RICE UNIVERSITY

Analytic Functionsin Computer Aided Geometric Design

by

Geraldine Morin

A THESIS SUBMITTEDIN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE

Doctor of Philosophy

APPROVED, THESIS COMMITTEE:

Ron Goldman, Professor, ChairComputer Science

Joe Warren, Professor,Computer Science

Michael Wolf, ProfessorMathematics

Houston, Texas

November, 2001

Analytic Functionsin Computer Aided Geometric Design

Geraldine Morin

Abstract

This thesis presents a new paradigm for geometric modeling based on analytic functions.

This model includes not only a representation of analytic curves and surfaces, but also tools

and algorithms to manipulate this representation.

Analytic functions on a given domain represent a very large class of infinitely smooth

functions, including trigonometric functions and functions with poles outside the domain.

Thus, the model is very rich; in particular the model is able to represent an object of optimal

smoothness as well as functions as close as desired to singularities.

The Bezier representation for polynomials generalizes to the Poisson representation for

analytic curves and surfaces. The coefficients in the Poisson basis not only characterize an

analytic function, but also are geometrically meaningful and intuitive control parameters

for the curve or surface the function defines – as the Bezier control points are for polyno-

mial shapes. Based on this Poisson representation, we derive standard geometric modeling

algorithms for analytic curves and surfaces, including subdivision, trimming, evaluation

and change of basis procedures. We also develop a notion of blossoming, as well as a

de Boor-Fix formula and a Marsden identity, for analytic curves. These algorithms and

tools provide an efficient and complete framework for using analytic functions in geomet-

ric modeling.

Acknowledgments

First of all, I want to thank Hans Hagen for his unconditional support and his encourage-

ment to keep working towards a doctorate degree through my hardest time.

The Rice community through its professionalism, competence and dynamism provided

me an incomparable working environment. Working under the supervision of Ron Gold-

man has been a great honor and a wonderful opportunity; thanks to his enthusiasm and

passion for research, his hard work and his kindness, being his student has been a very

instructive and fascinating experience. Several persons told me that Ron had been offering

them support and encouragement like a Ph.D. advisor; I had the great chance that Ron was

my advisor in the first place!

Joe Warren gave me the opportunity to work with him, and so to broaden my research

interests, and provided me very valuable comments and advise, always with much encour-

agement and concern for my interests. I am also deeply in debt to Jan Hewitt whose support

during this last year abroad has been invaluable; after assisting me to improve my writing,

she closely followed my progress and offered help, encouragement, and sensitive advise.

It has been a privilege to collaborate, interact with and work for the faculty from the

Rice computer science department. In particular, I would like to thank Matthias Felleisen

who let me take an active role in the teach-scheme project. Mike Wolf, from the Mathe-

matics department, has been always available for support and help.

Thank you to the administrative staff of the computer science department for their kind-

ness and efficiency. I wish to thank to Adria Baker and the OISS staff for making my time

at Rice such a valuable and enjoyable cultural experience. I want to thank my fellow

students in the department, in particular Eric Allen, for their support, their sympathy (by

etymology – suffering with) and their sym-happy (by extension – enjoying with). Finally,

iv

my internship at SDRC has also been interesting and motivating; I would like to thank the

“geometry” group for welcoming me, Tim Strotmam whom I very much enjoyed working

with, and Thomas Hermann for his comments on and interest in my work.

I must also thank my family (my parents, my favorite sister and my brother) and friends

(from Kaiserslautern, Grenoble, Clermont-Ferrand and Houston) that encouraged, sup-

ported and stood beside me through my long and sometimes painful path to this doctorate

degree.

I cannot end these acknowledgments without thanking Henrik for faithful, although

increasingly challenged, support since my coming to Rice. He has been sequentially a

very helpful mentor, an incomparably supportive boyfriend, a wonderful husband, and the

dedicated father of our little Kaspar.

Contents

Abstract ii

Acknowledgments iii

List of Illustrations viii

1 Introduction 11.1 The field of geometric modeling . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 A brief history of CAGD . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Successive models for curves and surfaces . . . . . . . . . . . . . . 3

1.2 Analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 The contribution of this dissertation . . . . . . . . . . . . . . . . . . . . . 4

1.4 Outline of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Background and Literature Review 82.1 In the beginning, were the polynomials . . . . . . . . . . . . . . . . . . . . 8

2.2 Generalization of the polynomial framework . . . . . . . . . . . . . . . . . 13

2.2.1 In analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 In probability theory . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.3 In approximation theory . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Limiting cases in CAGD . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 From limiting cases to a complete modeling framework . . . . . . . . . . . 17

3 The Poisson Representation for Analytic Functions 193.1 The Poisson basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Poisson curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

vi

3.3 Comparison of Taylor and Poisson approximations of analytic functions . . 25

4 Left Poisson Subdivision 284.1 Left Bezier subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Poisson subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3 Convergence of the Poisson subdivision scheme . . . . . . . . . . . . . . . 35

4.3.1 Proof of the convergence of the Poisson subdivision scheme . . . . 36

4.3.2 Smooth convergence of Poisson subdivision . . . . . . . . . . . . . 44

4.3.3 Order of approximation of left Poisson subdivision . . . . . . . . . 45

4.3.4 The variation diminishing property of Poisson curves . . . . . . . . 46

4.4 Examples and Implementation . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Right Poisson Subdivision 535.1 Right Poisson subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.1.1 Right Bezier subdivision . . . . . . . . . . . . . . . . . . . . . . . 54

5.1.2 Trimming: Right Poisson subdivision . . . . . . . . . . . . . . . . 55

5.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2.1 Approximation of a Poisson function inside the convergence

domain of its Poisson series . . . . . . . . . . . . . . . . . . . . . 64

5.2.2 Approximation of a Poisson function outside the initial

convergence domain . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2.3 Intersection of analytic functions . . . . . . . . . . . . . . . . . . . 68

6 The Analytic Blossom 746.1 The polynomial blossom . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.1.1 The polynomial blossom and its relation to dual functionals . . . . 75

6.1.2 Polynomial blossom and algorithms for Bezier curves . . . . . . . 78

6.1.3 Polynomial blossom, Marsden identity and de Boor-Fix formula . . 78

6.2 Definition, examples and properties of the analytic blossom . . . . . . . . . 80

vii

6.2.1 Definition, examples and diagonal properties of the analytic blossom 80

6.2.2 Existence of an analytic blossom . . . . . . . . . . . . . . . . . . . 81

6.2.3 The dual functional property . . . . . . . . . . . . . . . . . . . . . 87

6.2.4 Uniqueness of the analytic blossom . . . . . . . . . . . . . . . . . 89

6.3 Algorithms for Poisson Curves . . . . . . . . . . . . . . . . . . . . . . . . 90

6.3.1 Left subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.3.2 Right subdivision: trimming algorithm . . . . . . . . . . . . . . . 93

6.3.3 Evaluation algorithms for functions and derivatives . . . . . . . . . 94

6.3.4 Conversion between Poisson and Taylor basis . . . . . . . . . . . . 96

6.4 De Boor-Fix Formula, Marsden Identity and Blossoming for Analytic

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7 Conclusion 101

Bibliography 104

Illustrations

2.1 Two planar cubic Bezier curves over the interval with their control

polygons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Two bicubic Bezier surfaces with their control net. . . . . . . . . . . . . . 10

2.3 Two -continuous cubic B-splines with their control polygon. . . . . . . . 10

3.1 The Bernstein basis functions and and the Poisson basis

function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 The first four Poisson basis functions. . . . . . . . . . . . . . . . . . . . . 20

3.3 A dynamic programming algorithm computing Poisson coefficients. . . . . 22

3.4 A dynamic programming algorithm computing Taylor coefficients. . . . . . 23

3.5 Taylor and Poisson approximations of the functions and . . . . 24

4.1 A Bezier curve of degree 3, with its initial control points and its control

points after one round of subdivision. . . . . . . . . . . . . . . . . . . . . 28

4.2 The de Casteljau algorithm for a cubic Bezier curve. . . . . . . . . . . . . 29

4.3 The de Casteljau algorithm for a Poisson curve. . . . . . . . . . . . . . . . 32

4.4 The Poisson basis function , and its approximating subdivision polygons

of orders 1, 2 and 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5 The functions and and their approximating subdivision polygons

of orders 1, 4 and 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.6 The logarithmic spiral and its approximating

subdivision polygons of orders 0, 3 and 6. . . . . . . . . . . . . . . . . . . 47

ix

4.7 The archimedian spiral , and its approximating

subdivision polygons of order 0, 3 and 6. . . . . . . . . . . . . . . . . . . . 48

4.8 Approximation of the tensor product basis function by its

control polyhedra of orders 1, 2 and 3. . . . . . . . . . . . . . . . . . . . . 48

4.9 Approximation of the tensor product surface by its control

polyhedra of orders 1, 2, 4. . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.10 Control polyhedra of the sphere of orders 1, 2, 3 and 6. . . . . . . . . . . . 49

5.1 A de Casteljau-like algorithm to compute the points . . . . . . . . . . 54

5.2 A dynamic programming algorithm to compute the difference of the

Poisson control points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3 Approximation of the function over the interval . . . . . 63

5.4 Approximation of the function over the interval . . . . . . 65

5.5 Approximation of the function over the interval . . . . . 66

5.6 Successive trimming over a pole. . . . . . . . . . . . . . . . . . . . . . . . 67

5.7 Successive trimming on the real line. . . . . . . . . . . . . . . . . . . . . 68

5.8 The function over the interval and for values

greater than with piecewise linear approximations. . . . . . . . . . . . 69

5.9 Intersection points between the Archimedian spiral and a degree 3 Bezier

curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.10 Intersection between the function and the rational function

on the interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.1 Blossom interpretation of the de Casteljau algorithm for a cubic Bezier

curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.2 The analytic blossom computed from the Poisson

control points of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.3 Blossom interpretation of the left subdivision algorithm for Poisson curves. 91

x

6.4 Blossom interpretation of the trimming algorithm for Poisson curves. . . . . 93

6.5 Blossom interpretation of the computation of the difference of the Poisson

control points of a function analytic at . . . . . . . . . . . . . . . . . . 94

6.6 Blossom interpretation of the change of basis from Taylor to Poisson. . . . 96

6.7 Blossom interpretation of the change of basis from Poisson to Taylor. . . . 97

1

Chapter 1

Introduction

1.1 The field of geometric modeling

This computer science dissertation is in the field of geometric modeling. The goal of ge-

ometric modeling, or Computer Aided Geometric Design (CAGD), is to provide a frame-

work adapted to computations for modeling the geometric world.

The geometric world we want to represent includes both the physical world around us

and the virtual world in our minds – shapes that we are able to conceive before or without

a corresponding real, physical object. Examples of objects from the physical world that

we wish to represent are human bones for medical applications, buildings for architectural

design, or airplane bodies for physical simulations. Virtual geometric shapes may also be

created in order to plan their construction, for example designing a new car model using a

Computer Aided Design (CAD) system. But virtual geometric objects may be created as

well with no interest of realizing them, e.g. when a designer creates a character or an object

for an animated movie.

But what does modeling an object mean? First, modeling requires having a “good”

geometric representation of the object, where “good” may refer to physical realism, in the

case of the model of an existing geometry or a geometry to be manufactured, the metric

being chosen relative to the main interest (close in distance, or reliable topology...). In the

case of virtual objects, the quality of the model may be measured by an esthetic criteria

(for example, the smoothness of the shape) or by realism (close to some ideal geometry).

Apart from simply representing the geometry of the object, we also need to manipulate

this representation. Algorithms have to be developed to render, translate, scale, smooth,

cut or transform these objects, depending on the intention of the user. For example, a

2

physician may wish to explore the model of a human heart represented on his computer

screen, whereas an engineer may want to modify the geometry of a car body that he is

conceiving. The designer of a video game may wish to set some geometric invariants of

his virtual object, like the possible moves or degrees of liberty of the object. All these

operations on models require tools and manipulation algorithms for the model.

The field of geometric modeling needs to offer computation adapted representations for

physical or virtual objects, and to develop rapid and efficient algorithms for working with

these representations. In particular, people in this field are interested in representing the

geometric properties of an object, for example, its connectivity, topology, or smoothness,

and applying geometric operations to this representation, e.g. translating, trimming or

reshaping the object.

1.1.1 A brief history of CAGD

Engineers of the automotive industry created the first geometric models in the 1960’s, and

so initiated the domain of CAGD. Beginning in the 1960’s, engineers designing car body

parts started to use computers, an emerging technology, to represent the surfaces they were

to produce. Parametric surfaces which had been long known and studied by mathematicians

are well suited for representing these surfaces. Nevertheless finding appropriate numerical

representations for these surfaces and tools adapted to manipulate them was a completely

new task, and therefore developing these new representations was the first goal of the pio-

neers of the field. These new representations for surfaces, using intuitive numeric parame-

ters to control their shape, quickly made their way to other manufacturers: from car bodies,

to airplanes, to mechanical parts. The need for geometric modeling rapidly extended to

other domains – to medicine, to architecture, and to geology – following the growing in-

fluence of computers. More recently, virtual worlds have also used geometric models, for

the conception of physical shapes or simply for representing non physical objects. With the

role of computers always increasing, the need for geometric models is growing too. The

increase in computational power allows richer models to be developed, and users expect the

3

manipulation tools and algorithms to be increasingly efficient and to offer more and more

functionality. New challenges for models appear as new application domains develop, for

example the modeling of nanotechnologies for engineers and molecules for biochemists.

Thus, the need for geometric models adapted to computation increases as the influence of

computers becomes more and more pervasive in many domains of science and engineering.

1.1.2 Successive models for curves and surfaces

In this thesis, we are interested in modeling curves and surfaces. We work with parametric

curves and surfaces which are the types of models most used in CAGD. In this paragraph,

we review different parametric representations that have been used over the years. Then,

we propose our new model.

The first mathematical model to be used in CAGD to represent car body parts was poly-

nomials, which offered a quite flexible and smooth set of geometrical objects. Although

polynomials had been known in mathematics for a long time, a new representation adapted

to modeling with polynomials was developed. This representation provides an intuitive

characterization of polynomial curves and surfaces via control points. This very intuitive

representation was later extended to B-splines. B-splines are piecewise polynomial func-

tions that connect with a chosen degree of continuity. This B-spline model is similar to the

polynomial model, since it also admits a representation based on control points. Moreover,

the B-spline model has a property that the polynomials were lacking: locality (in Chapter 2,

we will explain this property). Because the representation based on control points was very

adapted to design, it was extended to yet another, richer family or curves and surfaces: the

non-uniform rational B-splines (NURBS). As their polynomial and B-spline counterparts,

NURBS curves and surfaces are represented by control points but to be entirely charac-

terized they also require a set of scalar weights, one weight for each control point. The

underlying mathematical model for NURBS is only slightly more complicated than the

mathematical model for B-splines, and therefore for polynomials. Moreover, these three

models are included in one another – that is, any polynomial object is a particular case of

4

a B-spline and any polynomial or B-spline curve or surface is a NURBS curve or surface.

Apart from the fact that NURBS offers a richer model than B-splines, they have the particu-

lar and very useful ability to represent arcs of circles and sections of spheres, and therefore

are able to model exactly circular objects, whereas polynomial and B-splines parameteri-

zation are not able to model sections of circular objects. Nevertheless, the parameterization

of an arc of a circle modeled by NURBS is highly non uniform.

The new model we propose in this thesis is based on analytic functions. As with the

three previous models, curve and surfaces are also represented parametrically and are char-

acterized by control points that have an intuitive interpretation. Moreover, analytic func-

tions include trigonometric functions, and therefore, this new model is able to represent

exactly circle and spheres with a uniform parameterization.

1.2 Analytic functions

Analytic functions are very well known and have been studied in much detail by math-

ematicians and analysts. These functions have also been used for applications in both

probability theory and approximation theory. Nevertheless till now no representation of

the analytic functions suitable for geometric modeling had been developed together with a

suite of adapted algorithms.

1.3 The contribution of this dissertation

This thesis presents a new paradigm for geometric modeling based on analytic functions.

This model includes not only a representation of analytic curves and surfaces, but also tools

and algorithms to analyze and manipulate this representation.

In this work, we shall consider parametric representations of curves or surfaces. The

underlying geometric space is an affine space, containing points and vectors. The tools,

or functionalities, of the model must be affine invariant. That is, applying an affine map

and then an operation or applying the operation and then the affine map must produce the

5

same result. This property insures consistent results when a user applies a procedure to

two translated, or scaled copies of a same object. We shall provide an intuitive geometric

representation for analytic curves or surfaces, so that potential users can manipulate, edit

or modify curves and surfaces, without needing to understand the underlying mathematics.

For this purpose, we introduce control parameters that have an intuitive geometric meaning.

Analytic functions on a given domain represent a very large class of infinitely smooth

functions, including trigonometric functions and functions with poles outside the domain

of convergence. Thus, the model we propose is very rich and, in particular, provides an uni-

fying representation for polynomial objects and uniformly parameterized circular shapes.

Instead of the classical Taylor representation, we choose to represent analytic functions

in the Poisson basis, because the Poisson basis is more adapted to geometric modeling.

The Poisson basis functions are defined as the limit of Bernstein polynomials, but they

can also be represented by simple explicit expressions. The Poisson control points are the

coefficients of an analytic function in the Poisson basis; they generate a control polygon,

similar to the Bezier control polygon in the polynomial setting. An analytic curve or surface

lies in the convex hull of its control polygon or polyhedra. The Poisson control points can

be computed efficiently from the Taylor coefficients of the function. A similar algorithm

converts the Poisson representation of an analytic function to its Taylor representation. The

Taylor and Poisson representations are equivalent; a function admits a convergent Taylor

series at an arbitrary parameter if and only if the function admits a convergent Poisson

series at . Therefore, Poisson curves and surfaces are exactly the same as analytic curves

and surfaces.

Based on the Poisson representation, we derive tools for analyzing the model. The

left Poisson subdivision algorithm, derived from the left Bezier subdivision algorithm, pro-

duces successive piecewise linear approximations starting from the Poisson coefficients.

This family of piecewise linear functions converges to the corresponding Poisson curve. In

order to represent an analytic curve on a finite interval, the approximations requires only a

finite number of the Poisson coefficients. For a specific subdivision parameter , or equiv-

6

alently a specific number of subdivision rounds at , and a particular domain , only

the first Poisson coefficients are necessary. These Poisson coefficients can

be computed from the first Taylor coefficients. Therefore, although the Poisson basis in-

cludes an infinite number of functions, a finite number of coefficients suffices to represent

an analytic curve on a finite domain.

The convergence of these piecewise linear approximations to the corresponding curve

is smooth. That is, the difference of the points defining the approximation approximates the

derivative of the curve. This result holds for the derivatives of any order; an approximation

of the derivative is computed by taking the difference of the points defining the

piecewise linear approximant.

Subdivision extends naturally from curves to tensor product surfaces by applying sub-

division subdivision is applied independently in each parameter direction to generate ap-

proximating polyhedra for an analytic surface.

The right part of the Bezier subdivision algorithm can also be generalized to the analytic

setting. The right Poisson subdivision algorithm produces a trimming algorithm. From the

Poisson representation of an analytic function , the trimming algorithm computes the

Poisson representation of the trimmed function , where is an arbitrary parameter

in the domain. Using a combination of the left and right Poisson subdivision algorithms, we

can render an analytic function on any interval from its Poisson representation at zero.

(We chose to start from the representation at zero for the seek of simplicity, but, equiva-

lently, we could start with the Poisson representation at any arbitrary parameter where the

function is analytic.) The trimming algorithm allows us to get away from singularities.

That is, by trimming, we can extend the domain of convergence of the Poisson series, and

therefore extend the domain where piecewise linear approximants are valid.

Using both the subdivision and the trimming algorithm, we develop a divide and con-

quer intersection algorithm for analytic functions. Since we can generate an arbitrarily

close piecewise linear approximation of an analytic function on any chosen interval, the in-

tersection points of two curves can be found by intersecting only segments. This algorithm

7

can be used to solve transcendental equations involving analytic functions. This method

presents advantages over the classical numerical Newton’s method, since, for example, the

interval in which we seek the root can be chosen.

We introduce a notion of blossoming for analytic functions. We prove that, as in the

polynomial setting, the analytic blossom exists and is uniquely defined by a simple set of

axioms. The analytic blossom offers a very intuitive and practical notation for expressing

the Poisson control points. The left and right subdivision algorithms can be interpreted

in terms of the blossom; each step of these algorithms follows from the symmetry and

multiaffinity of the blossom. By interpreting the left and right subdivision processes in

terms of the blossom, these algorithms appear as change of bases procedures. As in the

polynomial setting, the blossom interpretation is very enlightening. Related identities like

Marsden’s identity and the de Boor-Fix formula, known not only for polynomials but also

for splines, are also generalized here to the analytic setting.

1.4 Outline of the dissertation

Our goal for this dissertation is to present a model adapted to geometric modeling based

on analytic functions. This model includes not only a representation of analytic curves and

surfaces, but also tools and algorithms to manipulate this representation.

In Chapter 2, we present previous developments and work that led to our research.

Chapter 3 details the Poisson representation: a representation for analytic functions adapted

to geometric modeling. The following chapters present algorithms and tools for this new

model. In Chapter 4, a subdivision algorithm for analytic curves and surfaces is developed.

The next chapter, Chapter 5, presents a trimming algorithm. In both of these chapters,

examples illustrate the uses and applications of these algorithms. Chapter 6 extends the

notion of blossoming to analytic functions. We also give a de Boor-Fix formula and a

Marsden identity, tools closely linked with the standard notion of blossoming, for analytic

functions. Finally, Chapter 7 concludes and exposes open questions and problems for future

research.

8

Chapter 2

Background and Literature Review

The purpose of this thesis is to advance the field of geometric modeling adapted to compu-

tation, more commonly called the field of computer aided geometric design (CAGD). We

begin by recalling how this field originated, and what the motivations were of the pioneers

in this area. Initially polynomials fulfilled the requirements of the model. But polynomials

were replaced first by B-splines, piecewise polynomial curves and surfaces, to gain locality,

and later by NURBS (Non Uniform Rational B-splines) to get an exact representation of

circular arcs and spherical patches. We propose a model for curves and surfaces based on

analytic functions.

To understand our approach in the wider mathematical context, we consider in this

chapter the generalization of polynomials to the analytic setting in three mathematical do-

mains: analysis, probability theory, and approximation theory. We then review the previous

work on analytic functions in geometric design that led us to consider the Poisson basis as

the right basis for expressing analytic functions in geometric modeling. We also briefly

explain the specifics of our approach. We want not only to show that the theory of Bezier

curves and surfaces extends to the more powerful and more general analytic setting, but

also to build a practical framework for using analytic functions in geometric modeling.

2.1 In the beginning, were the polynomials

The field CAGD has its source in the sixties in the French automotive industry. Pierre

Bezier and Paul de Casteljau developed, independently at Renault and Citroen, similar

solutions to the same problem: finding a new, simple, yet powerful model to represent

freeform curves and surfaces [5, 6, 13, 14]. At that time, designers still drew curves and

9

surfaces by hand, which people working in production approximated to generate shapes

they were then able to manufacture. These different representations of a geometric object,

a physical part, led to a trial and error manufacturing process that was inefficient and un-

satisfactory. The entire production line, from the design to the manufacturing of a part,

needed a simple and unique model for geometric shapes. In order to be used by design-

ers, the model had to be intuitive. Also, computers, an emerging technology making the

transfer of precisely defined shapes not only possible but easy, needed to be a key tool

for unifying the model along the production line. However, in order to be understood by

computers, the model required a numeric representation. Apart from these simplicity and

compatibility constraints, the model had to be sufficiently powerful: that is, it had to be

able to represent a large variety of shapes adapted to the production of mechanical parts.

Car bodies, in particular, require the representation of a wide variety of smooth freeform

shapes. Thus, not only did the parameters of the model need to be intuitive, but also they

had to provide a lot of flexibility for generating shapes.

In CAGD, an intuitive model to generate parametric curves or surfaces uses control

points and blending functions. The designer selects a sequence of points

called the control points. The polyline joining this points, the control polygon, roughly

describes the shape of the desired curve. The points characterize a parametric curve

(2.1)

where the functions are the blending functions of the model. Goldman describes the

properties of curves desirable in geometric modeling and shows that these properties

follow from requirements on the blending functions [17]. The first requirement

is for the sum defining the parametric curve to be well defined independent of the

coordinates system. Since the points lie in an affine space, the right-hand-side of (2.1)

must be an affine combination. That is, the blending functions must form a partition of

unity, i.e.

for all in the parameter domain.

10

Other requirements on the blending functions include non negativity, smoothness, linear

independence, symmetry, and a recurrence formula. Depending on the blending functions,

the model interpolates or approximates the control points. For example, Lagrange poly-

nomials are a family of interpolating blending functions. In the following discussion we

are interested in approximating blending functions. In particular, the Bernstein polyno-

mials and the Poisson basis functions are both families of blending functions adapted for

geometric modeling, since they satisfy the requirements proposed by Goldman in [17].

Polynomials can represent a wide variety of shapes. Moreover, polynomials are well

known mathematical objects and are easy to represent numerically. The initial choice for

a CAGD model was polynomial curves and surfaces: a good family of polynomial blend-

ing functions (in the sense Goldman defines in [17]) that forms a basis for polynomials of

a given degree must provide an intuitive model that can be represented easily numeri-

cally. The Bernstein basis is such a polynomial basis, since it satisfies the requirements for

blending functions adapted to geometric modeling.

Forrest first showed that the Bezier representation of a polynomial [5, 6, 13, 14] is

equivalent to using the Bernstein basis as blending functions [16]. When polynomials are

represented in their Bezier form, the control points, also called Bezier points, are intuitive

and simple. No knowledge of analysis is required for design. Also this Bezier representa-

tion is more adapted to computations than the more classical representation in themonomial

basis; the de Casteljau algorithm provides a very powerful, efficient and stable algorithm

for evaluating Bezier curves.

We shall denote by

the Bernstein polynomial of degree . The Bernstein polynomials of degree

form a basis for the set of polynomials of degree less or equal to . Therefore, any

polynomial of degree less than or equal to defines a parametric curve

11

If the control points are chosen in , the polynomial defines a parametric curve in

. In the remainder of this manuscript, we shall identify the function and the parametric

curve it represents. Since the Bernstein basis functions form a partition of unity –that is,

they sum to one– and are positive on the interval , the Bezier curve on the interval

lies in the convex hull of its control points. This property is called the convex hull

property. Figure 2.1 illustrates that a Bezier curve not only has the convex hull property,

but also follows the shape of its control polygon. Moreover, the Bernstein basis is variation

diminishing: a polynomial curve oscillates no more than the control polygon formed by its

Bzier control points. That is, the number of intersections of the curve with any given line is

less than or equal to the number of intersections of the same line with the control polygon

of the curve. Carnicer and Pena proved that the Bernstein basis is the optimal variation

diminishing basis of blending functions for the space of polynomials of degree less than

or equal to [7]. Thus, the control polygon relative to the any other variation diminishing

polynomial basis would intersect a given line at least as many times as the control polygon

relative to the Bernstein basis does. Bezier surfaces generalize Bezier curves by employing

tensor products of Bezier curves –that is, a Bezier surface is a parametric polynomial in

two variables

Just as a Bezier curves has a control polygon, a Bezier surface has a control net generated

by the control points (Figure 2.2).

Bezier curves have been generalized to splines. Splines are also parametric polynomial

curves, but piecewise polynomial functions. That is, there exists a partition of the interval

where a spline curve is defined, such that the restriction of the spline curve to each segment

of the partition is a polynomial. Between two successive intervals, compatibility conditions

–restrictions of the function and its derivatives– insure the continuity and smoothness of the

curve. The B-spline basis functions form a new family of blending functions. In addition to

preserving the intuitiveness of the polynomial model, this piecewise polynomial approach

provides better locality by limiting the influence of a given control point on the curve.

12

Figure 2.1 : Two planar cubic Bezier curves over the interval with their control poly-gons. Between the two figures, only one control point has been moved. Note the effect ofmoving one control point on the corresponding Bezier curve.

Figure 2.2 : Two bicubic Bezier surfaces with their control net. Between the two figures, acorner control point has been moved, modifying the shape of the entire surface.

13

Figure 2.3 : Two -continuous cubic B-splines with their control polygon. Between thetwo figures, only one control point has been moved. Note that the effect of moving onecontrol point is local and does not affect the right part of the curve.

Changing one coefficient of a spline will not affect the curve on the entire domain. This

locality property is an advantage, since it avoids recomputation of the entire curve when

only one parameter is altered and allows a designer to make local changes (see Figure 2.3).

2.2 Generalization of the polynomial framework

Polynomials generalize to analytic functions. This extension appears in many different

fields of mathematics: in analysis, in probability theory, and in approximation theory. Since

polynomials are a strict subset of analytic functions, analytic functions offer a much richer

class of functions, and therefore, in probability and approximation theory, more general

models. We now recall this generalization in these three different contexts.

2.2.1 In analysis

Univariate polynomials are introduced to the high school student as a linear combination

of monomials. Later, the set of polynomials of degree less than or equal to is defined as a

vector space, whose canonical basis is the monomial basis, that is, the basis .

An analytic function is a function that can be expressed as an infinite weighted sum of

monomials, or series of monomials, or power series, or Taylor series. In order for the func-

tion to be well defined, these series have to converge. As with polynomials, the canonical

basis for analytic functions is the monomial basis, or the Taylor basis (these two bases are

14

the same up to a constant factor), although in the analytic setting the Taylor and monomial

bases are infinite. Most of the time, in analysis, analytic functions are expressed and stud-

ied as power series, that is, in the Taylor basis. But, as we have just seen, in CAGD the

canonical basis for studying polynomials is the Bernstein basis, not the monomial basis.

Similarly, we shall see that the Poisson basis and not the Taylor basis is the canonical basis

for expressing analytic functions for geometric modeling. Unfortunately, some results for

analytic functions are bound to the Taylor representation. For example, the convergence

of a Taylor series implies the absolute convergence of this series. We will not be able to

invoke these results directly.

Another point of divergence between our interest in analytic functions and the classical

approach of analysts is that we are interested in the domain Rwhereas analysts often prefer

to consider analytic functions over the domain C . Most of the results for analytic func-

tions are satisfied in both domains. Nevertheless, there are subtle and essential differences

between the notions of real analytic functions and complex analytic functions, so although

we are interested in real analytic functions, we will sometimes need to invoke results from

complex analysis, and therefore will sometimes have to work with complex variables (see

for example, Chapter 6 on blossoming).

2.2.2 In probability theory

Bernstein polynomials have a well known probabilistic interpretation (see, for example,

[9]). Repeat Bernoulli trials of an event with two possible outcomes, success or failure,

where success has a probability between and and failure a probability . If

denotes the probability of successes in trials, then

In probability theory , for , is the binomial distribution of order . In

approximation theory, is the Bernstein polynomial basis of degree .

The convergence of the binomial distribution to the Poisson distribution is a classical

result in probability theory ([9] page 63, Theorem 2.2.3): If goes to when goes to

15

infinity, then

In particular, for , the binomial distribution converges to the Poisson distri-

bution .

Bernoulli trials leading to the binomial distribution can be represented by an urn model

[17, 18]: each trial consists in picking a ball from an urn containing both white and black

balls and returning the ball to the urn. The ratio of white balls to the total number of balls in

the urn is number of white ballstotal number of balls ; when the ball is replaced, repeating this experiment

corresponds to the binomial distribution. Goldman studied other distributions generated by

urn models and their close relationship to the blending functions used in CAGD [17, 18]. A

generalization of the Bernoulli model, adding balls of the selected color after each drawing,

leads to Polya’s urn model, or Polya’s polynomial basis. Goldman shows that interesting

geometric features of the corresponding curves follow directly from probabilistic proper-

ties of these new blending functions. Moreover, experimental parameters in the trial, such

as the ratio of balls added to the urn to the initial number of ball in the urn, may provide

geometrically meaningful parameters for the corresponding modeling basis. The urn model

also generalizes by adding balls of the opposite color after each drawing. This new model

corresponds to B-spline blending functions [18]. Another probabilistic model for B-spline

basis functions, sums of independent random variables, is given by Dahmen and Michelli

in [10]. As an open problem, Goldman suggests using the Poisson distribution for defin-

ing new blending functions for CAGD [17]. He points out that many requirements for a

family of blending functions adapted to geometric modeling follow from the probabilistic

properties of the Poisson distribution. But although this is not an issue for the binomial

distribution, convergence problems may arise when using the Poisson distribution due to

the infinite number of basis functions.

16

2.2.3 In approximation theory

The Stone-Weierstrass approximation theorem states that any continuous function can be

approximated uniformly by polynomials on any compact interval. In the case of analytic

functions, partial sums of the Taylor development provide such an approximating family

of polynomials. Stone proved a extension of the Stone-Weierstrass theorem [40], stating

that any continuous function going to zero at infinity can be uniformly approximated by

a sequence of linear combinations of Poisson basis functions in any subinterval of

(the compactness of the domain is no longer necessary). In the case of analytic functions,

partial sums of the Poisson development again provide such an approximating family of

functions.

The Szaz-Mirakian operator

is also based on Poisson series. This operator associates approximating Poisson series to

a function . Hermann studied the order of approximation of the Szaz-Mirakian operator

[22]. Although the basis of the Szaz-Mirakian operator is the Poisson basis, the Poisson

sums and series we shall consider later in this dissertation to approximate analytic functions

are not the ones defined by this operator. The coefficients of the Szaz-Mirakian approxi-

mating Poisson series are values of at and are different from the Poisson control points

for .

2.3 Limiting cases in CAGD

In geometric modeling and approximation theory, limiting cases of polynomial or spline

bases have already been studied. As we saw in the section on probability, Goldman sug-

gested the use of the Poisson distribution [17], but he never seriously followed up on his

suggestion. In 1966, Curry and Schoenberg studied limits of B-splines [8]. For example,

they considered the spline with multiple knots at , on the interval – that is, exactly

the first Bernstein basis function of degree . Using their general result for limiting cases,

17

the authors proved that when goes to infinity this basis function goes to an exponential

function. This result is one limiting case that we shall use in our work. But these authors

considered only the limit of the first Bernstein basis function; we shall take limits of all the

Bernstein basis functions.

2.4 From limiting cases to a complete modeling framework

Our goal is to construct a framework for using analytic functions in geometric modeling

or CAGD. The work presented in the previous sections led us to the choice of the Pois-

son basis. The polynomial blending functions best adapted to geometric modeling are the

Bernstein basis [7]. From probability theory, under the right conditions, the binomial dis-

tribution (or Bernstein basis) converges to the Poisson distribution when the number of trial

goes to infinity. So, using the right scaling of the parameter domain, the Poisson basis is the

limit of the Bernstein basis, when the degree goes to infinity. The Szaz-Mirakian operator

and the interval of the B-spline function considered by Curry and Schoenberg led us to

consider the right scaling of the Bernstein basis. Moreover, we shall see that Poisson series

exactly model analytic functions. Most of our convergence results derive from analytic

function theory.

Our approach is very practical: we develop tools for manipulating analytic functions in

CAGD. Finding tools and algorithms for analytic functions is quite a natural task. The Pois-

son basis is the basis best adapted to geometric modeling for analytic functions just as the

Bernstein basis is for polynomials. Thus, by drawing the parallel between the well-known

polynomial situation and our new analytic framework, we develop modeling algorithms

and tools for analytic functions. The generalization of polynomial algorithms is intuitive:

we consider an analytic function to be like a polynomial of infinite degree and take into

account the appropriate scaling. Our generalization, however, requires considering limits:

the polynomial case needs only a finite basis and sums, but the analytic case requires an

infinite basis and infinite series. Thus although the algorithms are natural to develop, their

correctness is sometimes hard to prove. For example, convergence problems, like the con-

18

vergence of the subdivision algorithm, leads to switching limits. Hence, although when

implemented these algorithms are efficient (the convergence is rapid), the actual theoretical

proof of the convergence often requires a lengthy analysis.

Conclusion

We saw how polynomial curves, in Bezier form, have been used as a model in CAGD. On

the other hand, in many different domains of mathematics, polynomials have been extended

to analytic functions offering a much richer set of functions. In the past, in geometric mod-

eling, two extensions of the polynomial model to the analytic setting have been proposed.

Goldman left this development as an open problem [17], while Curry and Schoenberg gave

examples illustrating theoretical results on the limits of B-splines [8]. Both of these devel-

opments open the way for us, and suggest the right model, the limit case, to consider. We

shall study the extension from polynomial to analytic functions in great detail, addressing

not only theoretical results (proving that the desirable properties for a basis in CAGD hold

for the Poisson basis) but also providing tools (e.g. a Marsden formula) and algorithms

(e.g. a trimming procedure) to manipulate this new model.

19

Chapter 3

The Poisson Representation for Analytic Functions

Introduction

In this chapter, we present a representation for analytic functions adapted to geometric

modeling: the Poisson representation. We shall see that Poisson curves are to analytic

functions what Bezier curves are to polynomials.

Bezier curves are characterized by their control polygons, and their shape can be mod-

ified intuitively by moving their control points. Typically the control points of a Bezier

curve are the coefficients of a polynomial curve in the Bernstein basis over the interval

. However, the Bernstein basis can be scaled so that the domain of the Bezier curve

is an arbitrary interval . Bezier curves have numerous remarkable geometric features;

for example, they lie in the convex hull of their control points and satisfy the variation di-

minishing property [15]. Many algorithms have been developed to analyze Bezier curves

including the degree elevation and subdivision algorithms.

Here, we extend the notion of Bezier curves to Poisson curves. The Poisson basis is the

limit as tends to infinity of the Bernstein basis of degree over the interval . Thus

the Poisson basis consists of an infinite number of basis functions defined over the set of

positive real numbers. Hence, whereas Bezier curves are polynomials, Poisson curves are

analytic functions. Nevertheless, Poisson curves share many of the geometric properties

of Bezier curves and some of their analysis algorithms as well. Since the Poisson basis

consists of infinitely many analytic functions, some additional convergence issues arise for

Poisson curves that do not appear in the polynomial setting. In this chapter, we review the

Poisson basis and present some of its fundamental properties. We also show that any ana-

lytic function has a Poisson representation and we provide efficient algorithms to convert

20

between the Poisson coefficients and the Taylor coefficients of an analytic function. Then,

we compare the approximation of an analytic function by its Taylor polynomial and its

Poisson partial sum with the same number of terms and illustrate that for functions with

limit zero at infinity and for bounded functions the Poisson expansion provides a better

approximation to the function than the Taylor expansion.

3.1 The Poisson basis

The Poisson basis is a limiting case of the Bernstein basis. Let denote the th Poisson

basis function over the interval and let denote the th Bernstein basis function

of order over the interval . Then, by definition

for

for(3.1)

where denotes the binomial coefficient , and

(3.2)

The following theorem asserts that the convergence of the Bernstein functions to the cor-

responding Poisson functions is uniform on any closed disk containing . This result will

be used to establish the convergence of the subdivision algorithm for Poisson curves in

Chapter 4.

Theorem 3.1 Let be a positive constant, and let be a fixed index. Then con-

verges uniformly to on the closed disk of center and radius when goes to

infinity.

Proof. By definition

21

But, converges to , and the functions and , for any fixed ,

converge respectively to and uniformly on any closed disk of center . Thus, on any

closed disk of center , converges uniformly to .

2 4 6 8 10

0.1

0.2

0.3

0.4

Figure 3.1 : The Poisson basis is the limit of the Bernstein basis of degree scaled tothe interval when the degree goes to infinity. This figure illustrates this convergence,for , with the Bernstein basis functions and (dashed), and the Poissonbasis function (solid).

The explicit expression for the Poisson basis functions (Identity 3.2) provides a way to

derive many of the properties of Poisson curves. Nevertheless, in general, we prefer to use

the characterization of the Poisson basis as the limit of the Bernstein basis in order to take

advantage of the rich accumulated knowledge about Bezier curves.

3.2 Poisson curves

We call a Poisson function or a parametric Poisson curve on the interval [0,R), if for any

the function is defined by a Poisson series in a neighborhood of : that

22

2 4 6 8

0.2

0.4

0.6

0.8

1

Figure 3.2 : The first four Poisson basis functions. These functions satisfy the basic prop-erties of good blending functions: they are positive and bell-shaped.

is, if for any , converges . The vector of coefficients

is the vector of Poisson control points of at . Notice that whereas Bezier

curves are polynomials expressed as finite sums, Poisson curves are analytic functions

represented by an infinite series. The Poisson basis functions are positive on . This

result follow from Theorem 3.1. Moreover, like the Bernstein basis functions, the Poisson

basis functions form a partition of unity. Therefore Poisson curves are affinely invariant

and lie in the convex hull of their control points.

Any analytic function on the interval defines a Poisson curve on this same

interval. Indeed, since the function is analytic on the interval , the function

is also analytic on this interval. Therefore there exists such that

so

The author wants to thank Jean Gallier for suggesting this definition of a Poisson curve.

23

Moreover, for a given and , there exists such that, for all ,

for all

Thus, for all in ,

Hence,

Lemma 3.1 If is an analytic function on the interval , then is a Poisson curve

on this interval and the partial sums converge absolutely and uniformly

to on any interval where .

Moreover each coefficient of in the Poisson basis can be computed as a function of a

finite number of coefficients in the Taylor basis. Let for in ;

since is analytic, for all : , where denotes the derivative of .

Hence, since the function is also analytic:

so

(3.3)

expresses the Poisson coefficients as explicit functions of a finite number of Taylor

coefficients . These Poisson control points can also be computed recursively and

more efficiently from the Taylor sequence using dynamic programming (Figure 3.3).

Since the number of paths from to is exactly , the equality in equation (3.3) holds.

Conversely, we can also compute the Taylor coefficients from the Poisson coefficients.

Since the Taylor coefficients correspond to the derivatives of the function at zero, we need

only differentiate the Poisson series term by term. Let

24

Figure 3.3 : This dynamic programming algorithm computes Poisson coefficients ,appearing on the lateral edge of the diagram, from Taylor coefficients , given at thebase of the diagram. Each node in the diagram is computed by summing the values of thenodes at the origin of the two incoming arrows. This algorithm is quadratic in time, i.e.,

and linear in space , where is the amount of input (or equivalently output),that is, the number of Taylor coefficients given (or equivalently the number of Poissoncoefficients computed from these Taylor coefficients.)

Then

because . By induction

Thus

(3.4)

From the similarity between identities (3.3) and (3.4), we deduce a corresponding dynamic

programming algorithm to convert from the Taylor to the Poisson coefficients (Figure 3.4).

25

Figure 3.4 : This second dynamic programming algorithm efficiently computes Taylorcoefficients from Poisson coefficients. Here, the nodes in the diagram are computed byadding the value at the origin of an incident arrow labeled , and subtracting the value atthe origin of an incident arrow labeled . The complexity in time and space is similar tothe complexity of the algorithm in Figure 3.3.

3.3 Comparison of Taylor and Poisson approximations of analytic func-

tions

For some analytic functions, the Poisson partial sums provide better approximations than

the Taylor partial sums. The Taylor series converges uniformly to on any

subinterval of . Similarly, the Poisson series converges uniformly

to on any interval . But if and then the Poisson

series converges uniformly to on [40]. As an illustration, Figure 3.5 (top) shows

the function and its approximations by Taylor and Poisson sums of order ten –that

is, the first eleven terms. The Taylor approximation is a polynomial and diverges rapidly

(in this example around the value ). Moreover, the fact that the Taylor approximation

diverges and the Poisson approximation is bounded also gives an advantage to the Poisson

26

series for approximating bounded functions. The bottom part of Figure 3.5 illustrates the

approximation of the function by the first eleven terms of the Taylor and Poisson

series.

1 2 3 4 5 6

-0.4

-0.2

0.2

1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5 6

-1

1

2

3

4

1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

Figure 3.5 : The left figures show the functions at the top and at the bottom(solid) together with their Taylor (long dashes) and Poisson (short dashes) approximationsof order ten. The right figures show the difference between the function and the corre-sponding Taylor (long dashes) and Poisson (short dashes) approximations. Because theTaylor partial sum is a polynomial and the functions approximated are bounded, the Taylorapproximation diverges when the parameter goes to infinity.

Conclusion

We presented a representation for analytic functions, the Poisson representation, by gen-

eralizing the Bezier representation for polynomials. We proved the uniform convergence

of the Bernstein basis functions to the Poisson basis functions on closed disks of center

zero in C . The Poisson representation is equivalent to the classical Taylor representation

of analytic functions; we developed two dynamic programming algorithms to convert be-

tween these two bases. For analytic functions going to zero at infinity, and also for bounded

27

analytic functions, we showed that the Poisson partial sums provide better approximations

than the Taylor partial sums. In the next chapter, based on the Poisson representation and

using the uniform convergence of the Bernstein basis to the Poisson basis, we shall develop

geometric modeling algorithms for analytic functions, and prove their correctness.

28

Chapter 4

Left Poisson Subdivision

Introduction

Subdivision as a process to define geometric objects has been gaining increasing popular-

ity in the geometric modeling community, both in academia and more recently in indus-

try. Subdivision characterizes a curve or a surface as the limit of a sequence of piecewise

linear functions. This method presents many advantages: this process is recursive and dis-

crete, therefore adapted to computation. The simplicity of the approximations involved

–piecewise linear functions– leads to efficient and easy to program algorithms. More-

over, this description of geometric objects is refinable, and therefore, provides a multiscale

model, which is a great advantage since it enables the designer to work on the shape of

the object at different level of detail. Warren and Weimer present a general approach to

subdivision methods in [42].

A curve that is the limit of a subdivision process can be approximated by a polygonal

curve to any desired accuracy: for example, to the rasterization precision. By subdivision,

intersections of curves can also be determined within any given tolerance. Theoretical

properties of the subdivision scheme also characterize certain geometric features of the

curve. In particular, the variation diminishing property of Bezier curves, that is, Bezier

curves oscillate no more than their control polygons, is a consequence of the fact that

Bezier curves are limits of a recursive subdivision process based on corner cutting because

cutting corners reduces oscillations [15, 26].

In the previous chapter, we defined a basis for analytic functions adapted to geomet-

ric modeling, the Poisson basis, to express analytic functions. In this chapter, our ap-

proach generalizes standard subdivision techniques from polynomials to arbitrary analytic

29

functions, based on the representation of analytic functions in the Poisson basis. The de

Casteljau algorithm is both an evaluation technique and a subdivision procedure for Bezier

curves. By generalizing different interpretations of polynomial subdivision, in particular

the de Casteljau algorithm, we define a stationary, non-uniform, subdivision algorithm for

Poisson curves –that is, for analytic functions– and prove its convergence. Extensions of

this new subdivision scheme from curves to tensor product surfaces are also discussed. Fi-

nally, we present some examples of approximations of Poisson curves and tensor product

Poisson surfaces generated using this subdivision algorithm.

4.1 Left Bezier subdivision

First, we recall the well known Bezier subdivision algorithm. In the remainder of the

chapter, we generalize this polynomial algorithm to analytic curves.

For a degree Bezier curve

(4.1)

a step of subdivision consists of splitting the curve defined by into two Bezier curves. At

a fixed parameter , the first curve corresponds to the parameter range and

the second to the parameter range . Explicitly, these curves are respectively the Bezier

curves corresponding to and for in (see

Figure 4.1). Bezier curves are represented by their control points; a subdivision algorithm

is a procedure for finding the control points of and from the control points of .

In this section, we present two different ways of expressing Bezier subdivision. These two

interpretations are both useful for generalizing polynomial subdivision to Poisson curves.

For reasons that will become clear in the next section, we shall focus our attention on left

subdivision (first part of the curve).

There is an algebraic identity [19] for the Bernstein basis functions corresponding to

left subdivision:

(4.2)

30

Figure 4.1 : A Bezier curve of degree 3, with its initial control points (on the left) and itscontrol points after one round of subdivision (on the right). The first four of the new controlpoints are generated by the left subdivision algorithm, the last four by the right subdivisionalgorithm.

From this formula it follows that

with (4.3)

The points are generated by the de Casteljau algorithm, which provides not only

a procedural approach to subdivision, but also an engaging schematic interpretation for the

points . In Figure 4.2, the de Casteljau algorithm is applied to the control points of

a Bezier curve of degree 3 at the parameter . Since the control points are inter-

mediate points in the de Casteljau evaluation algorithm, they can be computed efficiently

using dynamic programming. Lane and Riesenfeld show that this polynomial subdivision

algorithm is quadratic in time and linear in space [25].

The algebraic expression for subdivision can also be written in terms of matrix multi-

plication. For convenience, the indices for all our matrices and vectors start from 0. With

this convention, equation (4.3) is equivalent to

where (4.4)

Thus, multiplying the vector of control points of the Bezier curve by the matrix

gives the vector of control points of the polynomial , the first segment of the Bezier

curve. Note that the matrix corresponds to the first levels of Pascal’s triangle

31

Figure 4.2 : The de Casteljau algorithm for a cubic Bezier curve. The Bezier control pointsat the bottom of the diagram are the input. Each interior node, or point, is

computed as an affine combination of the points at the origin of the two incident arrows,using the weight on the arrows. The output, the points , are thecontrol points of the left segment of the subdivided curve. This algorithm also computes,at the apex of the diagram, the value of the polynomial at parameter . This dynamicprogramming algorithm is quadratic in time and linear in space in the number of controlpoints, that is, in the degree of the polynomial curve.

normalized so that the entries of each row sum to one; that is,

... . . .

Proposition 4.1 The matrices form a group under matrix multiplication.

In particular, for any , in :

(1) ,

32

(2) , .

(3) .

Proof. follows immediately from equation (4.2).

, follow immediately from (1).

As a consequence, two rounds of subdivision at parameters and are equivalent to

an only round of subdivision at parameter . In practice, we shall emulate rounds of

subdivision at parameter simply by subdividing once at . Note that this property does

not apply to Bezier subdivision, because both left and right subdivision matrix must be

considered.

4.2 Poisson subdivision

These two interpretations of Bezier subdivision – algebraic and algorithmic – can be ex-

tended to define a stationary subdivision algorithm for Poisson curves. By stationary sub-

division we mean that the same subdivision rule is applied at each stage of the subdivision

process, independent of the number of iterations. Poisson subdivision is derived from and

is very similar to polynomial subdivision. As in the polynomial case, Poisson subdivi-

sion provides an efficient way to render curves and to compute intersections. Nevertheless,

there are several differences between Bezier and Poisson subdivision. First, there are an

infinite number of control points for a Poisson curve. The subdivision process is applied

recursively to all these control points and produces infinite families of control points. Next,

although only the subdivision process that generates the left segment of a Bezier curve

will be extended here (an extension of right Bezier subdivision to a trimming algorithm for

Poisson curves is presented in Chapter 5), this process corresponds to subdivision for the

entire Poisson curve. This feature is a direct consequence of the following observation:

Let be a Poisson curve on the interval . Define the Poisson function

, where , on the interval . Point for point, the functions and repre-

sent the same Poisson curve on their respective domains of definition albeit with different

33

parameterizations. Moreover, this entire curve lies in the convex hull both of the coeffi-

cients of and of the coefficients of in the basis because for all

and the basis functions form a partition of unity. In contrast, in the poly-

nomial case, if is a Bezier curve over the interval and , then for any

the convex hull property in the basis holds for the polynomials

and only on the interval . And, the curve represented by

on the interval is only a segment of the curve represented by on the interval ,

namely its restriction to the interval .

Finally, as a consequence of Proposition 4.1, we shall see that steps of Poisson sub-

division at can be simulated by step of Poisson subdivision at . This equivalence

does not hold in the Bezier setting, in which we need to consider both left and right Bezier

subdivision.

Each of the two approaches to Bezier subdivision presented in the preceding section can

be extended to Poisson curves. First, let us write the equivalent of the subdivision formula

(4.2) for the Poisson basis:

(4.5)

This result can be proved directly:

Thus for a Poisson curve

(4.6)

we have from equation (4.5), for

with (4.7)

34

As in the polynomial case, we can compute the points by applying an extension

of the de Casteljau algorithm for Bezier curves. The following figure is the analog of the

de Casteljau algorithm for Poisson curves; the initial control points , , at the base

of the diagram are the input to the algorithm, and the points , , that emerge

along the left lateral edge, the control points after one step of subdivision, are the output.

Note that in the Poisson case, this algorithm does not provide an evaluation procedure.

Figure 4.3 : The de Casteljau algorithm for a Poisson curve. The input is the set of Poissoncontrol points of the function at the bottom of the diagram. The interior points are computedas in the polynomial setting (Figure 4.2). This algorithm is also quadratic in time and linearin space in the amount of input. (In the application section, we justify that only finitelymany initial control points are necessary to represent a finite segment of a curve or surface.This observation leads to a finite number of input points to this algorithm.) However, in theanalytic setting, this algorithm does not provide an evaluation procedure.

Finally, we give the matrix expression for Poisson subdivision. By equation (4.7), the

Poisson subdivision matrix is simply the limit of the Bezier subdivision matrix, that

is: . The matrix has an infinite number of columns and rows.

However, since is lower triangular, each of its rows has only finitely many non-zero

entries. As in the polynomial case, for the mask applied to the control points, e.g.

, is exactly Pascal’s triangle, normalized so that the entries of each row sum to one.

35

One final observation: Let be the Poisson control points of and , ob-

tained from by one round of subdivision at , be the control points of . Then,

is a Poisson curve on the interval , if and only if is a Poisson curve on the in-

terval . Indeed, from equation (4.7), if is a Poisson curve on the interval ,

then is a Poisson curve on the interval . But from Proposition 4.1:

, the identity matrix. Thus, if the control points are derived

from the control points by subdividing at , then conversely, the control points

are derived from the control points by subdividing at . Consequently, if is a

Poisson function on the interval , then is a Poisson function on the interval

.

Poisson subdivision is an extension of Bezier subdivision. The control polygons gen-

erated by Bezier subdivision are known to converge to the original Bezier curve [25]. We

show next that a similar result is true for Poisson curves.

4.3 Convergence of the Poisson subdivision scheme

In the first part of this chapter, we presented a subdivision algorithm for analytic functions.

Any analytic curve can be expressed in the Poisson basis and its coefficients in

the Poisson basis form the Poisson control polygon (Chapter 3). The subdivision process

generates new control polygons for the same curve (see the algorithm

illustrated in Figure 4.3). The remainder of this section establishes the convergence of

this subdivision algorithm; that is, the control polygons converge to the

corresponding Poisson curve, when goes to zero.

Hermann proposes an alternative proof of a similar convergence result [23]: his method

applies to more general families of functions, and he considers the sequence of parameters

. The proof we develop here is longer than Hermann’s, but does not rely on any

previous results.

36

4.3.1 Proof of the convergence of the Poisson subdivision scheme

In this section, we prove the convergence of the stationary subdivision process defined by

. We begin by fixing our notation. The sequence denotes an initial

vector of control points. For all positive , define the vector of points

. Let , be a Poisson curve and

. It is easy to show that for all positive . Thus, we

have:

But since from equation (4.7) , we also have:

Thus, it is natural to associate with and with . Denote by the piece-

wise linear interpolant (control polygon) of the vector of points , so that

and is a straight line on for all non-negative

integers . We will now show that the uniform limit of the control polygons for

tending to zero is the Poisson function .

We shall establish this result in three steps: first, the convergence will be proved for the

basis functions. Then, we shall extend the result to functions defined by a finite number

of non-zero coefficients in the Poisson basis. Finally, the convergence of the subdivision

scheme will be established for any analytic function. We begin by establishing the conver-

gence of the subdivision scheme for the basis functions in a neighborhood of zero. In the

following two Lemmas, and are fixed parameters — is a non-negative integer and a

real number in .

Lemma 4.1 For any , there exists a real number and a positive value such

that for all positive :

for all

37

Proof. We are going to show that for sufficiently small , both and

(when is sufficiently small) are close to .

Let us fix an arbitrary . The function is continuous everywhere, so certainly it

is continuous at . Therefore there exists a real number such that:

for all (4.8)

The function is piecewise linear and defined by its values at the knots .

Since the initial vector of control points of the basis function is the canonical basis vector

, where the value appears at the th position (we start counting from

zero), the values defining are . Thus, we need to bound the scaled Bern-

stein polynomials because . The polynomials

converge uniformly to on any closed disk containing zero (Theorem 3.1 in chapter 3),

and therefore on any finite interval. Therefore, there exists an integer such that for all

for all (4.9)

Thus, from equations (4.8) and (4.9)

for all (4.10)

On the other hand, if , since is continuous, there exists a real number

such that for all in

(4.11)

Hence, if , then from equations (4.10) and (4.11), we have for

all :

for all (4.12)

because for all non-negative integers and .

Set . If both and are in , then . In the rest of the proof,

will denote an arbitrary subdivision parameter in . On the interval , the function

38

is defined by the values , where

. Therefore, to prove that

for all

it is sufficient to prove that

for all

But for

Since , from equation (4.12),

for all

Also ; thus from equation (4.8)

Thus, for all ,

and, as a consequence,

for all and in

The next Lemma establishes the convergence of the subdivision scheme applied to a

Poisson basis function on any bounded, strictly positive interval.

Lemma 4.2 When goes to zero, the collection of functions converges uniformly

to the basis function on any interval , where .

Proof.We fix arbitrary and such that , and consider an arbitrary in .

Let . Then, by the definition of the piecewise linear function ,

39

By rewriting

we obtain

Both of these two quantities are small when is small. First,

We shall bound the first term on the right hand side, using the uniform convergence

of the sequence to on any bounded interval. By definition

, so . Hence . Therefore if is in

the interval , then must be in the interval . Choose so that .

Then and for all we have . Therefore, if , then

for all . By the uniform convergence of to , there

exists an integer so that for all

for all

Select so that . Then, for all , . Thus,

for all (4.13)

that is, for all .

To bound the second term, , we use the continuity of . The function

is a continuous function everywhere, and therefore uniformly continuous on .

Thus, since , we can find a positive number so that for all ,

and therefore

(4.14)

40

As a result, for ,

In the same way, we can bound :

Indeed, since , we obtain from equation (4.13)

and from equation (4.14)

Hence, for all ,

for all

The convergence of the subdivision scheme for the basis functions is a direct conse-

quence of the two previous Lemmas:

Theorem 4.1 The function is the uniform limit of the collection of piecewise linear

functions on any bounded subinterval of , when goes to zero.

Proof. From Lemma 4.2, it is sufficient to prove uniform convergence on any interval

where . So let and . From Lemma 4.1 we can find and

such that

(4.15)

for any positive and . Moreover, from Lemma 4.2, we can find so

that the inequality (4.15) holds for all positive and . Thus, for any positive

, the inequality holds for all .

41

Since subdivision is a linear process and because Theorem 4.1 establishes the conver-

gence of Poisson subdivision for the basis functions, the convergence holds for any finite

linear combination of the basis functions:

Corollary 4.1 Let be a Poisson function defined by a finite sum — that is,

having only a finite number of non zero coefficients. Then the functions converge

uniformly to on any bounded subinterval of , when goes to zero.

Next we shall show that the uniform convergence of the Poisson subdivision scheme is

also valid for arbitrary Poisson functions. The following Lemma bounds the influence of

the high order coefficients.

Let be an arbitrary Poisson function on the interval . For any

integer , let the function be the th partial sum of relative to the

basis .

Lemma 4.3 Let . For any interval , where , there exists an integer

such that for all and for any

for all

Proof. If , since . Thus, in the rest of the

proof, we will assume .

On the interval , the function is defined by the first control points

obtained after one round of subdivision on at the parameter . Since

the function is piecewise linear, it is sufficient to show that for sufficiently

large for any . Note in particular that, for ,

42

. If , we have from equation (4.7) for

because and

Since is a Poisson function on the interval , the function is analytic on the interval

, and therefore so is the function . Thus

converges absolutely. Hence, there exists an integer such that for all

therefore

Thus, for any , for all .

Finally, we prove the convergence of the subdivision scheme in the most general case

— that is, for arbitrary analytic functions:

Theorem 4.2 Let be an analytic function on the interval . The

Poisson curve is the uniform limit of on any interval , where , when

goes to zero.

Proof.Due to Corollary 4.1, the theorem remains to be proved only for functions defined

by an infinite sum.

Let . Then we have:

43

Now fix an arbitrary . From Lemma 3.1 we can choose an integer so that for all

Moreover, from Lemma 4.3, there exists an integer such that for all and for

any in

Let us define . Then

and for any

Finally, by Corollary 4.1, since the function has a finite number of non-zero coefficients,

there exists a positive number such that for all :

Thus, for all , — that is, the Poisson curve is the uniform

limit of the piecewise linear functions on the interval , when goes to zero.

If is an analytic function, we have just proved the uniform convergence of the piece-

wise linear functions to , when goes to zero. If , these piecewise

linear functions are obtained by subdivision at the parameter using the initial

vector of control points .

Moreover, as we already noticed above, subdividing at the parameter is equivalent to

subdividing times at the parameter . When , the sequence goes to zero;

thus we have proved that the Poisson subdivision scheme applied recursively to the vector

of control points of converges uniformly to . Therefore, just as for Bezier curves, we

can define a convergent iterative subdivision process for analytic curves.

But the control polygon of a Poisson curve can also be deformed continuously into

the curve it defines. The curve is exactly the initial control polygon of , and

is the curve itself. Therefore we have a continuous process, as well as

44

a high order discrete iterative process, converging from a Poisson control polygon to the

corresponding analytic curve.

4.3.2 Smooth convergence of Poisson subdivision

This convergence result can be extended to derivatives: discrete derivatives of the piecewise

linear approximations converge to the corresponding continuous derivatives of the

function . This result holds as well for Bezier curves [32].

Any function that is defined by a Taylor series over the interval is

and its derivatives are also defined by a Taylor series on generated by differentiating

the original Taylor series term by term. Therefore, a Poisson function, defined by a Poisson

series on , is also and its derivatives are also Poisson functions defined

by Poisson series generated by differentiating the original series term by term. Moreover,

since , it is easy to show that the vector of Poisson control points

for the -th derivative, , of a Poisson function with vector of control points is

the discrete difference of . Thus, if denotes the difference matrix –i.e., ,

, otherwise – then are the control points of .

Moreover, subdivision and discrete differentiation commute. The following lemma

asserts the commutativity of these two linear processes:

Lemma 4.4

Proof. For , this identity follows by multiplying the matrices on both sides and

comparing the results. The general result then follows by induction on .

Since the discrete difference of the control points generates the control points of

, Lemma 4.4 implies that applying the subdivision process to the derivative of the

curve is equivalent to applying the discrete difference of order to the vector of control

points generated by subdivision. Thus, the control polygon generated by subdivision not

45

only provides a piecewise linear approximation of the original Poisson function, but also

intrinsically contains approximations for the derivatives. We define the discrete derivative

of of order to be the piecewise linear function defined over the same partition

as and whose values are the discrete difference of order of the values defining

, i.e., –we need to divide by because the distance be-

tween any two consecutive knots defining the piecewise linear function is . As a

consequence of Lemma 4.4, this discrete derivative of order is exactly and

therefore by Theorem 4.2 approximates the Poisson curve .

4.3.3 Order of approximation of left Poisson subdivision

Lane and Riesenfeld proved that the polynomial subdivision algorithm produces quadrati-

cally convergent piecewise linear approximation of a curve. That is, the order of approx-

imation of the piecewise linear approximant is in , where is the distance between

two consecutive knots of the partitioned domain of the piecewise linear approximant. In-

deed, this is the best possible order of approximation for a piecewise linear approximation.

When running the polynomial subdivision algorithm, the subdivision parameter is typi-

cally chosen to be for two reasons. First, since once step of subdivision is equivalent

to considering the left and right part of the initial curve over the intervals and ,

subdividing at allows us to keep a uniform partition of the parameter domain. More-

over, dividing by is an efficient operation, since it requires only a right bit shift. So,

one step of polynomial subdivision divides the distance between the knots of the piecewise

linear approximation by , and therefore at each step the upper bound between the piece-

wise linear approximation and the function is divided by . Thus, even though the order

of approximation of the piecewise linear interpolant is quadratic, the distance between the

approximants and the function is a decreasing geometric sequence.

The order of approximation of the left Poisson subdivision algorithm has so far only

been proved to be linear. The proof of Theorem 4.2, that is, of the convergence of the

piecewise linear approximants generated by left subdivision to the corresponding Poisson

46

function, relies on the convergence of the Bernstein polynomials to the Poisson basis func-

tions (Theorem 3.1). The rate of convergence of the basis functions has been proved to be

of order , where is the degree of the Bernstein polynomials (see, for example, [2]).

As a consequence, the piecewise linear approximation generated by subdividing at is in

. But in his alternative proof, Hermann derives a better upper bound for the order

of approximation when the parameter : he proves that the difference between the

piecewise linear approximant and the function is [23].

In practice, we shall run the subdivision algorithm at or, more generally, at

. (More details on the implementation are given in the application section; see

Section 4.4.) In this case, the sequence of subdivision parameters (or degree, for Hermann

who considers degree elevation) is a geometric sequence. Hence, at each step, the subdivi-

sion parameter, which is also the distance between two consecutive knots in the parameter

domain of the piecewise linear approximant, is divided by . Therefore, at each step the

distance between the Poisson function and its approximant is divided by . Therefore,

as in the polynomial case, the distance between the approximants generated by subdivision

and the original function decreases geometrically.

4.3.4 The variation diminishing property of Poisson curves

A curve scheme satisfies the variation diminishing property if no curve oscillates more than

its control polygon. That is, the number of intersections of the curve with any given line is

less than or equal to the number of intersections of the same line with the control polygon

of the curve. This property is important in geometric modeling and is one of the desirable

properties proposed by Goldman [17] for a family of blending functions in CAGD. Not

only does the variation diminishing property insure that a curve oscillates no more than

its control polygon, providing, for example, an upper bound on the number of inflection

points on a curve, but also, this property allows us to develop efficient divide and conquer

intersection algorithms for such curves –using the fact that if there is no intersection be-

tween a line and the control polygon, then there is no intersection between the same line

47

and the curve itself. An intersection algorithm for Poisson curves based on this strategy is

developed in the next chapter. In the following paragraph, we show that Poisson curves are

variation diminishing.

The variation diminishing property is a consequence of the existence of a convergent

subdivision scheme based on corner cutting. Therefore the variation diminishing property

is satisfied by Poisson curves. Corner cutting consists of replacing a vertex in a polygon by

two vertices belonging to its two adjacent edges. Each step of the de Casteljau algorithm,

for a subdivision parameter between and , is a corner cut. Therefore, subdividing is

equivalent to applying a sequence of corner cutting operations, an infinite sequence of such

operations in the case of Poisson subdivision. Corner cutting diminishes the oscillation of

a polygon: the number of intersections of any line with a polygon obtained after a corner

cutting operation is at most the number of intersections of the same line with the original

polygon. Therefore, subdivision diminishes the oscillations of the successive control poly-

gons. In particular, any control polygon obtained after rounds of subdivision oscillates

no more than the original control polygon corresponding to the Poisson coefficients. But

we have proved that as tends to infinity, the control polygons generated by subdivision

converge to the Poisson curve. Thus, a Poisson curve can oscillate no more than its control

polygon: the number of intersections of any line with a Poisson curve, counting only sim-

ple intersections, is at most the number of intersections of the same line with the control

polygon of the curve.

4.4 Examples and Implementation

In the following examples, each Poisson curve is represented together with three approxi-

mating control polygons, and the subdivision parameter is arbitrarily fixed to be . These

control polygons are computed by applying the de Casteljau algorithm for Poisson curves

to a subset of the initial control points. More precisely, to obtain the control polygon of

order over the interval , we apply one step of our subdivision algorithm at parameter

to the first coefficients of the Poisson curve. The resulting polygon approximates

48

2 4 6 8

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Figure 4.4 : The Poisson basis function (solid), and its approximating subdivision poly-gons of orders 1 (dots), 2 and 4 (dashed). Note the rapid convergence of the the controlpolygons to the curve.

the Poisson curve on the parameter interval . These polygons converge

to the Poisson curve because, as we have seen in the previous section, only the first

initial control points influence the control polygon of order over the interval .

Our first example, Figure 4.4, represents approximations of the Poisson basis function

, which corresponds to the initial vector of control points .

0.2 0.4 0.6 0.8 1

2

4

6

8

10

12

14 0.5 1 1.5 2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 4.5 : The functions and (solid), and their approximating subdivision poly-gons of orders 1 (dots), 4 and 8(dashed). Both these functions have a pole at , butonly presents a singularity at this point. In the left figure, the singularity is approxi-mated by the subdivision algorithm, in the right figure, the subdivision algorithm appearsto diverge for values greater than (the control polygon of order clearly diverges around).

49

Figure 4.5 illustrates the convergence of the subdivision scheme to an analytic function

with a finite radius of convergence. The left figure shows the function , which has a

pole at , and illustrates the approximation of the discontinuity. The right figure is

the function , which has no singularity at , but has radius of convergence due

to the pole at . In both cases we observe the divergence of the subdivision polygon

for values greater than the radius of convergence.

0.2 0.4 0.6 0.8 1

-0.1

0.1

0.2

0.3

0.4

Figure 4.6 : The logarithmic spiral (solid), and its approximat-ing subdivision polygons of orders 0 (dots) – the initial control polygon: the square

– 3 and 6 (dashed). The control polygon of order ap-proximates the curve closely.

If is an analytic function, the sequence is also the se-

quence of coefficients for the function in the Poisson basis. Figure 4.6 rep-

resents the logarithmic spiral : its vector of coefficients corresponds to

the Taylor coefficients of the parametric circle , that is, the vertices of the square

.

The following figure, Figure 4.7, represents the archimedian spiral on

the interval . The control coefficients of the spiral were computed from its Taylor

coefficients using the algorithm presented in Figure 3.3 in the previous chapter.

We can also generate Poisson surfaces by the standard tensor product construction; Fig-

ure 4.8 and Figure 4.9 represent the basis function and the function

with three approximating polyhedra. Figure 4.10 illustrates the use of the subdivision al-

gorithm to generate approximations of the sphere. Note that this algorithm is the only

50

-20 -10 10 20

-15

-10

-5

5

10

15

Figure 4.7 : The archimedian spiral , (solid), and its approximat-ing subdivision polygons of order 0 – initial control polygon – (dots), 3 and 6 (dashed). ThePoisson control points are computed from the Taylor coefficients of the functions using thealgorithm in Figure 3.3.

Figure 4.8 : Approximation of the tensor product basis function by its controlpolyhedra of orders 1, 2 and 3 and the basis function itself (right most figure).

known stationary subdivision scheme providing an exact representation of the sphere –that

is, such that the sphere is the limit of the subdivision process. Another subdivision scheme

for surfaces of revolution, non stationary but with finite support, is given in [33]. To com-

pute the initial coefficients of the sphere

, we first generate the Poisson control points and of the trigonometric

functions and from their Taylor coefficients by the algorithm proposed in Figure

3.3. Then, the left subdivision algorithm is applied to in

both parameter directions. That is, first one iteration of subdivision is applied to the points

51

Figure 4.9 : Approximation of the tensor product surface by its control polyhedraof orders 1, 2, 4 and the surface itself (right most figure).

for each , to generate the new control points . Then, subdividing the

family of control points for each generates the new control points . As

with the curves, the subdivision parameter is and the order of subdivision is the number

of iterations in each direction.

Figure 4.10 : A Poisson subdivision scheme converging to the sphere: control polyhedra of the

sphere of orders 1, 2, 3 and 6.

Conclusion

We developed a subdivision algorithm for analytic functions based on the Poisson repre-

sentation. We then proved the convergence of this subdivision algorithm; that is, on any

closed subinterval in the domain of convergence of the Poisson (or Taylor) series of the

function at zero, the piecewise linear approximations generated by subdivision converge

uniformly to the Poisson curve. Finally, we generated examples by implementing the sub-

division algorithm for curves and tensor product surfaces. These examples show the rapid

52

convergence of the piecewise linear curves or surfaces to the actual analytic functions.

53

Chapter 5

Right Poisson Subdivision

Introduction

Using the Poisson representation, two different subdivision algorithms for analytic func-

tions can be derived as extensions of the Bezier polynomial subdivision algorithm. The

first procedure, the left subdivision presented in the previous chapter, generates an approx-

imation algorithm; the second, the right subdivision that we shall develop in this chapter,

generates a trimming algorithm. Three applications of these two subdivision algorithms

are presented below: an evaluation algorithm, a method for extending the convergence do-

main of the Poisson representation of an analytic function, and an intersection algorithm

for analytic curves.

5.1 Right Poisson subdivision

In this section we construct the Poisson control points of a trimmed Poisson curve. That

is, if is our original Poisson function, we want to compute the Poisson control points

of where is an arbitrary but fixed parameter. But since these points are not

computable, they are expressed as the limit of order approximating control points. We

then prove that the Poisson functions defined by these approximating families of points

converge uniformly to the trimmed function. This convergence property is true for any

function defined by its Poisson expansion at zero, as well as for all its derivatives.

54

5.1.1 Right Bezier subdivision

Right Poisson subdivision is an extension of the well known de Casteljau subdivision algo-

rithm for Bezier curves, different from the extension we presented in the previous chapter.

In this subsection we first recall some classical results for polynomials, in order to be able

to refer to them later on.

A Bezier curve is a polynomial parameterized over an arbitrary interval . If is

the degree of the curve, the control points of the curve are the coefficients

of in the basis , that is

The polygonal curve defined by the control points is called the control polygon of the

Bezier curve, and is independent of the interval of parameterization . Therefore, a

common choice is to take . Here, however, we shall consider the param-

eterization over the interval for two reasons: First, the Poisson basis is a limiting

case of the Bernstein basis of degree over the interval . Second, we shall consider

the subdivision algorithm in two distinct ways by considering two different approaches to

subdivision, but over the interval , these two approaches merge into one.

In the polynomial setting, the de Casteljau algorithm is both an evaluation algorithm

and a subdivision algorithm [Figure 4.2]. This dynamic programming procedure offers

a nice way to visualize, and an efficient way to compute, subdivision. In the following

discussion, we consider a Bezier curve of degree parameterized over the interval .

The subdivision parameter can be defined in two different ways. Either we can choose

in and subdivide at , in which case the coefficients along the arrows in the de

Casteljau algorithm would be and , as in the previous chapter. Or we can choose an

absolute parameter , in which case the coefficients in the de Casteljau algorithm would be

and . While the first approach generates an algorithm for approximating a Poisson

curve, the second approach yields both an evaluation algorithm and a trimming algorithm.

In the next paragraph, we present algebraic identities characterizing this second approach

55

to subdivision in the polynomial setting.

Just as for left subdivision, we can derive algebraic identitiesmodeling right subdivision

[19]. In this second approach the subdivision parameter is a number in . One step of

this right subdivision is equivalent to computing the control points of the restriction

of the curve to the interval from the original control points over the interval .

The corresponding algebraic identities for the change of basis are

(5.1)

From these equalities, we obtain an expression for the new control points:

(5.2)

In the next section we present a generalization of this subdivision algorithm to the

Poisson setting, and in the remainder of this chapter we develop several applications of the

two subdivision algorithms for analytic functions.

5.1.2 Trimming: Right Poisson subdivision

Extension of the polynomial procedure

In this section, we extend right polynomial subdivision to analytic functions and show that

this extension generates a subdivision algorithm for Poisson curves. Equalities (5.1) and

(5.2) from the polynomial case generalize readily to the Poisson setting. Indeed, it follows

easily from the binomial theorem that

(5.3)

so, for in ,

where (5.4)

56

These equations provide a tool for trimming Poisson curves, since the points

are the Poisson control points of the curve parameterized by the analytic function

. We will discuss in detail, in the applications section, the domain where the Poisson

series at of converges – that is, where we can approximate by our left subdivision

algorithm. In this section, though, we consider only the interval , where

the function is defined by the control points : If ,

then and therefore equation (5.4) converges. Geometrically, the points

define a control polygon that provides a piecewise linear approximation to the

Poisson curve defined by over the interval . These approximations can be refined,

as desired, using the approximation algorithm presented in Chapter 4.

Approximation of right subdivision

Equation (5.4) defines the control points of the Poisson curve restricted to

the interval , but no algorithm to compute these control points is provided since the

control points are defined by an infinite sum (equation 5.4). In fact, it is not possible to

compute these points by extending the de Casteljau algorithm [Figure 4.2] directly, as in

the case of left subdivision. Since in the Poisson setting we start with an infinite number

of control points, the points corresponding to the points on the right lateral edge of the de

Casteljau diagram in the polynomial case recede to infinity. To obtain finite sums, for every

positive integer we define a computable sequence so that the Poisson curves

defined by the control points converge uniformly to the Poisson curve ,

that is, to the trimmed curve with control points .

Since the scaled Bernstein basis converges to the Poisson basis, we define

(5.5)

Once again, the de Casteljau algorithm allows us to visualize the construction of the points

and defines a dynamic programming algorithm to generate these points from

the initial control points [Figure 5.1].

57

.....

.....

.....

....P1 P2P0

1 - a_na_n 1 - a_n

a_n

* *

*

1 - a_na_n

1 - a_na_n 1 - a_

na_n

R (a)n0 R (a)n

1

nlevels

Figure 5.1 : A de Casteljau-like algorithm to compute the points .

The following lemma asserts that the points are the Poisson control points of

an analytic function, for any arbitrary positive integer . This result is useful for proving

the main theorem, Theorem 5.2.

Lemma 5.1 Let be a Poisson curve over the interval . Then for

any positive integer , the series defines a Poisson curve over the interval

, and

(5.6)

Proof.This result is proved by induction on . First, for , by hypothesis

, which is a Poisson function on the interval . For , observe that .

Using this identity, we can easily deduce that

(5.7)

58

From Chapter 3, both and are Poisson functions over the interval , and by

addition so is the function .

Equation (5.7) yields, by induction,

(5.8)

Thus, by induction, for any positive , the function is a Poisson function on

the interval .

Now consider the points and defined for a fixed parameter value in

.

Theorem 5.1

Proof.Note that, if we call and the first points of , then

and are the first points of the function with Poisson coefficients which

is analytic over the interval (from Lemma 5.1.) Thus, it is sufficient to prove that

, for a generic Poisson function, that is, for the function .

For in , it follows from Theorem 4.2 with that

But by construction and by definition

, so .

Before proving the convergence of the sequence , we need to verify that the series

converges, that is, the function is well defined:

But, from Lemma 5.1, converges on . Therefore, the infinite series

is well defined.

The following theorem establishes the convergence of the functions , defined by

the sequences , to the function , defined by the sequence , where

59

. Note that this proof does not use Theorem 5.1 in which had to be positive.

But first we prove the following general result that we will need in the proof of the theorem.

Lemma 5.2 The series

(5.9)

converges absolutely, therefore uniformly, in on any closed subinterval of ,

where is the radius of convergence of .

Proof. To establish this result, we need to consider not as a real analytic function, but

as a complex holomorphic function. By hypothesis, the magnitude of the pole closest to

is . Thus, for any in , the series (5.9) is absolutely convergent.

Let us take an arbitrary in so that . Let

; then , but . We are going to establish the uniform convergence

of the series (5.9) for any of magnitude less or equal to , that is, for in ,

the closed disk of center and radius . First, note that on the closed disk

the function admits a maximum . That is, for any , .

Moreover, from Cauchy’s integral formula for the derivatives (see, for example, [37])

where denotes the circle of center and radius . Since for any of magnitude

less or equal to , , it follows that . Moreover, on

, . Thus, for any in ,

This inequality establishes the absolute and uniform convergence of the series, relative to

, on any closed disk , where is chosen arbitrarily in . In particular,

this inequality implies the absolute and uniform convergence of the series for any real in

.

The author wants to thank Mike Wolf for his help with this proof.

60

This main theorem establishes that, not only does each control point converge to

the control point , as Theorem 5.1 asserts, but also the Poisson functions defined

by the control points converge to the function .

Theorem 5.2 When goes to infinity, the sequence converges uniformly to on any

closed subinterval of .

Proof. Let be the vector of control points for the Poisson function . We

apply on this initial set of control points one round of the subdivision algorithm at the

parameter as illustrated in Figure 5.1. Denote by the sequence of

points generated by left subdivision: that is, the points that appear on the left lateral edge

of the algorithm. In addition to and , we now introduce the Poisson functions and

defined by the control points and .

From section 4.2 on left subdivision, we have

(5.10)

and

(5.11)

Moreover, from the proof of Lemma 5.1,

(5.12)

From equations (5.10) and (5.12), we get

(5.13)

Thus by equations (5.13) and (5.11) and using the change of variable

(5.14)

61

But by Taylor’s theorem:

(5.15)

From equations (5.14) and (5.15) it appears already that converges point-wise to

. But to establish uniform convergence we must show that

goes to zero when goes to infinity for every in . For any integer ,

(5.16)

Let us fix . From Lemma 5.2, the series converges

uniformly relative to on any closed subinterval of . That is, there exists

so that, for all and ,

(5.17)

And, from the absolute and uniform convergence of this same sum, there exists so that

for all

(5.18)

Moreover, for any positive integers and ,

(5.19)

so from equations (5.18) and (5.19), for all ,

62

(5.20)

Finally, to bound the term , observe that

(5.21)

because

But

so

Now, let . Since converges absolutely and uniformly

on , there exists a constant so that, for all ,

(5.22)

Choose so that . From equations (5.21) and (5.22), for any

and

63

This last inequality implies

(5.23)

for all and .

Using equations (5.17) and (5.20) with and equation (5.23) for any ,

we can bound each term of the left hand side of equation (5.16) by . Thus converges

uniformly to on .

The following corollary extends Theorem 5.2 to the derivatives of . Thus, the func-

tions converge smoothly to the function .

Corollary 5.1 For any integer , the functions converge uniformly to on

, where is an arbitrary number in .

Proof. This corollary is a consequence of a general result on convergence for analytic

functions; that is, if we have a uniform approximation of an analytic function by analytic

functions, then the th derivative of the approximants also provides a uniform approxima-

tion of the th derivative of the original function. See Davis[11], p 112.

5.2 Applications

In this section we use the two subdivision algorithms, left subdivision leading to an ap-

proximation algorithm and right subdivision leading to a trimming algorithm, to develop

practical analysis algorithms for Poisson curves and surfaces. First, we present an evalu-

ation procedure for Poisson curves via an approximation algorithm that comes about as a

direct consequence of right subdivision. Then, we propose a rendering method for analytic

curves, on different intervals. In the case of analytic functions with poles, we consider more

closely than in the previous section the domain of convergence of a trimmed Poisson func-

tion; the domain where a Poisson curve can be approximated using a subdivision algorithm

is extended thanks to the trimming algorithm. Finally, we describe an intersection algo-

64

rithm for analytic functions, similar to the intersection algorithm induced by subdivision

for polynomial curves.

5.2.1 Approximation of a Poisson function inside the convergence domain of its Pois-

son series

Evaluation of a Poisson curve and its derivatives at a given parameter value

As in the polynomial setting, the de Casteljau algorithm provides an evaluation procedure

for analytic curves. At a given parameter in the domain of convergence, .

But, from Theorem 5.1, we have:

and can be computed by applying the de Casteljau algorithm at [Figure 5.1].

The derivatives of can also be approximated using the de Casteljau algorithm. From

Chapter 4 Section 4.3.2, the coefficients of the -th derivative of can be derived from

the coefficients of by taking discrete differences [Figure 5.2]. Then, using the same

procedure for as for , we can find a sequence of values converging to for

any in .

Approximation on a domain not containing zero

In Chapter 4, we presented an approximation algorithm for Poisson curves on any subinter-

val of the domain of convergence of the curve at . Here, using both the approximation

and trimming algorithms, we construct a more general approximation procedure; that is,

we approximate a Poisson function on , an arbitrary closed subinterval of

where is the radius of convergence at zero. From the control points of we com-

pute the control polygon that defines a function approximating the trimmed

function . We can then apply the approximation algorithm induced by left subdivision to

the control points . From Theorem 5.2, the convergence of the functions to

the function is uniform on the interval , and by Theorem 4.2 the control polygons

65

.....

..........

....P (m)0

mlevels

-1

-1 1 1-1

-1 1 1-1

-1 1 1-1 -1

P (m)1

P0

P,0

P2P1

P,1 P

,2

P1(2)

....

....

....

Figure 5.2 : This diagram describes a dynamic programming algorithm to compute thedifference of the Poisson control points , which are also the control points of, from the Poisson control points of the function . The time complexity of this

algorithm is where is the order of the derivation and is the number of controlpoints to compute. The space complexity is , since each row of control points in thediagram depends only on the previous (lower) row.

generated by the left subdivision algorithm also converge uniformly to the function on

the interval . Therefore, by combining these two algorithms, we can construct a se-

quence of control polygons converging uniformly to the function on the interval .

Figure 5.3 illustrates this algorithm.

This approximation depends on the two subdivision algorithms: for the trimming al-

gorithm, the relevant parameter is the degree , chosen here to be powers of two; in our

example is the trimmed function generated for . For the approximation

algorithm, the order is, as in the previous paragraph, the number of iterations at . In Fig-

ure 5.3 the order of the approximation algorithm is kept constant in order to observe the

convergence of the trimming algorithm.

The algorithm presented above is general enough to approximate entire functions – that

is, analytic functions having no poles – on any interval [a,b]. Nevertheless, for an analytic

function having a finite domain of convergence around zero, we are limited to

66

1.5 2 2.5 3 3.5 4

-1

-0.5

0.5

Figure 5.3 : Approximation of the function (solid line) over the intervalby the control polygons of order of the functions , , and (dashed

lines). Using both the trimming and the approximation algorithms, the cosine function canbe approximated on any closed interval.

intervals . In the next section, we shall see how to approximate outside

the domain of convergence of the function at . In particular, this new method will solve

the divergence problem illustrated in the previous chapter in Figure 4.5.

5.2.2 Approximation of a Poisson function outside the initial convergence domain

In this section, we address the approximation of analytic functions with poles. So far,

we have been able to approximate functions with poles on any closed subinterval of their

domain of convergence around zero. Below, trimming is used to extend the domain where

a Poisson function is defined by a Poisson series. The rendering process induced by left

subdivision then applies on this extended domain.

Analytic extension

Let us first consider, for reference, the well known case of an analytic function defined by

a Taylor series, and then generalize, in the next section, to the Poisson setting. Let

, for . The function has a radius of convergence

67

of at least , where . But may have a radius of convergence greater

than . For example, for , , but has radius of

convergence even though (see, for example, [28].)

Application to the Poisson setting

We shall see that analytic extension works as well for the Poisson representation. However,

in practice, we are not able to get all the possible cases of analytic extension working due

to error accumulation. For a function , having a convergent Taylor series is equivalent

to having a convergent Poisson series. Analytic extension applies as well in the Poisson

representation; that is, the radius of convergence of the Poisson series defining may

be greater than . Moreover, a Poisson series is associated with a rendering algorithm

on the interval . Thus, the function can be rendered on using the Poisson

control points of at and on using the trimming and rendering algorithm.

Figure 5.4 shows an approximation of the function on the interval . This function

could be approximated directly –that is, without trimming– only for values less than

[Figure 4.5].

Moreover, from Theorem 5.2 the trimming algorithm applies as well for .

Thus, the rendering domain can be extended not only to positive but also to negative values

[Figure 5.5].

More generally, if we consider as a function of a complex variable, the trimming

algorithm extends the domain when the trimming value is further than from

the poles. Since must be in the convergence domain, several rounds of trimming may be

necessary, as in the example in Figure 5.5.

Now, let us consider the case where is closer than to a pole:

Suppose is a pole of in . Can we approximate on an interval where

? Since any analytic function on can be extended on , the pole can be

avoided by going into the complex plane, as shown in Figure 5.6. Unfortunately we

found that, in practice, these successive trimming steps do not lead to the expected

68

0.6 0.8 1.2 1.4 1.6 1.8 2

0.35

0.45

0.5

0.55

0.6

0.65

Figure 5.4 : Approximation of the function over the interval (solid line)by the control polygons of order of the functions , , (dashed lines). Note thatthe singularity appearing in Figure 4.5 for has been eliminated.

result, probably due to error accumulation. Therefore, we did not manage to approx-

imate successfully on domains beyond a singularity.

The closest poles to are in . By successive

trimming, we can find a sequence of Poisson series defining the function on

for any . Using left subdivision, we can then render the function on

[Figure 5.7]. Figure 5.8 illustrates this procedure for the function ,

which has poles at and . Approximations to this functions are computed, first

on the interval where the Poisson series converges, and then, using successive

trimming at and , for values greater than .

5.2.3 Intersection of analytic functions

The recursive nature of the Bezier subdivision algorithm for polynomials leads to an ef-

ficient, divide and conquer intersection algorithm for polynomial curves [24]. From our

piecewise linear uniform approximation scheme for Poisson functions, we can generate a

similar divide and conquer intersection algorithm for analytic curves.

69

-2 -1.5 -1 -0.5 0.5 1

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Figure 5.5 : Approximation of the function over the interval (solidline) by the control polygons of order of the functions , , (dashed lines). Thedomain has been extended using two successive rounds of trimming: one at , one at .

To determine at a precision the intersection points of two analytic functions and

defined on the intervals and , the algorithm proceeds as follows:

Find control polygons and providing an -approximation of on and

of on .

where the function does:

if

return .

else if is a line segment

– if is a line segment

return

70

a0

a2a1 a3

Pole

a4

Figure 5.6 : Trimming successively at , , and , should, in theory, enable us to ren-der the function for values greater than the pole. Unfortunately, in practice, this algorithmis numerically unstable.

– else return

else

Note that this algorithm can be improvedwhen computing the intersection of an analytic

function and a polynomial by using for the polynomial curve the classical Bezier approach,

subdividing only when we need to split the curve into two parts [Figure 5.9]. The right

side of Figure 5.9 shows the discarded control polygons while searching for the left most

intersection in the left part of the figure.

The intersection algorithm can be used to find the roots of a transcendental equation on

a bounded interval, as illustrated in Figure 5.10. By using the trimming algorithm with

at and then the subdivision approximation algorithm of order , the root is found with

three significant digits of accuracy. The drawback of this algorithm compared to Newton’s

method is that the approximation order has to be decided in advance, whereas the result of

71

a0a2

a1

Poles

Figure 5.7 : By trimming successively at , ..., the function can be approximated by ourrendering algorithm on for any . This successive trimming can be done as longas there is no pole at any positive real value.

Newton’s iterativemethod can be refined. Nevertheless, the intersection algorithm based on

subdivision has two advantages over Newton’s method: first, we choose the interval where

we seek the root. The same example, treated by Newton’s method with an initial estimate

of zero for the root, does not lead to the same result, but rather to a root greater than

five. Moreover, a piecewise linear -approximation of a Poisson function computed once

can be used several times to solve different equations involving the same function. Note,

however, that the advantages of these two approximation algorithms can be combined:

The subdivision algorithm can be used to determine the initial guess used by Newton’s

approximation method. A first guess close enough to the expected root insures that the

Newton method converges to this particular root, and the refinability and the convergence

rate of the Newton method then insures efficient computation of an approximation of the

root at the desired precision.

72

0.5 1 1.5 2 2.5

-1

-0.5

0.5

1

1.5

2

1.5 1.75 2.25 2.5 2.75 3

-1

-0.5

0.5

1

1.5

2

2.5

Figure 5.8 : The left figure represents the function over the interval(solid line), with successive piecewise linear approximation of orders 1, 2, 4 and 6 (dashedlines). Note that the approximations converge to the curve only on the interval . Theright figure shows approximations to the same curve (solid line) of orders 3 and 6 (dashedlines) for values greater than , generated by trimming successively at 1 and .

Conclusion

In this chapter, we presented a second subdivision algorithm for Poisson curves by gen-

eralizing the right hand side of the polynomial subdivision algorithm. This right Poisson

subdivision algorithm produces approximations of the control points of a trimmed Poisson

function. Using the two subdivision algorithms for Poisson curves, we developed several

applications: Direct applications are generation of piecewise linear approximations of an-

alytic curves on a chosen interval, inside the initial domain of convergence of the analytic

function at zero, and in some cases also outside this domain. Less direct applications are

determining intersections of analytic functions and finding approximations of roots of tran-

scendental equations.

73

Figure 5.9 : On the left, we find the intersection points (green dots) between the Archime-dian spiral on the interval (red curve) and a degree 3 Bezier curve(blue curve). On the right, the control polygons successively discarded when seeking theleft intersection point are highlighted.

1.75 2.25 2.5 2.75 3

-1

-0.8

-0.6

-0.4

-0.2

Figure 5.10 : Using the intersection algorithm for Poisson curves, the intersection point(in green) between the function (red curve) and the rational function (bluecurve) on the interval – that is, the solution of the equation on thisinterval – is computed.

74

Chapter 6

The Analytic Blossom

Introduction

Blossoming is a powerful tool for studying and computing with Bezier and B-spline curves

and surfaces – that is, for the investigation and analysis of polynomials and piecewise

polynomials in geometric modeling. In this chapter, we define a notion of the blossom for

Poisson curves, that is, for analytic functions.

The blossom, or polar form, presented by Ramshaw [34, 35, 36] and de Casteljau [14]

is a powerful tool very adapted to working with the Bezier representation of a polynomial.

Not only can the dual functionals of the Bernstein basis be expressed simply in terms of

the blossom, but also algorithms like subdivision or change of basis can be explained and

justified easily using the multi-affinity and symmetry of the blossom [1]. The blossom

is related as well to other effective tools for analyzing Bezier and B-splines curves and

surfaces like the de Boor-Fix formula [1, 3, 12] and the Marsden identity [29, 39, 20].

As in the polynomial setting, the analytic blossom provides a simple, elegant and com-

putationally meaningful way to analyze Poisson curves. Here, we define the analytic blos-

som and interpret all the algorithms for Poisson curves presented in the previous chapters

– subdivision, trimming, evaluation of the function and its derivatives, and conversion be-

tween the Taylor and the Poisson basis – in terms of this analytic blossom. As in the

polynomial setting, many of these algorithms follow simply from the multi-affinity and

symmetry of the blossom. This chapter introduces the analytic blossom, and interprets the

algorithms for Poisson curves in terms of this blossom.

75

6.1 The polynomial blossom

In this section we recall the classical definition of the blossom [34]. To simplify our nota-

tion and to emphasize that a polynomial and its blossom are just different representations of

the same underlying object, we will denote a function and its blossoms by the same name.

For example, if is a polynomial of degree , then will denote the blossom

of evaluated at . More generally, for will denote the blossom

of considered as a polynomial of degree evaluated at . (Note that when using the

standard notation for the blossom of , the distinction between the polynomial blossoms

of different degrees is also determined implicitly by the number of arguments, or domain

of the function.) Later, we shall use to denote the

analytic blossom of an analytic function .

6.1.1 The polynomial blossom and its relation to dual functionals

Definition 6.1 The blossom of a polynomial of degree is the unique symmetric, multi-

affine, -ary function that satisfies the diagonal property: [34]. That

is,

,

.

The blossom is a linear operator which is easy to compute. For example, the blossom

of the monomial at parameters is

(6.1)

Note that expression 6.1 insures the existence of the polynomial blossom. Ramshaw also

proves the uniqueness of the polynomial blossom in [34].

76

The dual functional property links Bezier curves to the blossom. Let denote

the th Bernstein polynomial of degree and , , the coefficients (or Bezier

points) of the polynomial in the Bernstein basis of degree . Then the dual functional

property asserts that, for ,

Here we use the notation to represent .

The dual functional property is known to be an easy consequence of the three blos-

soming axioms (see Definition 6.1). What is not generally appreciated is that this dual

functional property can actually replace the diagonal property in the blossoming axioms.

Proposition 6.1 The blossom of a degree polynomial is the unique symmetric, multi-

affine, -ary function characterized by the dual functional property:

for all ,

where the points are the Bezier control points of the polynomial over the

interval – that is, the coefficients of in the Bernstein basis.

Proof. By the definition of the Bezier control points,

(6.2)

Moreover, by the symmetry and multi-affinity of the blossom (Figure 6.1 taking )

(6.3)

If we assume that the blossom satisfies the diagonal property, then from equation 6.3

So, from the uniqueness of the coefficients of in the Bernstein basis,

for all .

77

Hence the diagonal property implies the dual functional property.

Conversely, if the blossom satisfies the dual functional property, then

for all .

Using these identities in equations 6.2 and 6.3, we obtain . Hence the dual

functional property implies the diagonal property.

Figure 6.1 : Like Figure 4.2, this diagram represents the de Casteljau algorithm for a cubicBezier curve, although, here, the points have been labeled by their blossom values. As aconsequence, the coefficients on the arrows follow simply from the symmetry and multi-affinity properties of the polynomial blossom.

The knot vector corresponds to the Bernstein basis – that is, the blossom evalu-

ated over this knot vector (at every consecutive coordinates of the knot vector) provides

the dual functionals for the Bernstein basis.

Bezier curves and polynomial blossoms are affine invariant. Indeed an affine transfor-

mation of the domain does not affect a Bezier curve. The same Bezier curve is represented

by the polynomial over the parameter interval and by over the

interval . The Bezier control points are the coefficients of the polynomial

in the basis . The corresponding dual functional property is

(6.4)

78

so the knot vector associated to this scaled and translated parameterization is the scaled

and translated knot vector .

6.1.2 Polynomial blossom and algorithms for Bezier curves

Not only can the control points be interpreted in terms of the blossom, but so too can many

algorithms for Bezier curves: in particular, subdivision, conversion between monomial and

Bernstein form, and evaluation of the function and its derivatives [1]. For all these algo-

rithms, both input and output can be expressed by the blossom of -tuples. An algorithm,

a way to compute the output from the input, then follows by applying the properties of the

blossom.

For example, one step of subdivision splits a Bezier curve into two curves (Chapter 4).

The control points of these new curves are computed from the control points of the original

curve. From the dual functional property, the original control points over the

interval are given by the blossom evaluated over the knot vector at every

consecutive coordinates of the knot vector. For subdivision at in , we compute the

control points associated with the knot vectors and from the control points

associated with the knot vector . Using the multi-affinity and the symmetry of the

blossom, we can generate a dynamic programming algorithm for subdivision: this is the de

Casteljau subdivision algorithm (Figures 4.2 and 6.1).

6.1.3 Polynomial blossom, Marsden identity and de Boor-Fix formula

The blossom expresses Bezier control points and algorithms in a simple and elegant way.

Moreover, some other powerful tools for studying Bezier and B-splines curves, such as the

Marsden identity [29] and the de Boor-Fix formula [12] are closely linked to the blossom

[3, 39]. We shall briefly review these results and their connections here, and we will extend

these identities to the analytic blossom in Section 6.4.

79

Given two polynomials and of degree , define the bilinear form [20]:

Notice that the right hand side is a constant independent of since its derivative is zero.

The next proposition expresses evaluation of the function, its derivatives and its blossom

in terms of the bracket operator.

Proposition 6.2

(6.5)

(6.6)

(6.7)

Proof. Equations 6.5 and 6.6 hold because the explicit computation of

leads to the Taylor expansion of at . Equa-

tion 6.7 holds since the right hand side satisfies the three blossoming axioms: symmetry

and multi-affinity in the ’s are immediate from the definition and the diagonal property

follows from equation 6.5.

Equation 6.7 is exactly the de Boor-Fix expression for the blossom derived by Barry

[3]. This explicit expression for the blossom provides an alternative proof of the existence

of the blossom.

The function plays a key role with respect to this bracket operator [20]. Brack-

eting a polynomial with not only reproduces , but also bracketing with the

derivative or the blossom of , reproduces the derivative or the blossom of . The

expression of this same function in the Bernstein basis is exactly the Marsden

identity for the Bernstein basis [39]

This identity follows directly from the dual functional property of the polynomial blossom.

80

6.2 Definition, examples and properties of the analytic blossom

In a first subsection, we propose two alternative definitions the analytic blossom and give

some examples and properties of this blossom. Existence, equivalence of the two defini-

tions and finally uniqueness of the analytic blossom are proved respectively in the three

following subsections.

6.2.1 Definition, examples and diagonal properties of the analytic blossom

Let be an analytic function at an arbitrary parameter in R . The blossom of at is

the unique symmetric, multi-affine function defined on an infinite number of parameters

almost all of which are such that the following diagonal property holds:

converges uniformly to (6.8)

on an open disk of C containing . This diagonal property is affine invariant: the blossom

applies to a set of arguments composed of and . The value is an arbitrary

parameter in the affine domain, and is an affine combination of

the values and , and therefore also represents a point in the affine domain.

Now consider an arbitrary . Then, as in the definition of the polynomial blossom,

we shall see that the diagonal property 6.8 can also be replaced by a corresponding dual

functional property

(6.9)

where are the coefficients of in the basis .

Let , so that . Then on the interval

and on the interval represent the same Poisson curve when ,

where is the radius of convergence of at . Moreover, the Poisson control points of

relative to the basis and of relative to the basis are

Here are some examples of the analytic blossom:

81

The analytic blossom of the Poisson basis and the Taylor basis with their domain trans-

lated by and scaled by are given respectively by the polynomial blossom of the

Bernstein and monomial basis scaled to the interval

Moreover, these formulas extend by linearity to arbitrary analytic functions expressed

in terms of the Poisson or Taylor bases, even to infinite sums. (Note that if ,

.)

We can compute for any where is analytic. Since ,

Thus

(6.10)

This second diagonal property of the analytic blossom generalizes nicely the polynomial

diagonal property in the axioms characterizing the polynomial blossom: .

Nevertheless, this property does not appear to be sufficiently strong to replace the third

axiom (the limit diagonal property 6.8) in the definition of the analytic blossom.

6.2.2 Existence of an analytic blossom

The main point of this section is to establish the existence of an analytic blossom satisfying

the three axioms of the definition. This proof of existence is based on a relation between

82

the analytic and polynomial blossoms. In particular, a polynomial has both an analytic

and polynomial blossoms.

We start with the following lemma. This result is crucial to proving both the existence

and the uniqueness of the analytic blossom.

Lemma 6.1 Let be an analytic function with radius of convergence . Then

as

converges uniformly to

on a closed disk of center and radius .

Proof. It is sufficient to prove this proposition for and , since the general

result then follows by a change of variable. From Theorem 3.1, converges uniformly

to on any closed disk of center zero. Now, consider

(6.11)

Let . Both and are defined by their Taylor and Poisson

series at on an arbitrary closed disk of center and radius , thus for any we

can choose so large that

and (6.12)

for all . Thus, in particular, .

Moreover, since as , for any fixed , there exists

such that for all :

Finally, we need to bound . First observe that, for all , and

(6.13)

83

because

Now

But , so from equations 6.13 and 6.12

Since , and can each be bounded uniformly on , the result follows by equation

6.11.

The next lemma relates the analytic blossom of Poisson and Taylor basis functions

respectively to the polynomial blossom of the Bernstein polynomials and the Taylor mono-

mials.

Lemma 6.2

(6.14)

(6.15)

Proof. First, we need to check that the left hand sides of these two identities are well

defined. That is, we must have

(6.16)

(6.17)

84

By the dual functional property, equation 6.4 is satisfied for ; it

is then satisfied on any by symmetry and multi-affinity. Moreover, since

equation 6.17 holds. This proves that the two identities are well defined.

Now we need to check that the right hand sides of equations 6.14 and 6.15 are symmet-

ric, multi-affine, and satisfy the diagonal property of the analytic blossom. The symmetry

and multi-affinity properties follow from the corresponding properties of the polynomial

blossom.

From the diagonal property of the polynomial blossom

But, from Lemma 6.1, and the convergence is uniform on

any closed disk of center in the domain of convergence. Thus the diagonal property holds

for equation 6.14.

For the right hand side of equation 6.15, the diagonal property holds because

which converges uniformly to any closed disk of center .

The next theorem is the main result of this section; existence of the analytic blossom

follows directly from it. Equations 6.18 and 6.19 express an analytic blossom of Poisson

or Taylor series as the polynomial blossoms of Bernstein or Taylor polynomials.

Theorem 6.1

(6.18)

(6.19)

85

Proof. The proof of these two identities mimics the proof of Lemma 6.2. From Lemma

6.2 and by linearity the left hand sides of these identities are well defined. Moreover, from

the symmetry, multi-affinity and linearity of the polynomial blossom, the right hand sides

of these equations are symmetric and multi-affine.

From the diagonal property of the polynomial blossom

and from Lemma 6.1 the sum on the right hand side converges uniformly to

on any closed disk of center and radius , where is the radius of convergence

of the scaled Poisson series at . Thus the diagonal property for the analytic blossom is

satisfied and therefore equation 6.18 holds.

Finally, equation 6.19 follows from the equation 6.18. From equation 3.3 in Chapter 3

and a change of variable

where

and from [21]

Thus

Theorem 6.1 gives explicit expressions for the analytic blossom of Poisson and Taylor

series at in terms of the polynomial blossom. Thus, since any analytic function at

admits both a Poisson and a Taylor development, an analytic blossom certainly exists.

Note that the blossom of could as well be defined by another family of polynomials

if the following two conditions hold:

86

the uniform convergence property

the compatibility property verified in the proof of Lemma 6.2:

We shall encounter another such family of functions in Section 6.4, where we pro-

vide an alternative proof of existence by exploiting the functions and

.

The next lemma shows that the analytic blossom at in arguments not equal to is

the blossom of the th Poisson or Taylor partial sum at .

Lemma 6.3

(6.20)

(6.21)

Proof. These results follow immediately from Lemma 6.2 and Theorem 6.1 by linearity.

Lemma 6.3 says that if we want to compute the analytic blossom of an infinite Poisson

or Taylor expansion, we can always truncate to a finite sum. In effect, then, the linearity of

the analytic blossom holds even for infinite series, since

87

6.2.3 The dual functional property

We will now prove that, as in the polynomial case, the diagonal property can be replaced

by a dual functional property in the definition of the analytic blossom. This second char-

acterization of the blossom in terms of the dual functional property will be very useful for

many reasons; uniqueness will follow very easily from the dual functional property and, in

the analytic case, the dual functional property is simpler, more intuitive, and often easier

to apply than the diagonal property. The dual functional property also allows us to avoid

issues of uniform convergence over subsets of C .

The following lemma is the first step in proving the equivalence of the two definitions

of the analytic blossom.

Lemma 6.4 Let and on the open

disk . If

converges uniformly to on

then

for all

Proof. Since we know from Lemma 6.1 that converges uniformly

to on , it is sufficient to prove: if con-

verges uniformly to on , then for all . Moreover,

since we can do a change of variable, we can consider just the case and .

The functions are polynomials on and therefore analytic on . The uniform

convergence of a series of analytic functions to an analytic function (here the

zero function) on implies the uniform convergence of the derivatives to

the derivative on the same domain [11].

Using this strong convergence property, we shall show by induction that for all

. First because for any and . Thus, from the convergence

88

hypothesis of the lemma, converge to – that is, . Suppose that for

. Then,

Thus

Since the sequence converges to and

. Thus .

Proposition 6.3 The blossom of an analytic function is a sym-

metric, multi-affine function over an infinite number of arguments almost all of which are

equal to , characterized by the following dual functional property

for all

Proof. Let be a symmetric, multi-affine function. Then

(6.22)

Now suppose that is a blossom of . Then from the diagonal property

converges to (6.23)

uniformly on a disk where and is the radius of convergence of

. From equations 6.22 and 6.23

(6.24)

89

and this convergence is uniform on the disk . Thus, from equation 6.24 and

Lemma 6.4

for all

Conversely, if satisfies the hypotheses of the proposition, we need to prove that is

a blossom of – that is, satisfies the diagonal property. Since are the

Poisson coefficients of the analytic function , by Lemma 6.1,

converges uniformly to on .

Thus, (equation 6.22) converges uni-

formly to on . So satisfies the diagonal property and by definition is a

blossom of .

Corollary 6.1

6.2.4 Uniqueness of the analytic blossom

Now we have all the necessary tools to prove one of the main results of this chapter: as in

the polynomial case, the analytic blossom exists and is unique.

Theorem 6.2 For any analytic function there exists a unique func-

tion that takes an infinite number of arguments almost all of which are equal to , is symmet-

ric, multi-affine and satisfies the diagonal property , (or

equivalently the dual functional property ).

Proof. Existence has been proved already, as a direct consequence of Theorem 6.1.

Uniqueness follows from Proposition 6.3. The control points are uniquely defined;

therefore the values of any blossom at are fixed. Moreover, defining the blossom

at determines it everywhere: by the multi-affinity and symmetry properties, the

values of the blossom at are easily computed. By induction, so are the values

90

Figure 6.2 : The analytic blossom can be computed from the Poissoncontrol points of by inserting at level and using the symmetry and multi-affinityproperties of the blossom.

at and hence too at for arbitrary ’s (see Figure 6.2).

Therefore, the blossom of an arbitrary analytic function exists and is unique.

The existence and uniqueness of the blossom for a function analytic at are now es-

tablishes. In the following section, we shall show how to interpret the algorithms presented

in the previous chapters in terms of the analytic blossom. As in the polynomial case, this

blossom interpretation is very simple and intuitive. Although these algorithms have already

been derived, the blossom interpretation strongly helps the understanding and intuition be-

hind these processes: the computations involved in these algorithms and their correctness

follow directly from the fundamental properties of the blossom.

6.3 Algorithms for Poisson Curves

As in the polynomial setting, many of the algorithms acting on a Poisson curve can be

expressed in terms of blossoming. When both the input and the output of the algorithm are

blossom expressions, then the algorithm itself often follows from the multi-affinity and the

symmetry of the blossom. Some convergence results, but not all, follow from the diagonal

91

property.

6.3.1 Left subdivision

In this subsection, we interpret the left subdivision algorithm presented in Chapter 4 in term

of the analytic blossom. To mimic the algorithm presented in Chapter 4 and for simplicity,

we consider analytic functions at zero. Although by a change of variable and thanks to the

existence and uniqueness of the blossom at any value , the blossom interpretation can be

developed more generally at any real value .

We start with two corollaries of Theorem 6.2.

Corollary 6.2 Let and be two functions analytic at zero, such that, .

Then

(6.25)

Proof. By uniqueness, it is sufficient to verify that the right hand side of equation 6.25

satisfies the three properties characterizing the blossom. The symmetry and multi-affinity

properties of follow from those of . Moreover, for in

which by the diagonal property of converges uniformly to on any closed

disk , .

Corollary 6.3 If denotes the set of control points of the function ,

then

for all

Proof. By Proposition 6.3 and Corollary 6.2, .

92

As in the polynomial case, one can define a knot vector corresponding to the Poisson

basis: , the infinite vector containing first an infinite number of ’s, and then an

infinite number of ’s. The Poisson control points are generated by the blossom evaluated

over this knot vector; the first point is the blossom evaluated over all the zeros, and the

following points are given successively by moving one position to the right which adds

each time one more to the set of arguments. Similarly, the points are generated

from the knot vector , which corresponds to the basis . Indeed by definition

so by Corollary 6.3, we have

This change of basis correspond to the left subdivision algorithm. Other change of basis

algorithms are treated in the subsequent sections.

The left subdivision algorithm for Poisson curves given in Chapter 4 has a very elegant

blossoming interpretation. At the first step of left subdivision, the control points

of , where , are generated from the initial Poisson control points

of ; similarly, at the next stage of left subdivision, the points are computed

from the points , for any and in . Since (Corollary

6.3), the first step of left subdivision consists of getting from the knot vector

to the knot vector – that is, from the control points of in the Poisson basis

, to the control points of in the basis ; the next step goes from the

knot vector to the knot vector . These steps can be computed

easily using the multi-affinity and symmetry properties of the blossom (Figure 6.3). This

computation is exactly the algorithm proposed in Figure 4.3 which corresponds, in the

polynomial setting, to the left hand side of the de Casteljau algorithm (Figure 4.2). Theorem

4.2 asserts that on any finite interval , , where is the radius of convergence of

at , the control polygons defined by the control points converge uniformly to

93

the Poisson curve as goes to zero, although, as in the polynomial case, the convergence

of the left subdivision process does not follow from the blossom interpretation.

Figure 6.3 : Blossom interpretation of the left subdivision algorithm for Poisson curves.This diagram is the same as Figure 4.3, but here the points that are computed are interpretedin term of the blossom. The coefficients on the arrows now follow directly from the multi-affinity and symmetry properties of the blossom.

6.3.2 Right subdivision: trimming algorithm

The trimming algorithm for Poisson curves proposed in Chapter 5 also has a very nice

blossom interpretation, from which, using the multi-affinity and symmetry properties of

the blossom, the trimming algorithm is retrieved. The aim of the trimming algorithm is to

compute the Poisson control points of the function , where is

a fixed but arbitrary number in from the Poisson control points , the coefficients

of in the basis . Each control point is the limit of a family of control points

. The control points not only converge to the

control points of , but also characterize a collection of functions that converge

uniformly to on any compact subinterval of the domain as goes to infinity

(Theorem 5.2). The convergence of the points, proved in Theorem 5.1, also follows from

the blossom interpretation. First, from the diagonal property:

94

Second, and are the control points respectively of the func-

tions and (Lemma 5.1). Since, by Proposition 6.3,

, the blossom of is

Thus, from the diagonal property

Finally, by induction, . The convergence of the functions , how-

ever, does not follow from the blossom interpretation. Since , the

knot vector corresponding to the function approximating is .

The algorithm for computing the control points of is illustrated by Figure 6.4.

Figure 6.4 : The same trimming algorithm for Poisson curves as Figure 5.1, but here thepoints are interpreted in terms of the blossom. Just as for left subdivision, the computationof the intermediate points follows from the multi-affinity and symmetry of the blossom.

6.3.3 Evaluation algorithms for functions and derivatives

The blossom can also characterize the control points of the derivatives of , as in

the polynomial setting [1]. We shall see, though, that the corresponding equalities in the

analytic setting differ from the equalities in the polynomial setting by constant factors.

95

The analytic blossom is a polynomial in the parameters , so we can homoge-

nize the analytic blossom; indeed the homogenization of the analytic blossom is similar to

the homogenization in the polynomial setting. (Detailed descriptions of the homogeneous

blossom can be found in [34, 1].) Formally, the homogeneous blossom of function ana-

lytic at is the unique symmetric, multi-linear function that when dehomogenized reduces

to the multi-affine blossom of , that is,

Let denote the Poisson control points of at . Since

. Thus by Proposition 6.3

(6.26)

From equation 6.26 and the linearity of the homogenized analytic blossom

where

By induction, if denotes the control points of at , then

(6.27)

Thus, more generally

(6.28)

Using equation 6.27 and the linearity of the homogeneous blossom, we can compute the

control points of a derivative of any order of from the Poisson control points of (Figure

6.5).

Thanks to the blossom expression for the derivative, the blossom characterizes the de-

gree of continuity of two analytic curves meeting at , as in the polynomial setting [27].

That is,

96

Figure 6.5 : This diagram is equivalent to the first line of the diagram in Figure 5.2 forthe more general case of a function analytic at (and not necessarily at ). Here thepoints are represented together with their blossom values. The coefficients to compute theintermediate points ( or ) can be simply derived from the multi-linearity andsymmetry of the homogeneous blossom.

Proposition 6.4 If and are two analytic functions at , then the following statements

are equivalent:

for all ,

.

Proof. By equation 6.28, the first statement is equivalent to , for

all . By the linearity of the homogenized blossom, this equality is equivalent to

, for all and an arbitrary , since .

This last equality is a particular case of and induces the second statement, since new

parameters can be introduced using the symmetry and multi-affinity of the blossom as in

the proof of Theorem 6.2.

6.3.4 Conversion between Poisson and Taylor basis

In the polynomial setting, algorithms to convert from the Bernstein basis to the monomial

basis and back are given in [1]. As in the polynomial setting, a translated and scaled mono-

mial basis, the Taylor basis corresponds to the knot vector . Indeed,

97

from equation 6.28 and the second diagonal Property 6.10, . Thus, if

we denote by the -th Taylor coefficient of , . Note that the scaling of

the monomial basis in the analytic setting differs from the scaling in the polynomial setting,

where the knot vector corresponds to the basis .

We can now apply the linearity and the symmetry of the homogenized blossom to com-

pute the Taylor coefficients from the Poisson coefficients, and conversely, the Poisson co-

efficients from the Taylor coefficients (Figures 6.6 and 6.7). Figure 6.6 is exactly the al-

gorithm proposed in Figure 3.3 of Chapter 3 to compute Poisson coefficients from Taylor

coefficients.

Figure 6.6 : The blossom interpretation of the change of basis from Taylor to Poisson pre-sented Figure 3.3. The coefficients on the arrows follow from the linearity of the blossombecause i.e. .

6.4 De Boor-Fix Formula, Marsden Identity and Blossoming for Ana-

lytic Functions

Here in analogy with the polynomial setting, we present a bilinear bracket operator charac-

terizing de Boor-Fix dual functionals for analytic functions. Then, we derive expressions

98

Figure 6.7 : Similar to Figure 6.6, the blossom interpretation of the change of basis fromPoisson to Taylor presented in Figure 3.4. Here also, the coefficients on the arrows followfrom the linearity of the blossom because .

for an analytic function and its derivatives, as well as for the blossom in term of this bracket

operator.

Definition 6.2 Let be an analytic function at , with radius of convergence , and let

, where is a polynomial and is in . Then

Proposition 6.6 will establish that the bracket operator on such functions is well defined

– that is, that the right hand side always converges. Note that the bracket operators at

different points are related:

The following proposition introduces a Boor-Fix like expression for the analytic blos-

som. As in the polynomial case, this new expression provides an alternative proof of the

existence of the analytic blossom.

Proposition 6.5

(6.29)

99

Proof. First, note that the bracket operator between a function analytic at , and a

polynomial is well defined, since is a finite sum.

To prove equation 6.29, we check the three blossoming axioms: symmetry, multi-

affinity and the dual functional property. The expression is

certainly symmetric and multi-affine in the ’s. Thus, so is the expression

by the linearity of the bracket operator. For the dual functional property,

we need to prove (for )

(6.30)

where are the control points of in the basis .

In Chapter 3, equation 3.3 gives an explicit expression for the Poisson control points in

terms of the Taylor coefficients at . For a translated and scaled version of these two basis,

the following expression still holds:

(6.31)

But

(6.32)

Equation 6.30 follows from equations 6.31 and 6.32.

By direct verification of the blossom axioms,

(6.33)

Thus, the function plays the same role for the analytic bracket operator as

the function plays for the polynomial bracket operator . The next proposition

states this result more precisely.

Proposition 6.6 If is an analytic function with radius of convergence at , then for

any in

100

,

,

.

Proof. The right hand side of the first two equations is exactly the Taylor development

of and at . The last equation follows immediately from equation 6.29 in

Proposition 6.5 and equation 6.33.

By linearity and from Proposition 6.6 the series defining con-

verges, so the bracket operator is well defined.

Similar to the polynomial setting, the expansion of the reproducing function

in the Poisson basis generates a Marsden identity for analytic functions

This Marsden identity follows easily from the dual functional property of the analytic blos-

som, since , where

from equation 6.33.

Conclusion

In this chapter we have defined the analytic blossom and established its existence and

uniqueness. Like the polynomial blossom for Bezier curves, the analytic blossom is a

simple, powerful and elegant tool for analyzing Poisson curves. All the algorithms pre-

sented in this chapter had already been developed in Chapters 3, 4 and 5, but each required

a different approach. From the blossoming interpretation given here, these algorithms all

follow directly from the multi-affinity and the symmetry of the blossom (or the linearity of

the homogenized blossom).

101

Chapter 7

Conclusion

This thesis presents a new paradigm for geometric modeling based on analytic functions.

This development represents analytic functions in a geometrically intuitive way and derives

classical CAGD algorithms and mathematical analysis tools for this new representation.

First, we introduced a geometrically meaningful representation for analytic functions;

analytic functions are characterized by their coefficients in the Poisson basis. These co-

efficients, or Poisson control points, have a geometric interpretation similar to the Bezier

control points in the polynomial setting. Thus, the Poisson control points provide intuitive

control parameters for the geometrical object the function represents. Efficient algorithms

convert the Poisson representation of an analytic function to and from the classical Taylor

representation, or Taylor expansion. Second, we proposed two algorithms, subdivision and

trimming, based respectively on the left and right subdivision algorithm. Both algorithms

are proved correct, that is, their convergence is established. We also study their theoret-

ical complexity and convergence rates and, by implementing them, show evidence that

the convergence is in practice even better than expected. The first algorithm, left Poisson

subdivision, generates sequences of approximating control polygons for analytic curves –

control polyhedra for analytic surfaces – from the Poisson coefficients of these functions.

The second algorithm, or right Poisson subdivision, provides a trimming algorithm. These

two algorithms lead to several applications: approximation of analytic functions on an arbi-

trary closed interval or product of intervals inside the domain of convergence, a divide and

conquer procedure to find the intersection of analytic curves, a procedure for approximating

the roots of a transcendental equation. Finally, we also developed classical mathematical

analysis tools for this analytic function based model. A blossoming notion for analytic

102

functions provides, as in the polynomial framework, a very intelligible notation for Pois-

son control points and a clear interpretation for the Poisson subdivision algorithms. De

Boor-Fix formulas and a Marsden identity are also given for this new paradigm.

This model provides a unifying representation for a rich variety of shapes. Analytic

functions offer a very rich set of functions to represent geometric shapes. Most of the usual

mathematical functions are indeed analytic. In particular, polynomial and trigonometric

functions are analytic. Therefore, this model provides a unique representation for polyno-

mial shapes of any degree as well as for circular shapes. Moreover, functions with poles

are analytic on their domains. Thus, functions with poles can also be represented by this

same model inside their domain of convergence, as close to a singularity as one wishes.

In order to develop a new framework for analytic functions in CAGD, we took the limit

of the well known polynomial framework. But, this generalization from polynomial to

analytic involves going from a finite to an infinite space of functions. As a consequence, we

addressed convergence issues, and established that our procedures and algorithms require

only a finite number of both input and output and execute in finite amount of time.

This work addresses a subject that was almost untouched. Although the Poisson ba-

sis had been known for long to be a limiting case of Bernstein basis, analytic functions

had never been studied as a CAGD model. This thesis proposes manipulation algorithms

and analytic tools for this new model. But others tools or notions, for example triangular

patches, remain to be developed. Moreover, Ramshaw has just published a new theory

embedding his former work on blossoming in a more general framework by considering

point multiplication. Possibly a similar theory for analytic functions could be developed.

The main drawback of the Poisson representation is its lack of locality: moving only one

control point changes the whole curve or surface. Similar to the Bernstein polynomials for

which they are the limit, the Poisson basis functions are non zero on the whole domain.

B-splines are often preferred to polynomials because they have better locality properties,

i.e. B-spline basis functions vanish outside a finite interval. A new analytic model with

more locality may perhaps be derived from the B-spline based model by following in our

103

foot steps, generalizing from the polynomial to the analytic paradigm.

104

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