1 / 44

Using the Power Diagram to Computing Implicitly Defined Surfaces Michael E. Henderson IBM T.J. Watson Research Center Y

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.

alpha
Download Presentation

Using the Power Diagram to Computing Implicitly Defined Surfaces Michael E. Henderson IBM T.J. Watson Research Center Y

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 Locating point easy Merge hard • Could: • +Select from fixed grid • Allgower/Schmidt • Rheinboldt • +Advancing front • Brodzik • Melville/Mackey

  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

  8. 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. Cover a cube 2500

  22. Cover a cube 5000

  23. Cover a cube 7476

  24. When not flat : Charts

  25. Cover a circle

  26. Cover a circle

  27. Cover a circle

  28. 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

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

More Related