Ball-map: Homeomorphism Between Compatible Surfaces
Rossignac, Jaroslaw R.
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We introduce the ball-map, BM [subscript S],[subscript T], between two manifolds, S and T. It maps each point x of S to a point x = BM[subscript S],[subscript T](x) of T. Its inverse is BM[subscript T],[subscript S]. We define conditions for BM[subscript S],[subscript 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[superscript k] (n-1)-manifolds in R[superscript n], BM[subscript T],[subscript S] is a C[superscript k-1] diffeomorphism and defines a C[superscript 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.