BALL-MAP: HOMEOMORPHISM BETWEEN COMPATIBLE SURFACES

作者：Chazal Frederic^*; Lieutier Andre; Rossignac Jarek; Whited Brian

来源：International Journal of Computational Geometry and Applications, 2010, 20(3): 285-306.

DOI：10.1142/S021819591000330X

• Homeomorphism
• surface
• frechet

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, BM(S,S)', between two such manifolds, S and S', maps each point x of S to a point x' = BM(S,S)' (x) of S'. BM(S)'(,S) is the inverse of BM(S),(S)', hence BM is symmetric. We also show that, when S and S' are C(k)(n - 1)-manifolds in R(n), BM(S,S)' is a C(k-1) diffeomorphism and defines a C(k-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'.

• 出版日期2010-6