1. Using the Power Diagram to Computing Implicitly Defined Surfaces Michael E. Henderson IBM T.J. Watson Research Center Yorktown Heights, NY Presented at DIMACS Workshop on Surface Resconstruction May 1, 2003 An Implicitly Defined Surface M is the set of points Find the component "connected" to Restrict to a finite region Find: A set of points on M A set of charts -

2. Continuation Methods

3. Mesh or Tiling Could: +Select from fixed grid Allgower/Schmidt Rheinboldt +Advancing front Brodzik Melville/Mackey Locating point easy Merge hard

4. Covering Locating point hard Merge easy

5. The boundary of a union

6. Can form the boundary from pairwise subtractions

7. Pairwise Subtractions - Spheres The part of a sphere that doesn't lie in a spherical ball

9. Pairwise Subtraction, Spherical Balls

10. Instead of part not in another ball Part in a Finite Convex Polyhedron

11. Boundary -> on Sphere and in Polyhedron

12. Power Diagram a.k.a. "Laguerre Voronoi Diagram" Restricted to the interior of the balls is same as the polyhedra.

13. Finding a point on the boundary If all vertices of the polyhedron lie inside the ball

14. Finding a point on the boundary If a vertex of the polyhedron lies outside the ball "All" we have to do is find a point u in both. If ratio of radii close to one can use origin. One sqrt gives bnd. pt.

15. Continuing Find a P w/ ext. vert. Get pt. on dM P=cube Find overlaps Remove 1/2 spaces

16. Cover a square

17. Cover a Square 120

18. Cover a Square 240

19. Cover a Square 368

20. Cover a cube

21. 2500

22. Cover a cube 5000

23. Cover a cube 7476

24. When not flat : Charts

25. Cover a circle

29. Cover a Torus 20

30. Cover a Torus 700

31. Cover a Torus 1400

32. Cover a Torus 2035

33. Implementation Data Stuctures: List of "charts" (center, tangent, radius, Polyhedron) Basic Operations: Find a list of charts which overlap another Hierarchical Bounding Boxes - O( log m ) Subtract a half space from a Polyhedron Keep edge and vertex lists (Chen, Hansen, Jaumard). Find a Polyhedron with an exterior vertex Keep a list, as half spaces removed update.

34. Coupled Pendula

36. Flexible Rod Clamped at Ends Sebastien Neukirch (Lausanne)

37. Flexible Rod Clamped at Ends Sebastien Neukirch (Lausanne) These are all configurations of the Rod

38. Flexible Rod Clamped at Ends Sebastien Neukirch (Lausanne)

39. Rings

40. Planar Untwisted Ring Layer 2+

41. Planar Untwisted Ring Layer 3-

42. Planar Untwisted Ring Layer 4-

43. Summary Start with a point on M Add a neighborhood of a point on dM Based on the boundary of a union of spherical balls. Each ball has a polyhedron If P has vertices outside the ball, then part of the sphere is on dM Complexity O(m log m) Resembles incremental insertion algorithm for Laguerre Voronoi. Points not closer than R not further apart than 2R

44. Preprints on TwistedRod http://lcvmsun9.epfl.ch/~neukirch/publi.html References Multiple Parameter Continuation: Computing Implicitly Defined k-manifolds, Int. J. Bifurcation and Chaos v12(3), pages 451-76 My Home page -- http://www.research.ibm.com/people/h/henderson/