Thaw: A Tool for Approximating Cut Loci on a Triangulation of a Surface

Abstract

The cut locus from a point on the surface of a convex polyhedron is a tree containing a line segment beginning at every vertex. In the limit of infinitely small triangles, the cut locus from a point on a triangulation of a smooth surface therefore tends to become dense in the smooth surface, whereas the cut locus from the same point on the smooth surface is also a tree, but of finite length. We introduce a method for avoiding this problem. The method involves introducing a minimal angular resolution and discarding those points of the cut locus on the triangulation for which the angle measured between the shortest geodesic curves meeting at these points is smaller than the given angular resolution. We also describe software based upon this method that allows one to visualize the cut locus from a point on a surface of the form $(x/a)^{n}+(y/b)^{n}+(z/c)^{n}=1$, where $n$ is a positive even integer. We use the software to support a new conjecture that the cut locus of a general ellipsoid is a subarc of a curvature line of the ellipsoid.