Problems in curves and surfaces M. Ramanathan Problems in curves and surfaces.

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)

<p – C(t), C’(t)> = 0Constraint equation

p

C(t)


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


Common tangent lines


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

http://www.cs.technion.ac.il/~irit


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

3

4

6

5


Bi-Tangents and Convex hull


CH of closed surfaces


Minimum enclosing circle

• smallest circle that completely contains a set of points


Minimum enclosing circle – two curves


Minimum enclosing circle – three curves


MEC of a set of closed curves


Kernel problem• Given a freeform curve/surface, find a point

from which the entire curve/surface is visible.


Kernel problem (contd.)

Solve


Kernel problem in 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')

http://en.wikipedia.org/wiki/Transformation_(geometry)

http://en.wikipedia.org/wiki/Incidence_(geometry)

http://en.wikipedia.org/wiki/Relation


Point-Line Duality


Common tangents


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.


Skeleton – Voronoi diagram

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


Voronoi diagram (illustration)

P1 P2

B(P

1,P

2)

P1

P2

P3B(P1,P3)

B(P1,P2)

B(P2,

P3)

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.


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.


Skeletons – medial axis


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)


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)


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

p

q

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


Definition (Voronoi diagram)

The Voronoi diagram is the union of the Voronoi cells of all the free-form curves. C0(t)


Skeleton-Bisector relation


Bisector for simple curves


Bisector for simple curves (contd)


Point-curve bisector


Curve-curve bisector

C0(t)

C1(r)

LL

LR

RL

RR


Outline of the algorithm

tr-space

Lower envelope algorithm

Implicit bisector function

Euclidean space

C0(t)

C1(r)

Limiting constraints

Splitting into monotone pieces


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

rCtC

rCtCrtPrtF

)(0 tC

)(1 rC ),( rtP

q

P(t,r) - q


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.


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


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


The orientation constraints (Contd.)

C0(t)

C1(r)

LL

LR

RL

RR


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)


Effect of the curvature constraint

1)(),(),()( tCttPtNt ijiiiii


Application of curvature constraint

Before After


Lower envelopes

t

D

(a)

t

D

(b)

t

D

(c)


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


Result I

C0(t)

C1(r1)

C0(t)

C1(r1)

C0(t)

C1(r1)C2(r2)


Result I (Contd.)

C0(t)

C1(r1)

C2(r2)

C0(t)

C1(r1)

C2(r2)


Results II

C0(t)C1(r1)

C2(r2)

C3(r3)

C0(t)C1(r1)C2(r2)

C3(r3)

C4(r4)


Results III

C0(t)

C1(r1)

C2(r2)

C0(t)

C1(r1)

C2(r2)C3(r3)

C4(r4)


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 r

parametric directions.


Bisectors in 3D


Bisector in 3D


Bisectors in 3D


Bisector in 3D (space curves)


Bisectors in 3D


Surface-surface bisector


Constraints


-sector Constraints

Y-axis


-sector


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

http://www.cs.technion.ac.il/~irit

Problems in curves and surfaces M. Ramanathan Problems in curves and surfaces.

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