Geodesic as limit of geodesics on PL-surfaces
Résumé
We study the problem of convergence of geodesics on PL-surfaces and in particular on subdivision surfaces. More precisely, if a sequence of PL-surfaces converges in distance and in normals to a smooth surface S and if C_n is a geodesic of T_n (i.e. it is locally a shortest path) such that converges to a curve C, we wonder if C is a geodesic of S. Hildebrandt et al. [11] have already shown that if C_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 geodesics. We give a result of convergence for geodesics 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 geodesics avoid extraordinary vertices.