Problems in curves and surfaces M. Ramanathan 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) Problems in curves and surfaces = 0 Constraint equation p C(t)

Point-curve tangents Problems in curves and surfaces Given a point p and a parametric curve C(t), find the tangents from p to C(t)

Common tangent lines Problems in curves and surfaces

The IRIT Modeling Environment 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

Convex hull of a point set Problems in curves and surfaces  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)

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 (contd.) Problems in curves and surfaces Solve

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')geometric transformationsincidence relations 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) Problems in curves and surfaces P1P2 B(P1,P2) P1 P2 P3 B(P1,P3) B(P1,P2) B(P2,P3) 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.

Skeleton – Medial Axis Problems in curves and surfaces 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 Problems in curves and surfaces

Definition (Voronoi Cell) Given - C 0 (t), C 1 (r 1 ),..., C n (r n ) - disjoint rational planar closed regular C 1 free-form curves. The Voronoi cell of a curve C 0 (t) is the set of all points closer to C 0 (t) than to C j (r j ), for all j > 0. Problems in curves and surfaces C1(r1)C1(r1) C2(r2)C2(r2) C3(r3)C3(r3) C4(r4)C4(r4) C0(t)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. Problems in curves and surfaces C0(t)C0(t) C1(r1)C1(r1) C2(r2)C2(r2) C3(r3)C3(r3) C4(r4)C4(r4) C3(r3)C3(r3) C 0 (t), C4(r4)C4(r4)

Definition (Voronoi cell (Contd.)) The above definition excludes non-minimal-distance bisector points. This definition excludes self- Voronoi edges. Problems in curves and surfaces r2r2r2r2 r3r3r3r3 r1r1r1r1 t r4r4r4r4 C0(t)C0(t) C1(r)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. Problems in curves and surfaces C0(t)C0(t)

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

Curve-curve bisector Problems in curves and surfaces

Outline of the algorithm Problems in curves and surfaces tr-space Lower envelope algorithm Implicit bisector function Euclidean space C0(t)C0(t)C0(t)C0(t) C1(r)C1(r)C1(r)C1(r) Limiting constraints Splitting into monotone pieces

The implicit bisector function Given two regular C 1 parametric curves C 0 (t) and C 1 (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 F 3 is defined by: Problems in curves and surfaces q P(t,r) - q

The untrimmed implicit bisector function Problems in curves and surfaces t r F 3 (t,r) Comment Comment: Note we capture in the (finite) F 3 the entire (infinite) bisector in R 2.

Splitting the bisector, the zero-set of F 3, into monotone pieces Problems in curves and surfaces r t Keyser et al., Efficient and exact manipulation of algebraic points and curves, CAD, 32 (11), 2000, pp

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: Problems in curves and surfaces

The orientation constraints (Contd.) Problems in curves and surfaces

The curvature constraints Problems in curves and surfaces 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/κ1N1/κ1 P(t 1, t 2 ) C1(t1)C1(t1) C2(t2)C2(t2)

Effect of the curvature constraint Problems in curves and surfaces

Application of curvature constraint Problems in curves and surfaces Before After

Lower envelopes Problems in curves and surfaces tD(a) tD(b) tD(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. ||D i (t, r i )|| 2 = ||D j (t, r j )|| 2, F 3 (t, r i ) = 0, F 3 (t, r j ) = 0. Compare ||D i (t, r i )|| 2 and ||D j (t,r j )|| 2 at the parametric values. Split F 3 (t, r i ) = 0 at the tr i - parameter. Problems in curves and surfaces Distance function D defined by D i (t, r i ) = || P(t, r i ) - C i (t) || General Lower Envelope VC Lower Envelope

Result I Problems in curves and surfaces C0(t)C0(t) C1(r1)C1(r1) C0(t)C0(t) C1(r1)C1(r1) C2(r2)C2(r2)

Result I (Contd.) Problems in curves and surfaces C0(t)C0(t) C1(r1)C1(r1) C2(r2)C2(r2) C0(t)C0(t) C1(r1)C1(r1) C2(r2)C2(r2)

Results II Problems in curves and surfaces C0(t)C0(t) C1(r1)C1(r1) C2(r2)C2(r2) C3(r3)C3(r3) C0(t)C0(t) C1(r1)C1(r1) C2(r2)C2(r2) C3(r3)C3(r3) C4(r4)C4(r4)

Results III Problems in curves and surfaces C0(t)C0(t) C1(r1)C1(r1) C2(r2)C2(r2) C0(t)C0(t) C1(r1)C1(r1) C2(r2)C2(r2) C3(r3)C3(r3) C4(r4)C4(r4)

Results IV (For Non-Convex C 0 (t)) Problems in curves and surfaces C0(t)C0(t) C3(r3)C3(r3) C2(r2)C2(r2) C1(r1)C1(r1) C0(t)C0(t) C1(r1)C1(r1) C2(r2)C2(r2) Voronoi cell is obtained by performing the lower envelope on both t and r parametric directions.

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 Constraints Problems in curves and surfaces Y-axis

 -sector Problems in curves and surfaces

References Gershon Elber and Myung-soo Kim. The convex hull of rational plane curves, Graphical Models, Volume 63, , 2001 J. K. Seong, Gershon Elber, J. K. Johnstone and Myung-soo Kim. The convex hull of freeform surfaces, Computing, 72, , 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; p. 1–10. ELBER, G., AND KIM, M.-S Bisector curves for planar rational curves. Computer- Aided Design 30, 14, 1089–1096. ELBER, G., AND KIM, M.-S The bisector surface of rational space curves. ACM Transaction on Graphics 17, 1 (January), 32–39. FAROUKI, R., AND JOHNSTONE, J 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, 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 Problems in curves and surfaces