Ball-Map: Homeomorphism between Compatible Surfaces

Homeomorphisms between curves and between surfaces are fundamental to many applications of 3D modeling, graphics, and animation. They define how to map a texture from one object to another, how to morph between two shapes, and how to measure the discrepancy between shapes or the variability in a class of shapes. Previously proposed maps between two surfaces, S and S′, suffer from two drawbacks: (1) it is difficult to formally define a relation between S and S′ which guarantees that the map will be bijective and (2) mapping a point x of S to a point x′ of S′ and then mapping x′ back to S does in general not yield x, making the map asymmetric. We propose a new map, called ball-map, that is symmetric. We define simple and precise conditions for the ball-map to be a homeomorphism. We show that these conditions apply when the minimum feature size of each surface exceeds their Hausdorff distance. The ball-map, BMS,S′, between two such manifolds, S and S′, maps each point x of S to a point x′ = BMs,s′(x) of S′. BMS′,S is the inverse of BMS,S′, hence BM is symmetric. We also show that, when S and S′ are Ck (n - 1)-manifolds in ℝn, BMS,S′ is a Ck-1 diffeomorphism and defines a Ck-1 ambient isotopy that smoothly morphs between S to S′. 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 x′ of S′ along a broken line that is normal to S at x and to S′ at x′.