Using the Power Diagram to

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 -gt 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 Pcube 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.epf
l.ch/neukirch/publi.html
My Home page -- http//www.research.ibm.com/people
/h/henderson/