problems in curves and surfaces
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Problems in curves and surfaces. M. Ramanathan. Simple problems. Given a point p and a parametric curve C(t), find the minimum distance between p and C(t). C(t). = 0. Constraint equation. p. Point-curve tangents. - PowerPoint PPT PresentationTRANSCRIPT
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Problems in curves and surfaces
Problems in curves and surfaces
M. Ramanathan
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Problems in curves and surfaces
Simple problems
• Given a point p and a parametric curve C(t), find the minimum distance between p and C(t)
<p – C(t), C’(t)> = 0Constraint equation
p
C(t)
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Problems in curves and surfaces
Point-curve tangentsGiven a point p and a parametric curve C(t), find the tangents from p to C(t)
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Problems in curves and surfaces
Common tangent lines
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Problems in curves and surfaces
The IRIT Modeling Environment
• www.cs.technion.ac.il/~irit• More like a kernal not a software – code can be downloaded from the same webpage.• Add your own functions and compile with
them (written in C language)• User’s manual as well as programming
manual is available
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Problems in curves and surfaces
Convex hull of a point set
· Given a set of pins on a pinboard· And a rubber band around them· How does the rubber band look
when it snaps tight?· A CH is a convex polygon - non-
intersecting polygon whose internal angles are all convex (i.e., less than π)
0
2
1
34
6
5
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Problems in curves and surfaces
Bi-Tangents and Convex hull
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Problems in curves and surfaces
CH of closed surfaces
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Problems in curves and surfaces
CH of closed surfaces
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Problems in curves and surfaces
Minimum enclosing circle
• smallest circle that completely contains a set of points
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Problems in curves and surfaces
Minimum enclosing circle – two curves
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Problems in curves and surfaces
Minimum enclosing circle – three curves
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Problems in curves and surfaces
MEC of a set of closed curves
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Problems in curves and surfaces
Kernel problem• Given a freeform curve/surface, find a point
from which the entire curve/surface is visible.
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Problems in curves and surfaces
Kernel problem (contd.)
Solve
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Problems in curves and surfaces
Kernel problem in surfaces
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Problems in curves and surfaces
Duality
• duality refers to geometric transformations that replace points by lines and lines by points while preserving incidence properties among the transformed objects. The relations of incidence are those such as 'lies on' between points and lines (as in 'point P lies on line L')
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Problems in curves and surfaces
Point-Line Duality
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Problems in curves and surfaces
Common tangents
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Problems in curves and surfaces
Voronoi Cell (Points)• Given a set of points {P1, P2, … , Pn}, the
Voronoi cell of point P1 is the set of all points closer to P1 than to any other point.
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Problems in curves and surfaces
Skeleton – Voronoi diagram The Voronoi diagram is the union of the
Voronoi cells of all the free-form curves.
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Problems in curves and surfaces
Voronoi diagram (illustration)
P1 P2
B(P
1,P 2
)
P1P2
P3B(P1,P3)
B(P1,P2)
B(P2,P
3)
Remember that VD is notdefined for just points but for any set e.g. curves, surfaces etc. Moreover, the definition is applicable for any dimension.
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Problems in curves and surfaces
Skeleton – Medial Axis
The medial axis (MA), or skeleton of the set D, is defined as the locus of points inside which lie at the centers of all closed discs (or balls in 3-D) which are maximal in D.
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Problems in curves and surfaces
Skeletons – medial axis
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Problems in curves and surfaces
Definition (Voronoi Cell)
• Given - C0(t), C1(r1), ... , Cn(rn) - disjoint rational planar closed regular C1 free-form curves.
• The Voronoi cell of a curve C0(t) is the set of all points closer to C0(t) than to Cj(rj), for all j > 0.
C1(r1)
C2(r2)
C3(r3)
C4(r4)
C0(t)
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Problems in curves and surfaces
Definition (Voronoi cell (Contd.))
• Boundary of the Voronoi cell.
• Voronoi cell consists of points that are equidistant and minimal from two different curves.
C0(t)
C1(r1)
C2(r2)
C3(r3)
C4(r4)C3(r3)C0(t),
C0(t), C4(r4)
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Problems in curves and surfaces
Definition (Voronoi cell (Contd.))
• The above definition excludes non-minimal-distance bisector points.
• This definition excludes self-Voronoi edges.
r2r3
r1
t
r4
C0(t)
C1(r)r
pq
“The Voronoi cell consists of points that are equidistant and minimal from two different curves.”
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Problems in curves and surfaces
Definition (Voronoi diagram)
The Voronoi diagram is the union of the Voronoi cells of all the free-form curves. C0(t)
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Problems in curves and surfaces
Skeleton-Bisector relation
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Problems in curves and surfaces
Bisector for simple curves
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Problems in curves and surfaces
Bisector for simple curves (contd)
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Problems in curves and surfaces
Point-curve bisector
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Problems in curves and surfaces
Curve-curve bisector
C0(t)
C1(r)
LL
LR
RL
RR
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Problems in curves and surfaces
Outline of the algorithmtr-space
Lower envelope algorithm
Implicit bisector function
Euclidean space
C0(t)
C1(r)
Limiting constraints
Splitting into monotone pieces
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Problems in curves and surfaces
The implicit bisector function• Given two regular C1 parametric curves C0(t) and
C1(r), one can get a rational expression for the two normals’ intersection point: P(t,r) = (x(t,r), y(t,r)).
• The implicit bisector function F3 is defined by: 0)()(,
2)()(),(),( 10
103
rCtCrCtCrtPrtF
)(0 tC
)(1 rC ),( rtP
q
P(t,r) - q
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Problems in curves and surfaces
The untrimmed implicit bisector function
tr
F3(t,r) C1(r)
C0(t)
Comment: Note we capture in the (finite) F3 the entire (infinite) bisector in R2.
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Problems in curves and surfaces
Splitting the bisector, the zero-set of F3, into monotone piecesr
t
Keyser et al., Efficient and exact manipulation of algebraic points and curves, CAD, 32 (11), 2000, pp 649--662.
t
r
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Problems in curves and surfaces
Constraints - orientation• Orientation Constraint – purge regions of the untrimmed bisector that do not lie on the proper side.• LL considers left side of both curves as proper:
0)(),(),( 21 iLiii tNtCttP
)( 22 tC
)( 11 tC),( 21 ttP
)( 22 tN L
)( 11 tN L
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Problems in curves and surfaces
The orientation constraints (Contd.)
C0(t)
C1(r)
LL
LR
RL
RR
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Problems in curves and surfaces
The curvature constraints
1)(),(),()( tCttPtNt ijiiiii
Curvature Constraint (CC) – purge away regions of the untrimmed bisector whose distance to its footpoints (i.e., the radius of the Voronoi disk) is larger than the radius of curvature (i.e., 1/κ) at the footpoint.
N1/κ1
P(t1, t2)C1(t1)
C2(t2)
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Problems in curves and surfaces
Effect of the curvature constraint
1)(),(),()( tCttPtNt ijiiiii
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Problems in curves and surfaces
Application of curvature constraint
Before After
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Problems in curves and surfaces
Lower envelopes
t
D
(a)t
D
(b)
t
D
(c)
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Problems in curves and surfaces
Lower envelope algorithm
• Standard Divide and Conquer algorithm.
• Main needed functions are: – Identifying intersections of
curves.– Comparing two curves at a
given parameter (above/below).
– Splitting a curve at a given parameter.
• ||Di (t, ri)||2 = ||Dj (t, rj)||2 , F3(t, ri) = 0, F3(t, rj) = 0.• Compare ||Di (t, ri)||2 and ||
Dj(t,rj)||2 at the parametric values.
• Split F3(t, ri) = 0 at the tri-parameter.
Distance function D defined by Di(t, ri) = || P(t, ri) - Ci(t) ||
General Lower Envelope VC Lower Envelope
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Problems in curves and surfaces
Result I
C0(t)
C1(r1)
C0(t)
C1(r1)
C0(t)
C1(r1)C2(r2)
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Problems in curves and surfaces
Result I (Contd.)
C0(t)
C1(r1)
C2(r2)
C0(t)
C1(r1)
C2(r2)
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Problems in curves and surfaces
Results II
C0(t)C1(r1)
C2(r2)C3(r3)
C0(t)C1(r1)C2(r2)
C3(r3)
C4(r4)
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Problems in curves and surfaces
Results III
C0(t)
C1(r1)
C2(r2)
C0(t)
C1(r1)
C2(r2)C3(r3)
C4(r4)
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Problems in curves and surfaces
Results IV (For Non-Convex C0(t))
C0(t)
C3(r3)
C2(r2)
C1(r1)
C0(t)C1(r1)
C2(r2)
Voronoi cell is obtained by performing the lower envelope on both t and rparametric directions.
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Problems in curves and surfaces
Bisectors in 3D
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Problems in curves and surfaces
Bisector in 3D
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Problems in curves and surfaces
Bisectors in 3D
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Problems in curves and surfaces
Bisector in 3D (space curves)
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Problems in curves and surfaces
Bisectors in 3D
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Problems in curves and surfaces
Surface-surface bisector
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Problems in curves and surfaces
Surface-surface bisector
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Problems in curves and surfaces
Constraints
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Problems in curves and surfaces
-sector Constraints
Y-axis
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Problems in curves and surfaces
-sector
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Problems in curves and surfaces
References
• http://www.cs.technion.ac.il/~irit• Gershon Elber and Myung-soo Kim. The convex hull of rational plane curves, Graphical Models, Volume 63,
151-162, 2001• J. K. Seong, Gershon Elber, J. K. Johnstone and Myung-soo Kim. The convex hull of freeform surfaces,
Computing, 72, 171-183, 2004• Elber Gershon, Kim Myung-Soo. Geometric constraint solver using multivariate rational spline functions. In:
Proceedings of the sixth ACM symposium on solid modeling and applications; 2001. p. 1–10.• ELBER, G., AND KIM, M.-S. 1998. Bisector curves for planar rational curves. Computer-Aided Design 30,
14, 1089–1096.• ELBER, G., AND KIM, M.-S. 1998. The bisector surface of rational space curves. ACM Transaction on
Graphics 17, 1 (January), 32–39.• FAROUKI, R., AND JOHNSTONE, J. 1994. The bisector of a point and a plane parametric curve. Computer
Aided Geometric Design, 11, 2, 117–151.• Ramanathan Muthuganapathy, Gershon Elber, Gill Barequet, and Myung-Soo Kim, "Computing the
Minimum Enclosing Sphere of Free-form Hypersurfaces in Arbitrary Dimensions" , Computer-Aided Design, 43(3), 2011, 247-257
• Iddo Hanniel, Ramanathan Muthuganapathy, Gershon Elber and Myugn-Soo Kim "Precise Voronoi Cell Extraction of Free-form Rational Planar Closed Curves ", Solid and Physical Modeling (SPM), 2005, MIT, USA, pp 51-59