1. 1 Precise Voronoi Cell Extraction of Free-form Rational Planar Closed Curves Iddo Hanniel, Ramanathan Muthuganapathy, Gershon Elber Department of Computer Science, Technion, Israel. & Myung-Soo Kim School of Computer Science and Engineering, Seoul National University, Korea.

2. Center for Graphics and Geometric Computing, Technion 2 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.  Currently, our implementation assumes C 0 (t) to be convex. C1(r1)C1(r1) C2(r2)C2(r2) C3(r3)C3(r3) C4(r4)C4(r4) C0(t)C0(t)

3. Center for Graphics and Geometric Computing, Technion 3 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)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)

4. Center for Graphics and Geometric Computing, Technion 4 Definition (Voronoi cell (Contd.))  The above definition excludes non-minimal-distance bisector points.  This definition excludes self- Voronoi edges. 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.”

5. Center for Graphics and Geometric Computing, Technion 5 Definition (Voronoi diagram) The Voronoi diagram is the union of the Voronoi cells of all the free-form curves. C0(t)C0(t)

6. Center for Graphics and Geometric Computing, Technion 6 Existing approaches for VD of free-forms   Farouki and Ramamurthy 1999 – V.D. and medial axis. Numerically traces the bisectors and trim them to get V.D. or MAT.   Ramanathan and Gurumoorthy 2003 – Medial Axis (MA). Numerically traces the MAT segments directly.   Elber and Kim 1999 – Bisector for planar rational curves. An implicit representation of bisector.  Linear/circular arc approximation [Gursoy and Patrikalakis 1992, Held 1998, Ju-Hsein Kao 1999].

7. Center for Graphics and Geometric Computing, Technion 7 Our approach  Using the original representation of the input curves (with no linear / circular approximation).  Generate an accurate implicit representation of the Voronoi cell. ExactApproximated

8. Center for Graphics and Geometric Computing, Technion 8 Outline of the algorithm Implicit bisector function Application of constraints Lower envelope algorithm Splitting into monotone pieces tr-space Euclidean space C0(t)C0(t) C1(r)C1(r)

9. Center for Graphics and Geometric Computing, Technion 9 Implicit bisector function  Given two regular C 1 parametric curves C 0 (t) and C 1 (r), we get an expression for a normal-intersection point: P(t,r) = (x(t,r), y(t,r)).  The implicit function F 3 is defined by: q

10. Center for Graphics and Geometric Computing, Technion 10 Computing points on the bisector  Points on the bisector are calculated using the (t,r) pairs of the zero-set of F 3 (t,r).  For every pair (t,r) the corresponding Euclidean point P is computed using the mapping: P(t,r) = (x(t,r), y(t,r)).

11. Center for Graphics and Geometric Computing, Technion 11 Implicit bisector function – untrimmed bisector t r F 3 (t,r) C 1 (r) C 0 (t)

12. Center for Graphics and Geometric Computing, Technion 12 Splitting the zero-set of the function into monotone pieces t t r r Keyser et al., Efficient and exact manipulation of algebraic points and curves, CAD, 32 (11), 2000, pp 649--662.

13. Center for Graphics and Geometric Computing, Technion 13 Constraints - orientation Orientation (LL) Constraint – purge away points of the untrimmed bisector that do not lie on the desired side (we assume left side of both the curves as the desire side).

14. Center for Graphics and Geometric Computing, Technion 14 LL-constraint Untrimmed bisector C0(t)C0(t) C0(t)C0(t) C1(r)C1(r) C1(r)C1(r) Trimmed bisector LLLLLLLL LRLRLRLR RLRLRLRL RRRRRRRR

15. Center for Graphics and Geometric Computing, Technion 15 Constraints - curvature Curvature Constraint (CC) – purge away points 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/κ1N1/κ1N1/κ1

16. Center for Graphics and Geometric Computing, Technion 16 Illustration of curvature constraint

17. Center for Graphics and Geometric Computing, Technion 17 Application of curvature constraint BeforeAfter

18. Center for Graphics and Geometric Computing, Technion 18 Illustration of lower envelope t D t t D D (a)(b) (c)

19. Center for Graphics and Geometric Computing, Technion 19 Lower envelope algorithm  Standard Divide and Conquer algorithm.  The 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. General Lower Envelope Voronoi Lower Envelope D i (t, r i ) = || P(t,r i ) – C 0 (t)|| A distance function D defined by

20. Center for Graphics and Geometric Computing, Technion 20 Result1 C0(t)C0(t) C0(t)C0(t) C1(r1)C1(r1) C1(r1)C1(r1) C2(r2)C2(r2)

21. Center for Graphics and Geometric Computing, Technion 21 Result1 (Contd.) C0(t)C0(t) C0(t)C0(t) C1(r1)C1(r1) C1(r1)C1(r1) C2(r2)C2(r2) C2(r2)C2(r2)

22. Center for Graphics and Geometric Computing, Technion 22 Results C0(t)C0(t) C0(t)C0(t) C1(r1)C1(r1) C1(r1)C1(r1)C2(r2)C2(r2) C2(r2)C2(r2) C3(r3)C3(r3) C3(r3)C3(r3) C4(r4)C4(r4)

23. Center for Graphics and Geometric Computing, Technion 23 Results C0(t)C0(t) C0(t)C0(t) C1(r1)C1(r1) C1(r1)C1(r1) C2(r2)C2(r2) C2(r2)C2(r2) C3(r3)C3(r3) C4(r4)C4(r4)

24. Center for Graphics and Geometric Computing, Technion 24 Discussion  Input rational curves are represented as Bezier/B-spline curves. The input curves are approximated neither by linear segments nor by circular arcs.  Implementation 1.Using IRIT software in C, developed at the Technion. 1.The bivariate function F 3 (t,r) is obtained using the symbolic library. 2.Constraints are solved using the multivariate library. 2.Uses floating-point arithmetic.  Computation took from several seconds to two minutes.

25. Center for Graphics and Geometric Computing, Technion 25 Future work  Extend to open or non-C 1 curves – extending the lower envelope algorithm to point-curve bisectors.  Generating Voronoi diagram and medial axis transform (MAT) of free-form curves efficiently using the implicit representation and solver, and possibly using the lower envelope algorithm.  Implementation using exact arithmetic.

26. 26 The End