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 - Continuation Methods Mesh or Tiling Could: +Select from fixed grid Allgower/Schmidt Rheinboldt +Advancing front Brodzik Melville/Mackey Locating point easy Merge hard Covering Locating point hard Merge easy The boundary of a union Can form the boundary from pairwise subtractions Pairwise Subtractions - Spheres The part of a sphere that doesn't lie in a spherical ball Pairwise Subtraction, Spherical Balls Instead of part not in another ball Part in a Finite Convex Polyhedron Boundary -> on Sphere and in Polyhedron Power Diagram a.k.a. "Laguerre Voronoi Diagram" Restricted to the interior of the balls is same as the polyhedra. Finding a point on the boundary If all vertices of the polyhedron lie inside the ball 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. Continuing Find a P w/ ext. vert. Get pt. on dM P=cube Find overlaps Remove 1/2 spaces Cover a square Cover a Square 120 Cover a Square 240 Cover a Square 368 Cover a cube 2500 Cover a cube 5000 Cover a cube 7476 When not flat : Charts Cover a circle Cover a Torus 20 Cover a Torus 700 Cover a Torus 1400 Cover a Torus 2035 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. Coupled Pendula Flexible Rod Clamped at Ends Sebastien Neukirch (Lausanne) Flexible Rod Clamped at Ends Sebastien Neukirch (Lausanne) These are all configurations of the Rod Flexible Rod Clamped at Ends Sebastien Neukirch (Lausanne) Rings Planar Untwisted Ring Layer 2+ Planar Untwisted Ring Layer 3- Planar Untwisted Ring Layer 4- 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 Preprints on TwistedRodReferences Multiple Parameter Continuation: Computing Implicitly Defined k-manifolds, Int. J. Bifurcation and Chaos v12(3), pages My Home page --