function A(t) ∈ C(t) such that P(t) = Q(A(t)). The degree of thisrational function is known as tracing index of the parametrization [8]. Moreover, P(t) is a proper parametrizationif, and only if, there exists a linear rational function B(t) ∈ C(t) such that P(t) = Q(B(t)).
Given a not proper parametrization P of C, the problem of finding a proper parametrization Q and a rationalfunction A such that P(t) = Q(A(t)), for all t ∈ C, is known as the reparametrization problem for the curve C.
26.2.1 Subresultants for rational curves
For a rational the curve C, given in parametric reduced form P =“p11(t)p12(t)
, p21(t)p22(t)
”, we define the following polynomials:
HP(x; t) = p11(t) − xp12(t) = am(x)tm + · · · + a1(x)t+ a0(x) ∈ C[x][t]
GP(y; t) = p21(t) − yp22(t) = bn(y)tn + · · · + b1(y)t+ b0(y) ∈ C[y][t]
where ai(x) ∈ C[x], for all 1 ≤ i ≤ m, deg(am) ≤ 1, and bi(y) ∈ C[y], for all 1 ≤ i ≤ n, deg(bi) ≤ 1. Whenever theparametrization P is clear from the context we shall simply write H(x; t) and G(y; t).
The standard subresultant sequence of the polynomials H(x; t) and G(y; t) is defined by the following way.For all i = 0, . . . ,minm,n, let Si(H(x; t),m,G(y; t), n) =
Ph
k=0∆ktk ∈ C[x, y][t] where ∆k (k = 0, . . . ,minm,n)
is the determinant of the submatrix intersection of the the first n+m− 2h− 1 columns, and the (n+m− h− k)-thcolumn of the matrix
Syli(H(x; t),m,G(y; t), n) =
0BBBBBBBB@
am(x) . . . a0(x)
. . .. . .
am(x) . . . a0(x)bn(y) . . . b0(y)
. . .. . .
bn(y) . . . b0(y)
1CCCCCCCCA
Note that the subresultant sequence depends on the parametrization P of the rational curve C. In particular,let us denote by u(x, y)t + v(x, y) = S1(H(x; t),m,G(y; t), n) ∈ C[x, y][t], the subresultant of degree one, and byr(x, y) = S0(H(x; t),m,G(y; t), n), the subresultant of degree zero. Notice that r(x, y) is exactly the resultant ofthe polynomials H(x; t) and G(y; t). Let us also denote by si(H(x; t),m,G(y; t), n) the coefficient of the term ti
in Si(H(x; t),m,G(y; t), n). A subresultant Si(H(x; t),m,G(y; t), n) is called regular, if deg(Si) = i (i.e., if si 6= 0).Otherwise, Si is called defective. Subresultant theorem (see [4]) provides an algorithm for computing the subresultantsequence of the polynomials H(x; t) and G(y; t) in O(maxm,n2) operations.
26.3 The Proper Case
We show some properties of the subresultant sequence for a proper parametrization of the curve C.
Proposition 2. Let P be a parametrization with tracing index s ≥ 1 of the rational curve C with implicit equationh(x, y) ∈ C[x, y]. Then, for almost every point (xo, yo) ∈ C, the subresultant Ss(H(x0; t),m,G(yo; t), n) is regular
and the s− 1 last elements in the subresultant sequence Si(H(x0; t),m,G(y0; t), n)minm,ni=0 are zero. Moreover, the
subresultant S0(H(x; t),m,G(y; t), n) is, up to a nonzero field constant, equal to h(x, y)s, and Ss(H(x; t),m,G(y; t), n)is regular.
Corollary 3. If P is a proper parametrization of the rational curve C, then the subresultant S1(H(x0; t),m,G(y0; t), n)is regular, for almost every point (x0, y0) ∈ C.
In the proper case, the regularity of the first subresultant provides an answer to the inversion problem for thecurve.
118 Ignacio Fernandez Rua and Laureano Gonzalez–Vega
Proposition 4. If P is a proper parametrization of the rational curve C, and (x0, y0) ∈ C is a point such that
S1(H(x0; t),m,G(y0; t), n) = u(x0, y0)t+ v(x0, y0) is regular, then (x0, y0) =“p11(t0)p12(t0)
, p21(t0)p22(t0)
”, where t0 = − v(x0,y0)
u(x0,y0).
That is, inversion is given by the equation
S1(H(x0; t),m,G(y0; t), n) = 0.
As a consequence of the previous results we have the following
Corollary 5. If P is a proper parametrization of C, and S1(H(x; t),m,G(y; t), n) = u(x, y)t + v(x, y), then a point(x0, y0) 6∈ V (u) is in the curve C if and only if
eH (x,−v(x, y), u(x, y)) = 0 and eG (y,−v(x, y), u(x, y)) = 0
where eH and eG denote the homogenized polynomials obtained from H and G, respectively.
26.4 The Non-Proper Case
Non-proper parametrizations of the rational curve C are considered in this section. Our first results provide a proper-ness criterium for a parametrization P.
Proposition 6. Let P be a parametrization with tracing index s ≥ 1 of the rational curve C with implicit equationh(x, y) ∈ C[x, y]. Then, for all 0 ≤ i ≤ s−1, the factor hs−i divides the subresultant Si = Si(H(x; t),m,G(y; t), n), andthis is the highest power of h dividing Si. In particular, h does not divide the subresultant Ss = Ss(H(x; t),m,G(y; t), n).Moreover, the subresultant Ss−1 = Ss−1(H(x; t),m,G(y; t), n) is never identically zero.
Corollary 7. Let P be a parametrization of the rational curve C, and let S1(H(x; t),m,G(y; t), n) = u(x, y)t +v(x, y), S0(H(x; t),m,G(y; t), n) = r(x, y) be the last terms in the subresultant sequence. Then, P is proper if and onlyif gcd(u(x, y), v(x, y), r(x, y)) ∈ C∗.
Next we consider the problem of reparametrization, inversion and implicitation of a not proper parametrizationP of degree s. We will consider the s-th subresultant Ss(H(x; t),m,G(y; t), n).
Proposition 8. Let Q be a proper parametrization of the rational curve C, for which the last elements in the subre-sultant sequence are
S1(HQ(x; t),m,GQ(y; t)) = u(x, y)t+ v(x, y)
and S0(HQ(x; t),m,GQ(y; t), n) = r(x, y). If P(t) = Q(A(t)), where A(t) = M(t)/N(t) is a rational function of degree
s thenD(x, y)Ss(H
P(x; t),ms,GP(y; t), ns) = (u(x, y)M(t) + v(x, y)N(t))Rs(x, y) + r(x, y)P (x, y; t) (26.1)
where the polynomials D(x, y) and Rs(x, y) in C[x, y] are the leading coefficients of u(x, y)M(t) + v(x, y)N(t) andSs(H
P(x; t),ms,GP(y; t), ns) w.r.t. t, and P (x, y; t) ∈ C[x, y][t].
From this proposition we have the following “inversion” consequence.
Corollary 9. Let P be a not proper parametrization of the rational curve C, with tracing index s. Then, for almostevery point (x0, y0) ∈ C the set of values t1, . . . , ts ∈ C such that P(ti) = (x0, y0) is given by the solutions to theequation Ss(H
P(x0; t),ms,GP(y0; t), ns) = 0.
We shall use this fact to consider a proper parametrization of the curve C whose inverse map will be easilyobtained from (26.1). Our approach follows the path of [7]. In our case we want to use the subresultant sequence ofthe polynomials H(x; t) and G(y; t) to obtain the inversion of a proper parametrization of the curve. Doing this weavoid new computations with the new proper parametrization. So, we assume that for P, a not proper parametrizationof C of tracing index s, we have P(t) = Q(A(t)), where Q is a proper parametrization of C for which the last terms inthe subresultant sequence are S1(H
Q(x; t),m,GQ(y; t)) = u(x, y)t+v(x, y) and S0(HQ(x; t),m,GQ(y; t), n) = r(x, y),
and A(t) = M(t)/N(t) is a rational function of degree s such that M(t) = csts+ · · ·+ c0, N(t) = dst
s+ · · ·+ d0 ∈ C[t]with cs 6= 0 or ds 6= 0. Let us denote θj(x, y) = cju(x, y) + djv(x, y), for all 0 ≤ j ≤ m, and am 6= 0 or bm 6= 0. Thefollowing result is [7, Theorem 1] in our setting.
26 Subresultants and Implicit Equations of Rational Curves 119
Proposition 10. For all 0 ≤ i, j ≤ s, the rational map
Θij(x, y) =θi(x, y)
θj(x, y)=ciu(x, y) + div(x, y)
cju(x, y) + djv(x, y)
is invertible if and only if θi(x, y), θj(x, y) are not associated polynomials, i.e., if cidj 6= dicj. Moreover, if M(t) andN(t) are coprime then there exist 0 ≤ i, j ≤ s such that θi(x, y), θj(x, y) are not associated.
Let assume that we can find 0 ≤ i, j ≤ s such that Θij(x, y) is invertible. The next result shows that there exists aproper parametrization of the curve C such that inversion is given by the equation t = Θij(x, y) (cf. [7, Theorem 2]).
Proposition 11. If Θij(x, y) is invertible for some 0 ≤ i, j ≤ s, then there exists a proper parametrization U of therational curve C such that inversion is given by t = Θij(x, y).
Let Ss(HP(x; t),ms,GP(y; t), ns) =
Ps
i=0 Sis(x, y)t
i be the s-th subresultant for the parametrization P, whereSis(x, y) ∈ C[x, y], for all 0 ≤ i ≤ s. From equation (26.1) there is direct relation between θi(x, y) and Sis(x, y), and sowe have the following result.
Proposition 12. For all 0 ≤ i, j ≤ s and for almost all (x0, y0) ∈ C, we have
Sis(x0, y0)
Sjs(x0, y0)=θi(x0, y0)
θj(x0, y0). (26.2)
Moreover, for all 0 ≤ i, j ≤ s such that the polynomials Sis(x, y), Sjs(x, y) are not associated, the polynomials
θis(x, y), θjs(x, y) are also not associated.
Corollary 13. If gcd(M(t), N(t)) = 1 then there exists 0 ≤ i, j ≤ s and a proper parametrization U =“u11(t)u12(t)
, u21(t)u22(t)
”
of the rational curve C such that inversion is given by t = Sis(x, y)/Sjs(x, y).
As we can see, for a not proper parametrization P of degree s, we can obtain, from the subresultant SPs (HP(x; t),ms,GP(y; t), ns),
an inversion equation for a proper parametrization U . Our final task is to find the polynomials defining this properparametrization.
Proposition 14. Let U(t) = (u11(t)u12(t)
, u21(t)u22(t)
) be the proper parametrization of the curve C (with implicit equation
h(x, y)) for which inversion is given by the rational mapSi
s(x,y)
Sjs(x,y)
. Then we have:
Resy(h(x, y), Sis(x, y) − tSjs(x, y)) = (u11(t) − xu12(t))
l, up to constants in C[x]∗,
Resx(h(x, y), Sjs(x, y) − tSjs(x, y)) = (u21(t) − yu22(t))
l, up to constants in C[y]∗,
where l = degy(Sis(x, y)/S
js(x, y)).
References
1. C. Alonso, J. Gutierrez, T. Recio: An implicitization algorithm with fewer variables. Computer Aided GeometricDesign 12, 251–258, 1995.
2. D. Cox, J. Little, D. O’Shea: Ideals, Varieties and Algorithms. Undergraduate Texts in Mathematics, Springer-Verlag, 1993.
3. X. S. Gao, S. C. Chou: Implicitization of rational parametric equations. Journal of Symbolic Computation 14,459–470, 1992.
4. J. von zur Gathen, J. Gerhard: Modern computer algebra. Cambridge University Press, 2003.5. C. M. Hoffmann: Geometric and Solid Modeling: An Introduction. Morgan Kaufmann Publishers, 1989.6. P. Luroth: Beweis eines Satzes uber rationale Curven. Math. Annalen 9, 163–165, 1876.7. S. Perez-Diaz: On the problem of proper reparametrization for rational curves and surfaces. Computer Aided
Geometric Design 23, 307–323, 2006.8. J. R. Sendra, F. Winkler: Tracing index of rational curve parametrizations. Computer Aided Geometric Design
18, 771–795, 2001.
27
Detecting Real Singularities of a Curve from a RationalParametrization
R. Rubio∗, J.M. Serradilla∗, and M. Pilar Velez∗
Escuela Politecnica Superior, Universidad Antonio de Nebrija, 28040 Madrid, [email protected], [email protected], [email protected]
Summary. In this paper we give an algorithm that detects real singularities and counts local branches of real rationalcurves without knowing an implicitization. The main idea behind this is a generalization of the D-resultant (see [9])to n rational functions. This allows us to describe all the singularities as solutions of a system of polynomials in onevariable.
27.1 Introduction
The presence of self-intersections and cusps becomes an obstacle in CAGD when plotting a curve. Consequently, thedetection of the singular points of an algebraic curve is important to understand its geometry. CAD programs use,generally, parametric representation of curves and this is why the study and manipulation of curves from a parame-trization is so interesting. We propose here a method to detect singularities of rational curves without implicitizating.
The classical approach to detect singularities deals with implicit representation and in many cases has been donefor planar curves (see [10]). The previously known methods, via a parametrization, are limited to planar or cubiccurves (see [5, 4, 3]). Nevertheless in [6] is introduced a method which computes the singularities of affine polynomialcurves over fields of characteristic zero. In this work we extend the main results in [6] to any affine rational curve.Furthermore, we count the local branches and adapt these issues to the real case.
In Section 2 we review some facts on proper parametrizations and generalized resultants. Section 3 contains themain results to detect singularities over the complex numbers and compute them using generalized resultants. Thelast section studies real singularities from a real parametrization and provides an algorithm to detect singular pointsand count the local branches.
27.2 On Proper Parametrizations and Generalized Resultants
A rational parametrization of a curve C in Cn is a map ψ such that
ψ : C 99K Cn
t 7−→ (f1(t), . . . , fn(t))
where fi ∈ C(t), ∀i = 1, . . . , n. Then, C = Im ψ (Zariski closure in Cn of the image of ψ).Without loss of generallity we can suppose that fi is not constant for i = 1, . . . , n.
Definition 1. Let f ∈ C(t). We define the bivariate polynomial associated to f as:
g(s, t) =fN(s)fD(t) − fN(t)fD(s)
s− t
where f =fN
fD
, fN , fD ∈ C[t] and gcd(fN , fD) = 1.
We denote by g1, . . . , gn ∈ C[s, t] the bivariate polynomials associated to the rational functions f1, . . . , fn of theparametrization.
∗Partially supported by MTM2005-02865
27 Detecting Real Singularities of a Curve from a Rational Parametrization 121
Definition 2. The generalized resultants of g1, . . . , gn ∈ C[s, t] are the polynomials hα(s) defined by Rest(g1, u2g2 +· · · + ungn) =
Pα hα(s)uα where u = (u2, . . . , un) are new variables.
For more details on generalized resultants see [2].
Definition 3. The map ψ is a proper parametrization of C if there exist two Zariski open sets U ⊂ C, V ⊂ C suchthat ψ|U : U −→ V is bijective.
A proper rational parametrization ψ of a curve C is almost injective (except for a finite number of points) and
C \ Im(ψ) contains at most one point P∞ =
„N (1)
D(1), . . . ,
N (n)
D(n)
«, where N (i) and D(i) are the coefficients of degree di
of fiN and fiD respectively and di = max deg( fiN ), deg( fiD ) (see [1]).
It is always possible to reparametrizate a rational curve in order to have a proper parametrization, see [8]. Thereis a characterization of properness using generalized resultants, see [7].
Theorem 4. A rational parametrization ψ is proper if and only if Ress(g1, u2g2 + · · · + ungn) 6≡ 0.
The generalized resultants of the bivariate polynomials associated to ψ not only give a test of properness for ψ,but they also describe the singularities of C as we show in the next section.
27.3 Singularities of Rational Curves
In this section we compute the singularities of a rational curve from a proper parametrization over the complexnumbers.
The affine variety V = V (g1, . . . , gn) ⊂ C2 defined by the bivariate polynomials gi associated to fi is essential forthe becoming results.
Lemma 5. The variety V can be rewritten as V = Aψ ∪Bψ ∪ Cψ whereAψ =
˘(a, b) ∈ C2/ a 6= b, gi(a, b) = 0, fiD (a) 6= 0 ∀i = 1, . . . , n
¯,
Bψ =˘(a, a) ∈ C2/ gi(a, a) = 0, fiD (a) 6= 0 ∀i = 1, . . . , n
¯,
Cψ = T 2 ∩ V where T = t ∈ C / fiD (t) = 0 for some i = 1, . . . , n .
From now on let π1 and π2 be the projections of C2 over the first and second coordinate, respectively.
Remark 6. If ψ is a proper parametrization the set ψ(π1(Aψ)) consists of all points P , except at most P∞, such thatthere exist different parameters a1, . . . , am ∈ C with ψ(a1) = · · · = ψ(am) = P . In other words, P is a singular pointbelonging to m local branches.Moreover, the set ψ(π1(Bψ)) contains every point Q, except at most P∞, such that there exists a parameter a ∈ C
satisfying that ψ(a) = Q and ψ′(a) = 0 (see Lemma 1.7 in [3], gi(s, s) = fiD (s)2f ′i(s)). That is, Q is a singular point
belonging to a cusp–type branch.
From this remark, the next theorem follows.
Theorem 7. If ψ is proper, then V is a finite set and
ψ(π1(V )) = ψ(π1(Aψ ∪Bψ)) = ψ(π2(Aψ ∪Bψ)) ⊂ C
describes all the singular points of C except at most P∞.
The result below allow us to detect the points in π1(V ) as solutions of univariate polynomial systems.
Theorem 8. Let W = a ∈ C : lct(g1(s, t))(a) = 0, lct(G(s, t, u))(a, u) = 0, where G(s, t, u) = u2g2(s, t) + · · · +ungn(s, t). If ψ is proper,
V (hα) \W ⊂ π1(V ) ∪ t0 ∈ C | ψ(t0) = P∞ ∈ Cn ⊂ V (hα)
122 R. Rubio, J.M. Serradilla, and M. Pilar Velez
Proof. Since ψ is proper, by Theorem 4, Rest(g1, G)(s, u) 6≡ 0. Take a ∈ V (hα) \W . By the behaviour of the resultantunder the evaluation homomorphism (see [11]), Rest(g1(a, t), G(a, t, u)) = 0. So there exists b ∈ C such that g1(a, b) = 0and G(a, b, u) = 0. Hence gi(a, b) = 0 for all i and a ∈ π1(V ).
For the other inclusion, let a ∈ π1(V ). Then, there exists b ∈ C such that gi(a, b) = 0 for all i. On the otherhand, there exist p, q ∈ C[s, t, u2, . . . , un] such that D(s, u) = Rest(g1, G) = pg1 + qG. If we evaluate this equality in(a, b, u2, . . . , un) we have D(a, u) = 0 and hα(a) ≡ 0 for all α. If ψ(t0) = P∞ ∈ Cn, then the leading coefficients of thegi’s vanish at t0. 2
Theorem 9. There is a proper parametrization eψ of C such that eψ(π1(eV )) describes all the singularities of C, whereeV is the zero set of the bivariate polynomials associated to eψ.
Proof. Consider a proper parametrization ψ(t) = (f1(t), . . . , fn(t)). Let t0 6∈ V (hα). Theorem 8 implies that ψ(t0) is
not a singular point of C. Then, it suffices to take a change of parameter σ(t) =t0t+ b
t− t0. 2
27.4 Detecting Real Singularities
Theorems 8 and 9 suggest a method to detect singularities of rational curves over the complex numbers computingthe generalized resultants of the bivariate polynomials associated to the parametrization. In this section we adaptthese ideas to the real case.
Take a proper parametrization ψ(t) = (f1(t), . . . , fn(t)) with fi ∈ R(t) for i = 1, . . . , n, then the Zariski closure ofψ(R) in Rn is a real curve CR. By Proposition 42 in [1], ψ(R) contains every point of CR except at most the isolatedones and possibly one extra point P∞ = lim
t→∞ψ(t) ∈ CR.
Applying the previous results to ψ : C 99K Cn and taking C = ψ(t) : t ∈ C ⊂ Cn we can recover this “missingpoints” as follows:
• To recover P∞ it suffices to change the parameter as in Theorem 9, taking for instance t =t0t+ 1
t− t0with t0 6∈ V (hα),
t0 ∈ R.• The isolated points correspond to pairwise complex conjugated branches. Thus they can be detected in Aψ as real
points corresponding to pairwise complex conjugated parameters.
The following algorithm computes the singularities of a real rational curve using a real parametrization.
Algorithm 1 (Detecting singular points). .
Input: ψ = (f1, . . . , fn) ∈ R(t)n a proper parametrization.
Output: The real singular points of the curve defined by (f1, . . . , fn) and the number of real ordinary branches andcusp–type branches.
Step 1: Compute g1, . . . , gn the bivariate polynomials associated to f1, . . . , fn.Step 2: Compute S = V (hα) ⊂ C where Rest(g1, u2g2 + · · · + ungn) =
Pα hα(s)uα.
Step 3: If deg fiN > deg fiD for some i then go to Step 5.
Step 4: Let t0 ∈ R \ S and σ =t0t+ 1
t− t0. Let ψ = ψ(σ), g1, . . . , gn the associated bivariate polynomials and S =
σ(S) ∪ t0.Step 5: Let P1, . . . , Pl = ψ(S) ∩ Rn.Step 6: For each i take ti1, . . . , timi ∈ S ∩ R such that ψ(tij) = Pi for all j
— If there is no tij for Pi, return Pi is an isolated point.Step 7: For each j = 1, . . . ,mi, compute slij = (g1(tij , tij), . . . , gn(tij , tij)).
Let ci = # j ∈ 1, . . . ,mi | slij = (0, . . . , 0).— If mi = 1 and ci = 0 take another i.— Return Pi is singular with mi − ci ordinary branches and ci cusp–type branches.
Step 7: If no Pi is returned, then the curve is non singular.
Proof. In Step 4 we reparametrizate the curve, if it is necessary, in order to have a new proper parametrization suchthat the point corresponding to t = ∞ is not singular. In Step 6 we collect all the parameters that give the same pointand in Step 7 count the cusp–type branches and discard the non singular points (i.e. points that are neither nodes,mi = 1, nor cusps, ψ′ 6= 0). 2
27 Detecting Real Singularities of a Curve from a Rational Parametrization 123
References
1. C. Andradas and T. Recio. Plotting missing points and branches of real parametric curves. To appear in AAECC,2006.
2. D. A. Cox, J. Little, and D. O’Shea. Ideal, Varieties and Algorithms. UTM, Springer-Verlag, New York, 1997.3. J. Gutierrez, R. Rubio, and J.-T. Yu. D-resultant of rational functions. Proc. Amer. Math. Soc., 130(8):2237–2246,
2002.4. Y.-M. Li and J.C. Cripps. Identification of inflection points and cusps on rational curves. Comput. Aided Geom.
Design, 14:491–497, 1997.5. D. Manocha and J.F. Canny. Detecting cusps and inflection points in curves. Comput. Aided Geom. Design,
9:1–24, 1992.6. H. Park. Effective computation of singularities of parametric affine curves. J. Pure Appl. Algebra, 173:49–58,
2002.7. R. Rubio, J.M. Serradila, and M.P. Velez. Parametrizaciones birracionales y singularidades de curvas racionales.
In Actas del 7o Encuentro de Algebra Computacional y Aplicaciones, EACA 2004, pages 255–260, 2004.8. T. W. Sederberg. Improperly parametrized rational curves. Comput. Aided Geom. Design, 3:67–75, 1986.9. A. van den Essen and J.-T. Yu. The D-resultant, singularities and the degree of unfaithfulness. Proc. Amer.
Math. Soc., 25(3):689–695, 1997.10. R.J. Walker. Algebraic Curves. Princeton University Press, Princeton, NJ, 1950.11. F. Winkler. Polynomial Algorithms in Computer Algebra. Springer, 1996.
28
Computing Minkowski Sums via Support Function Representation
Zbynek Sır∗
Johannes Kepler University, Institute of Applied Geometry, Altenberger Str. 69, 4040 Linz, [email protected]
Summary. We define a class of (locally) quasi-convex curvex/surfaces represented (locally) by support functiondefined on (a subset) of the unit circle /sphere. We investigate in particular the support functions which are restrictionsof polynomials to the sphere. After reviewing some geometric and metric properties of support function representationwe focus on its application for the computation of Minkowski sums and convolutions of curves and surfaces.
28.1 Definition of Pseudoconvex Curves/Surfaces.
Let U ⊆ Sn be an open subset of the n dimensional unit sphere and h : U → R a real C1 function. We define themapping xh : U → Rn+1 as
xh(n) = h(n)n + ∇Snh(n),
where ∇Snh is the gradient of h on Sn. The function h will be called support function.The mapping
E : h 7→ xh : C1(U,R) → C0(U,Rn+1)
is a linear operator. We will interpret xh ∈ C0(Sn,Rn+1) as an “oriented” shape, i.e. as a set Im(xh) ⊂ Rn+1 togetherwith normal vectors n attached at xh(n) for all n. The oriented shapes obtained as Im(xh) from C1 support functionswill be called oriented quasi-convex shapes (curves, surfaces) - see Figure 28.1. The space of quasi-convex shapes will
–4
–2
0
2
4
–4 –2 0 2 4
Fig. 28.1. Quasi convex curve and surface corresponding to the support function defined on thecircle and sphere as a restriction of the polynomial 11y4 − 8y2 + 1 considered on R2 and R3.
be denoted Qn and the operator E can be understood as a mapping from support functions to quasi-convex shapes
∗The contribution is based on the results of a joint research with J. Gravesen and B. Juttler.
28 Computing Minkowski Sums via Support Function Representation 125
E : h 7→ Im(xh) : C1(Sn,R) → Qn.
This is still a linear operator with respect to the standard addition and multiplication by constants on C1(Sn,R) andwith respect to Minkowski sum (as addition) and scaling (as multiplication by a constant) on Qn.
28.2 Properties of Support Function Representation.
Geometric operations (including offseting and Minkowski sums) on quasi-convex shapes are reflected by algebraicoperations on corresponding support functions. This correspondence is summarized in the Table 28.1.
Geometric operation Modified support
Translation by vector v hv(n) = h(n) + n · vRotation by matrix µ ∈ SO(n+ 1) hµ(n) = h(µ−1(n))Scaling by factor c ∈ R hc(n) = c h(n)
Offseting with distance d hd(n) = h(n) + dChange of orientation (all normals reversed) h−(n) = −h(−n)
Table 28.1. Geometric operations and corresponding changes of the support function.
The operator E also enjoys nice metric and analytic properties. For example if the support function h has at somepoint the analytic continuity of order k then the corresponding quasi-convex shape has (at the corresponding point)geometric continuity of order k (with an exception for cusps). Also the Sobolev distance (taking into account thefunction and its first derivative) of two support function functions yields an upper bound on the Hausdorff distanceof the corresponding quasi-convex shapes.
28.3 Minkowski Sum Computation
We present several support function based algorithms for Minkowski sums computation. The original shapes/bodies arefirst approximated/decomposed into (piecewise) quasi-convex shapes. For these shapes it is very simple to compute theMinkowski sum, which is realized by adding the support functions - see Figure 28.2. Also the trimming or determinationof the outer-most boundary of the Minkowski sum is simplier comparing to standard representations due to reducednumber of involved parameters.
References
1. Farouki, R.T. and Hass, J., Algorithms for evaluating the boundary and covering degree of Minkowski sums andother geometrical convolutions, preprint http://www-mae.engr.ucdavis.edu/ farouki/index.html
2. Pottmann, H., Rational curves and surfaces with rational offsets, Computer Aided Geometric Design 12, 175–192(1995).
3. Sabin, M., A Class of Surfaces Closed under Five Important Geometric Operations, Technical repport, Britishaircraft corporation Ltd., 1974.
4. Sır, Z., Gravesen, J. and Juttler, Curves and surfaces represented by polynomial support functions, in preparation.
126 Zbynek Sır
–2
–1
0
1
2
3
–4 –3 –2 –1 0 1 2 3
L
–2
–1
0
1
2
3
–4 –3 –2 –1 0 1 2 3
=
–2
–1
0
1
2
3
–4 –3 –2 –1 0 1 2 3
–2
–1
0
1
2
–4 –2 0 2 4
L
–2
–1
0
1
2
–2 –1 0 1 2
=
–4
–2
0
2
4
–6 –4 –2 0 2 4 6Fig. 28.2. Examples of the Minkowski sums of two planar regions.
29
Morse theory, Mayer-Vietoris sequence and the Computation ofthe Topology of Hypersurfaces of dimension ≤ 3
Carlo Traverso
Dipartimento di Matematica, Universita di Pisa. [email protected]
29.1 Introduction
The computation of the topology of a semialgebraic set (a set defined by equations and inequalities) is an importantproblem for several applications. It contains the problem of counting the number of connected components, or findingif a connected component is a disk, a sphere, a cylinder or a torus.
Topology cannot be completely defined in finite terms; a triangulation defines a topology, but finding a triangu-lation may be hard (using a Cylindrical Algebraic Decomposition is polynomial in fixed dimension, but the degree isdoubly exponential in the dimension) and deciding homeomorphism from triangulations is in general undecidable.
Better complexity, in theory and practice, can be achieved through Morse theory, that can yeld a cell decomposition.However a cell decomposition is not a finite description, since the maps are not defined in finite terms. Homologycan be defined in finite terms, and can be computed from a triangulation, but using a triangulation in general is toocostly.
In this paper we want to study the use of homology, through Morse theory and Mayer-Vietoris sequences.The resulting algorithm for curves and surfaces is substantially the same described in [FGPT] and before in [GT];
indeed the same approach is already contained substantially in [GT], in a more obscure form; the new contributionof this paper is a consideration of threefolds in 4 space, that clarifies the general setting that is specialized for curvesand surfaces. The idea of mapping the homology of a (compact) hypersurface in a semispace into the homology of thecomplementary, and retracting it to a piecewise linear subset is the new idea of this paper. If this can be applied indimension 4 or higher is still not clear, but is sufficient to design an algorithm for threefolds.
Threefolds are currently not used in CAGD, but extending CAGD to objects varying with time is a naturalextension, and anyway the clarification on the general setting can be used to better understand the case of surfaces.
The limitation to compact varieties and to Morse functions (isolated non-degenerate critical points with one criticalpoint for critical value) is purely technical, extending the same ideas to degenerate position (without generic changeof variables) and adding inequalities can be handled with the same ideas.
Extending the analysis to subvarieties of codimension larger than 1 might be possible too.
29.2 Morse Theory in a Nutshell
Morse theory studies compact differential manifolds through a Morse function f that spans the manifold. This canalways be chosen as a linear function (a coordinate function) of an embedding. The condition of being a Morse functionis that the differential vanishes only at isolated points, (critical points) each with a different value of the function(critical value) and at the critical point the Hessian ∂f
∂xi∂xjis non singular (non-degenerate critical point).
Define Mt being the set of points where f(x) < t. Morse theory describes the changes to the topology of Mt whent passes through a non-degenerate critical value: the topology of Mt does does not change in an interval that has nocritical point, and changes at a critical point in the same way in which it changes for quadric defined by the hessianmatrix, i.e. up to homotopy Mc+ε corresponds to adding a k-cell to Mc−ε, k being the index of the Hessian, througha map of the boundary of the cell to the boundary of Mc−ε; up to diffeomorphism, the cell has to be multiplied by an− k cell, if n is the dimension of the manifold. Hence the manifold can be described through successive additions ofcells.
The standard reference is [M].
128 Carlo Traverso
29.3 Homology, Mayer-Vietoris Sequence
The topology being difficult to classify, it is usually studied through algebraic functors. Homology is such a functor,with values in graded modules (or groups), and can be defined in different ways; these satisfy some axioms, thatguarantee the uniqueness of homology if we study a manifold. These axioms in particular require that the homologyof a cell is Z in degree 0 and 0 otherwise, and that a n-sphere (the boundary of a n+1-cell) has Z as 0- and n-dimensionalcomponents, and no other component. The homology gives informations on the topology like the connection degreeand the number of “holes”. The rank of the zero homology is the number of connected components.
The Mayer-Vietoris exact sequence is the basic tool to compute the homology of a hypersurface through a Morsecoordinate function. It relates the homology of a union to the homology of the two parts and the intersection.
The Mayer-Vietoris sequence can define inductively the homology provided that some ambiguity is resolved, andthis requires to be able to describe explicitly the maps in homology associated to some maps described in Morsetheory. In particular, this requires to be able to follow the homology not only when it changes, passing through acritical point, but also when it does not change, passing through non-critical points, and to explicitly describe thehomology map induced by the inclusion of the boundary of Mt into Mt at a non-critical value t.
The Mayer-Vietoris long exact sequence connects the homology of X, Y , X ∪ Y and X ∩ Y :
Hi(X ∩ Y ) → Hi(X) ⊕Hi(Y ) → Hi(X ∪ Y ) → Hi−1(X ∩ Y )
and is true under conditions that are verified in the situation of Morse theory.In this case, passing through a critical point c, X = Mc−ε is the set immediately before the critical point, Y is a
d-dimensional cell added (up to homotopy) passing through a critical point of index d, X ∩ Y is the boundary of Y(hence is a Sd−1) and X ∪ Y is (up to homotopy) is Mc+ε
The Euler-Poincare characteristic is hence uniquely defined by the sequence of the index of the critical points, butthe Betti numbers are not defined, unless we know the map Hi(X ∪ Y ) → Hi−1(X ∩ Y )
The zero-homology is simple to discuss, so we concentrate in higher homology.Since Y is contractible, the homology is trivial, and the Mayer-Vietoris sequence becomes
Hi(Sd−1) → Hi(Mc−ε) → Hi(Mc+ε) → Hi−1(Sd−1)
so we have isomorphism of Hi(Mc−ε) and Hi(Mc+ε) unless either Hi(Sd) or Hi−1(Sd) are zero; said differently,unless i = d or i = d− 1. To simplify the discussion, we use rational coefficients (although the full discussion can bemade with integer coefficients)
0 → Hd(Mc−ε) → Hd(Mc+ε) → Q → Hd−1(Mc−ε) → Hd−1(Mc+ε) → 0
and to compute the homology of Mc+ε from Mc−ε we have thus only to decide if the boundary of the added cell (thatis the generating cycle of the homology) goes to 0 (i.e. if the cycle is a boundary in Mc−ε or not. Remark that theinterection lives in the boundary of Mc−ε, i.e. the map can be factored through Mc−ε
We will discuss the issue with some examples in the next section.
29.4 Curves, Surfaces, Threefolds
29.4.1 Curves
In the case of curves (in the plane) the index of a critical point can be 0 or 1; in the first case the critical point isa local minimum, so passing through it adds a component, i.e. a free generator to H0. In the case of index 1, it is alocal maximum, and we add a “bridge” between two points. This may have two effects:
• If the two points lie in the same connected components of Mc−ε, i.e. if the two points are the two boundary pointsof a segment, then the bridge closes a cycle, and the component no longer comes into play for the rest of thecomputation: both it and its inside live under the critical level.
• If the two points lie in different connected components (different segments) then the bridge connects two segmentsinto one; the zero-th Betti number decreases, since two connected components are identified.
So the only problem is to identify the two components that are identified. This can be recognized, since the pointsof the curve in every Mt are totally ordered, and if the critical point is in the n-th position in Mc the two pointsjoined are the n-th and the n+ i-th in Mc−ε.
29 Morse Theory and Mayer-Vietoris sequence 129
29.4.2 Surfaces
The critical points may have index 0, 1 or 2. Index 0 here too means that a new component appears (this is generalfor every dimension).
If the index is 1 here too we are building a 1-dimensional bridge between two boundary points of Mc−ε. But whilethis is correct up to homotopy, the differential picture is rather that two surfaces are joined locally through a strip.
The two ends of the bridge might be in different components of the set Mc−ε, that hence become the same inMc+ε (in this case in the Mayer-Vietoris sequence the left map is 0 and the right map is injective) or the two endsare in the same component, and in that case Mc+ε acquires an extra generator in H1.
The key to decide what happens is hence the ability of deciding in which connected component a boundarycomponent of Mc−ε is; a connected component can be identified through a critical point of index 0; the bridge indeedmay join two points in the same boundary component, or in two different boundary components. In the second case itis critical to decide whether the two are in the same connected component or not. In the next subsection we describe(by hand-waving...) an example in which the topology at critical points alone is unable to decide the topology, andassociating connected components of the boundary to connected components of Mc−ε where c is essential.
Tire and sausage
Consider a hypersurface composed of a “tire” and a “sausage” and assume that the critical points (the tire has aminimum, a maximum and two saddle points, the sausage a minimum and a maximum) correspond to critical valueswith increasing index (first the two minima, then the saddles, then the maxima).
We can build a different surface cutting the tire and the sausage, and grafting the two half sausages to the tire.The critical points do not change, nor does the topology at the critical values, but the global topology changes.
This example is already discussed in [GT].
29.4.3 Threefolds
Here the situation is more complicated. We still have that index 0 creates a new connected component, index 1 setsa bridge between two boundary points (hence one has to be able to associate a boundary component to a connectedcomponent), and index 3 points add a small disk whose boundary 2-sphere matches a boundary component (here toosphere homology and/or homotopy shows that the map is defined up to the sign that is irrelevant); but a point ofindex 2 gives a problem.
Morse theory says that we are adding a 2-cell, the intersection being a circle. Looking at the Mayer-Vietorissequence it appears that passing from Mc−ε to Mc+ε one either adds a H2 or kills a H1, and this depends if the imageof the circle; the first case happens if the 1-cycle of Mc−ε to which the circle is mapped is a border, and the second ifit defines a non-zero homology (that is hence killed).
The holes of a donut
In dimension 4, it is difficult to make drawings. It is possible however to consider time (a special coordinate, but herewe have a special coordinate that is the Morse function). This means that instead of a drawing one has to make amovie. Inserting 3D drawings on paper is already difficult, inserting a movie is harder. So use your immagination.
There are two ways to make a donut: either make a long piece of dough, and join the ends, or make a flat pieceof dough and carve a hole in the middle.
The result is different: the donut at time 1 (i.e. the boundary component of M1) has a 2-dimensional H1, generatedby an “horizontal” cycle and a “vertical” one; but in M1 only one of the two is non zero: in the first case it is thehorizontal one, and in the second it is the vertical one.
We can first make a donut, then undo it. If we make a type-1 hole and we destroy it with a type-1 procedure, thehorizontal cycle survives; if we make and destroy the hole with a type-2 hole, the vertical cycle survives; if we mixtypes, no 1-cycle survives.
It is easy to make a donut with two holes, one of the first type and one of the second. And we can build a complexfigure, creating the two holes (t from 0 to 1) and later destroying them (t from 2 to 3). The problem is what happensfrom 1 to 2.
Assume thet the left hole is created and destroyed with a type-1 procedure, and the right one with a type-2procedure. Then the result has two 1-cycles. But if the holes are created and destroyed with different types, no1-homology survives.
From 1 to 2 we have no critical points, but we just have some space to manipulate the donut: if we do not followthe figure, the two holes may be interchanged. So at level 2 we do not know whether the left hole has a horizontal ora vertical H1.
130 Carlo Traverso
We need to mark one of the handles (for example with an handcuff) to follow it from 1 to 2. If a handle has ahorizontal H1 and we fix this H1 with the handcuff, even if we don’t look at it from 1 to 2, but we are sure that thehandcuff has not been released, we can identify the handle (and the relative H1 again at level 2.
This metaphor will be made explicit in the discussion of the next session.
29.5 Transporting Homology
We have seen that computing the homology of Mc−ε is not enough, and we need to map the homology of Mc−ε tothe homology of Mc−ε. The theory says that one can retract Mc1 to Mc2 if c2 < c1, but this is computationally hard,since it involves integrating a vector field.
The solution that we envisage consists is the following:
• Map the homology of Mt to the homology of its complement M ′t ; this is an isomorphism, by the Mayer-Vietoris
sequence, except at zero homology, where a component of M ′t corresponds to its boundary components and
conversely.• Retract M ′
t to a PL (piecewise-linear) skeleton;• Show how to recover the necessary information from the skeleton.
While we still do not know if this is sufficient in every dimension, we can prove that it is sufficient for threefoldsin 4-space.
A complete algorithm can be designed, but requires to handle degenerate critical points and critical values withmore than one critical point (without changing coordinates generically, that increase the practical complexity intoler-ably); one can also handle inequalities in addition to one equation (“hypersurface in a box”). This requires to handleother issues that are too long to describe in an introductory talk.
References
[FGPT] E. Fortuna, P. Gianni, P. Parenti, C. Traverso, Algorithms to compute the topology of orientable real algebraicsurfaces, Journal of Symbolic Computation, num. 3-4, vol. 36, pp. 343-364, 2003
[GT] Gianni, P., and Traverso, C. Shape determination for real curves and surfaces. Ann. Univ. Ferrara Sez. VII(N.S.) 29 (1983), 87–109.
[M] Milnor, J. Morse theory. Princeton University Press, Princeton, N.J., 1963.
30
Subdivision Method for Computing an Arrangement of ImplicitPlanar Curves
Julien Wintz and Bernard Mourrain
GALAAD, INRIA, BP 03, 06902 Sophia Antipolis, France.mourrain,[email protected]
In this work, we address the problem of computing an arrangement structure of algebraic curves in a specified domain[3]. Algebraic curves are compact representations that have numerous advantages over parametric ones, such as aneasy determination of inside/outside. This is particularly useful when we have to determine logical structures (union,substraction etc...) between two or more such objects. It has direct applications in Geometric Modeling, where objectsare defined by Constructive Solid Geometry operations. In this paper, we describe a new subdivision method toconstruct the arrangement of implicit planar curves defined by polynomials.
30.1 Overview
Since arrangement are relevant for numerous application fields, we propose an incremental dynamic algorithm whichcan maintain the solution to the problem as the input objects are inserted, with no preliminary knowledge of theinput data.
We define the problem in terms of objects, regions and conflicts between objects and regions. An object is anelement of S, in our case, an implicit planar curve. A region can be of different type among cell, face, edge andvertex. A conflict exist between two regions when they define a same part of the space. To describe an arrangement ofobjects, these regions are organized within an influence graph which is a directed, acyclic and connected graph thatrepresents the regions created by the arrangement algorithm during the incremental construction and can be used todetect conflicts between those regions and a new object. It possesses a single root and its nodes correspond to theregions created by the algorithm with potential additional information. Such a structure features several operationslike query, location, addition or deletion so that it can be directly and efficiently used in different application fields,either within static or dynamic algorithms.
Each time an object o is inserted into the set S its arrangement Ao is computed (figure 30.1) and merged (figure30.2) with the current arrangement A. If a region Ro of Ao conflicts with a region R of A, then we subdivide R andRo into R∩Ro and the remaining regions of R and Ro following the boundaries of the intersection (figure 30.3). Oncethe structural information computed, the new regions are inserted into A as children of either R, Ro or both of them(figure 30.4 with R = B4 and Ro = B5).
Rather than using a classical hierarchical quadtree to partition the R2 space, we base our subdivision on thebounding boxes of the input set of objects. This allows us to keep the advantages of the quadtrees : the hierarchicalstructure allows to terminate (stop further subdivisions) early when a cell is deemed regular or irrelevant, but, drivingthe subdivision process with input bounding boxes improves the subdivision by focusing on the cells of interest forour specific purpose, this choice saves computation efforts.
30.2 Regularity Criterion
Let us describe here the regularity test which allows to decide whether a box is relevant or not for deducing thestructural information.
It is widely known that the Bernstein polynomial basis is a useful tool in CAD for approximating implicit curves[2]. In conjunction with the Descartes’ Law of Sign, it allows the root information to be determined efficiently in agiven domain. In this proposal, we use the multivariate version of Bernstein’s basis in conjunction with Descartes’ lawfor efficiently computing an arrangement of curves (see [1]).
132 Julien Wintz and Bernard Mourrain
B1B2
B3
B
B4
C1 : f(x, y)
RB2
1
RB1
2
RB1
1
RB2
2
RB3
1
RB3
2
RB4
1
RB4
2
Fig. 30.1. Each time an object is inserted, its logical structure is computed
B
B1 B2 B3 B4
RB1
1R
B1
2R
B2
2R
B2
1R
B3
2R
B3
1R
B4
1R
B4
2
Fig. 30.2. This structure is then inserted into the influence graph defining the arrangement A
C1
C2
B1 B2
B3 B4
B6
B7 B8
B5
B4∩5
Fig. 30.3. When conflicts between regions are detected, more subdivision steps are performed
By subdivision, we can divide a larger domain into smaller ones and focus on those with uncertified arrangementstructure. Larger unpredictable domains are broken down to smaller predictable domains in which the structuralinformation can be easily inferred.
A crucial problem is how to efficiently and reliably deduce this structural information in a given subdivision box.Curve segments with regards to a given domain are considered as regular if their topological structure can be certifiedwithin that domain, from their intersection points with the boundary. In the case were there is no intersection pointwe use the following criterion [4] :
30 Subdivision Method for Computing an Arrangement of Implicit Planar Curves 133
B
B3 B4 B5 B6. . . . . .
B4∩5
B4∩5
1B
4∩5
2B
4∩5
3B
4∩5
4
. . .. . .
. . . . . .
. . .
. . .
Fig. 30.4. The resulting structural information is merged into A
B4∩5
1B
4∩5
2
B4∩5
3B
4∩5
4
RB
4∩5
1
1
RB
4∩5
1
2
RB
4∩5
2
1R
B4∩5
2
2
RB
4∩5
2
3R
B4∩5
2
4
RB
4∩5
3
1R
B4∩5
3
2
RB
4∩5
3
3R
B4∩5
3
4
RB
4∩5
4
1
RB
4∩5
4
2
Fig. 30.5. Boxes resulting from B4 ∩B5
Proposition 1. If ∂yf(x, y) 6= 0 (resp. ∂xf(x, y) 6= 0) in a domain D = [a0, b0] × [a1, b1] ⊂ R2, the curve C definedby f = 0 is regular on D.
This criterion can be verified from the sign of the coefficients of ∂xf (resp. ∂yf) in the Bernstein basis on the domainD.
We generalized this criterion to domains containing singular points or intersection points as follows.
Definition 2. A set of curves C1, . . . , Ck will be regular in a domain D, if
• for each curve intersecting D, there exists a point p ∈ D, such that the curve in D is homotopic to segmentsconnecting p to points of the curves on the border of D (also called star-shape from p).
• if two curves intersect in D, their union is star-shape from the intersection point.
If the box D contains only one smooth part of a curve Ci, this criterion implies that Ci ∩D is homotopic to a segmentconnecting the two points of Ci on the border of D. If the box contains only one curve Ci which is not smooth in D,
134 Julien Wintz and Bernard Mourrain
B4∩5
1B
4∩5
2
B4∩5
3B
4∩5
4
RB
4∩5
1
1 RB
4∩5
1
2R
B4∩5
2
1R
B4∩5
2
2 RB
4∩5
2
3R
B4∩5
2
4
RB
4∩5
3
1R
B4∩5
3
2R
B4∩5
3
3R
B4∩5
3
4R
B4∩5
4
1R
B4∩5
4
2
Fig. 30.6. Corresponding nodes to be inserted in the influence graph
there is only one singular point p and Ci is star-shaped from p. If the domain contains several curves, two curves haveat most one intersection point and the curves should be star-shape from this point.
In order to verify this criterion for singular or intersection points, we use degree theory [5], to compute the numberof branches at these points. This way, we obtain a topological description of cells in the given domain, associated topolynomial inequalities.
30.3 Conclusion
This work is a first attempt at using subdivision methods in a field where sweep methods have been heavily used.This technique proved reliable and can be easily extended to higher dimensions and different kind of objects, threedimensional algebraic curve or algebraic surfaces for example as well as the different algebraic criteria used in thealgorithm.
Experiments have been experienced using the algebraic computation library Synaps for implementation and thealgebraic modeling platform Axel for interaction and applications.
References
1. L. Alberti, G. Comte, and B. Mourrain. Meshing implicit algebraic surfaces: the smooth case. In L.L. SchumakerM. Maehlen, K. Morken, editor, Mathematical Methods for Curves and Surfaces: Tromso’04, pages 11–26. Nashboro,2005.
2. G. Farin. Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, 3rd Ed. Academic Press,1993.
3. D. Halperin. Arrangements. In Jacob E. Goodman and Joseph O’Rourke, editors, Handbook of Discrete andComputational Geometry, chapter 21, pages 389–412. CRC Press LLC, Boca Raton, FL, 1997.
4. Chen Liang, B. Mourrain, and J.P. Pavone. Subdivision methods for 2d and 3d implicit curves. In ComputationalMethods for Algebraic Spline Surfaces. Springer-Verlag, 2006. To appear.
5. N. G. Lloyd. Degree theory. Cambridge University Press, Cambridge, 1978. Cambridge Tracts in Mathematics,No. 73.
31
An Evolution–Based Approach for the ApproximateParameterization of Implicitly Defined Curves by ParametricSpline Curves
Huaiping Yang
Institute of Applied Geometry, Johannes Kepler University, Altenberger Str. 69, 4040 Linz, [email protected]
Summary. We propose a novel approach for approximate parameterization of an implicitly defined curve in theplane by B-spline curves. The method produces the parameterization result of a planar implicitly defined curve (notnecessarily algebraic) within a bounding box, without knowing any a priori information about its topology.
As the basic idea, we use an evolution process (see Figure 31.1): Starting from the bounding box of the domainof interest, a closed B-spline curve is moved gradually towards the given implicit curve. The evolution is governed bya differential equation which is derived from the function defining the given curve. In order to capture interior loops,a combination of different evolution laws is needed.
The evolving parametric B-spline curve may split itself into multiple B-spline curves, thereby adapting its topologyto the given implicitly defined curve. In order to handle different types of singularities more accurately, we also providean additional post-processing step to refine the parameterization result after the evolution has stopped. Experimentalexamples are presented to demonstrate the effectiveness of our approach.
The speaker is supported by Marie Curie grant of the European commission (project 022073 ISIS). This talk isbased on joint work with B. Juttler and L. Gonzalez–Vega.
Keywords: evolution, approximation, parameterization, implicit curves
136 Huaiping Yang
(a) (b)
(c) (d)
Fig. 31.1. Approximate parameterization by cubic B-spline curves. The implicit curve is defined by −3 + 12y2 +2y4 − 12y6 + y8 + 12x2 − 28y2x2 + 12y4x2 + 4y6x2 − 18x4 + 20y2x4 + 2y4x4 + 12x6 − 4x6y2 − 3x8 = 0. The figuresshow the initial B-spline curve (a), the intermediate B-spline curve (b), the evolution result (before post-processing)(c) and the final result (after post-processing) (d).
