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
Locating point easy
Merge hard
Could:
+Select from fixed grid
Allgower/Schmidt
Rheinboldt
+Advancing front
Brodzik
Melville/Mackey
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
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
Cover a cube
2500
Cover a cube
5000
Cover a cube
7476
When not flat : Charts
Cover a circle
Cover a circle
Cover a circle
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
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
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/