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 it 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, BM S,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′ areCk (n−1)-manifolds in Rn, BMS,S′ is aCk−1 diffeomorphism and defines an 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 circular arc that is normal to S at x and to S ′ at x′.