Geodesic as Limit of Geodesics on PL-Surfaces

摘要

We study the problem of convergence of geodesics on PL-surfaces and in particular on subdivision surfaces. More precisely, if a sequence (T{sub}n){sub}(n∈N) of PL-surfaces converges in distance and in normals to a smooth surface S and if C{sub}n is a geodesic of T{sub}n (i.e. it is locally a shortest path) such that (C{sub}n){sub}(n∈N) converges to a curve C, we wonder if C is a geodesic of S. Hildebrandt et al. [11] have already shown that if C{sub}n is a shortest path, then C is a shortest path. In this paper, we provide a counter example showing that this result is false for geodesies. We give a result of convergence for geodesies with additional assumptions concerning the rate of convergence of the normals and of the lengths of the edges. Finally, we apply this result to different subdivisions surfaces (such as Catmull-Clark) assuming that geodesies avoid extraordinary vertices.