Medial axis

Here is a nontrival polyhedron. It has two opposing grooves, which come very close together at the center.

Here is its medial axis, rendered as trimmed Bezier patches.

It's "very non-manifold"--at most seams, there are three incident sheets. If you remove all of the sheets that touch the boundary of the original polyhedron, you're left with this shape, which happens to be homeomorphic to a disk. All of the curved surfaces visible in this shot are sections of elliptical cones.

A tiny saddle surface lives at the center. It is the part of the medial axis which comes from the bisector surface of the two grooves. Zooming in it 200 times:

The surface has an extremely high curvature, and the gluNurbs rendering software can't do it very well. Here is what it should look like. (The boundaries of the saddle surface are hyperbolas.)

A medial axis with such small features is very difficult to compute using only floating-point arithmetic.

More pictures.

Tim Culver
Last modified: Fri May 12 09:59:06 EDT 2000