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 TwistedRod References Multiple Parameter Continuation: Computing Implicitly Defined k-manifolds, Int. J. Bifurcation and Chaos v12(3), pages My Home page --