Abstract:
We introduce the ball-map, BM_S,T, between two manifolds, S and T. It maps each point x of S to a point x = BM_S,T(x) of T. Its inverse is BM_T,S. We define conditions for BM_S,T to be a homeomorphism. We show that they hold when the minimum feature size of each surface exceeds their Hausdorff distance. We show that, when S and T are C^k (n-1)-manifolds in Rⁿ, BM_T,S is a C^k-1 diffeomorphism and defines a C^k-1 ambient isotopy that smoothly morphs between S to T. In practice, the ball-map yields an excellent map for transferring parameterizations and textures between ball compatible curves or surfaces. Furthermore, it may be used to define a morph, during which each point x of S travels to the corresponding point y of T along a circular arc that is normal to S at x and to T at y.

Keywords: mapping, morphing, Hausdorff Distance, distortion, isomorphism, isotopy, ball map

You can access this technical report via: PDF