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Simple problems

- Given a point p and a parametric curve C(t), find the minimum distance between p and C(t)

C(t)

<p – C(t), C’(t)> = 0

Constraint equation

p

Problems in curves and surfaces

Point-curve tangents

Given a point p and a parametric curve C(t), find the tangents from p to C(t)

Problems in curves and surfaces

Common tangent lines

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

Problems in curves and surfaces

5

3

6

2

1

0

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)

Problems in curves and surfaces

Bi-Tangents and Convex hull

Problems in curves and surfaces

CH of closed surfaces

Problems in curves and surfaces

CH of closed surfaces

Problems in curves and surfaces

Minimum enclosing circle

- smallest circle that completely contains a set of points

Problems in curves and surfaces

Minimum enclosing circle – two curves

Problems in curves and surfaces

Minimum enclosing circle – three curves

Problems in curves and surfaces

MEC of a set of closed curves

Problems in curves and surfaces

Kernel problem

- Given a freeform curve/surface, find a point from which the entire curve/surface is visible.

Problems in curves and surfaces

Kernel problem in surfaces

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')

Problems in curves and surfaces

Point-Line Duality

Problems in curves and surfaces

Common tangents

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.

Problems in curves and surfaces

Skeleton – Voronoi diagram

The Voronoi diagram is the union of the Voronoi cells of all the free-form curves.

Problems in curves and surfaces

Voronoi diagram (illustration)

P3

B(P2,P3)

B(P1,P3)

P1

P2

P2

P1

Remember that VD is not

defined for just points but

for any set e.g. curves, surfaces etc. Moreover, the definition is applicable for any dimension.

B(P1,P2)

B(P1,P2)

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.

Problems in curves and surfaces

Skeletons – medial axis

Problems in curves and surfaces

Definition (Voronoi Cell)

C2(r2)

- 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)

C0(t)

C3(r3)

C4(r4)

Problems in curves and surfaces

Definition (Voronoi cell (Contd.))

C2(r2)

- Boundary of the Voronoi cell.
- Voronoi cell consists of points that are equidistant and minimal from two differentcurves.

C1(r1)

C0(t)

C3(r3)

C3(r3)

C0(t),

C4(r4)

C0(t),

C4(r4)

Problems in curves and surfaces

Definition (Voronoi cell (Contd.))

“The Voronoi cell consists of points that are equidistant and minimal from two differentcurves.”

- The above definition excludes non-minimal-distance bisector points.
- This definition excludes self-Voronoi edges.

r3

r4

r2

r

C1(r)

r1

q

p

t

C0(t)

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)

Problems in curves and surfaces

Skeleton-Bisector relation

Problems in curves and surfaces

Bisector for simple curves

Problems in curves and surfaces

Bisector for simple curves (contd)

Problems in curves and surfaces

Point-curve bisector

Problems in curves and surfaces

C1(r)

C0(t)

Splitting into monotone pieces

Limiting constraints

Lower envelope algorithm

Outline of the algorithmtr-space

Implicit bisector function

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:

q

Problems in curves and surfaces

C1(r)

t

r

C0(t)

The untrimmed implicit bisector functionComment: Note we capture in the (finite) F3 the entire (infinite) bisector in R2.

Problems in curves and surfaces

t

t

Splitting the bisector, the zero-set of F3, into monotone piecesr

Keyser et al., Efficient and exact manipulation of algebraic points and curves, CAD, 32 (11), 2000, pp 649--662.

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 leftside of both curves as proper:

Problems in curves and surfaces

C1(t1)

P(t1, t2)

C2(t2)

The curvature constraintsCurvature 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.

Problems in curves and surfaces

Effect of the curvature constraint

Problems in curves and surfaces

Lower envelope algorithm

General Lower Envelope

VC Lower Envelope

Distance function D defined by Di(t, ri) = ||P(t, ri) - Ci(t) ||

- 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.

Problems in curves and surfaces

C0(t)

C2(r2)

Results IV (For Non-Convex C0(t))Voronoi cell is obtained by performing the lower envelope on both t and r

parametric directions.

C3(r3)

C2(r2)

C0(t)

C1(r1)

Problems in curves and surfaces

Bisectors in 3D

Problems in curves and surfaces

Bisector in 3D

Problems in curves and surfaces

Bisectors in 3D

Problems in curves and surfaces

Bisector in 3D (space curves)

Problems in curves and surfaces

Bisectors in 3D

Problems in curves and surfaces

Surface-surface bisector

Problems in curves and surfaces

Surface-surface bisector

Problems in curves and surfaces

Constraints

Problems in curves and surfaces

-sector

Problems in curves and surfaces

References

- http://www.cs.technion.ac.il/~irit
- GershonElber and Myung-soo Kim. The convex hull of rational plane curves, Graphical Models, Volume 63, 151-162, 2001
- J. K. Seong, GershonElber, J. K. Johnstoneand Myung-soo Kim. The convex hull of freeform surfaces, Computing, 72, 171-183, 2004
- ElberGershon, 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.
- RamanathanMuthuganapathy, GershonElber, 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
- IddoHanniel, RamanathanMuthuganapathy, GershonElber 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

Problems in curves and surfaces

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