Curves and surfaces for computer aided geometric design: a practical guide
Curves and surfaces for computer aided geometric design: a practical guide
Finding shortest paths on surfaces
Proceedings of the international conference on Curves and surfaces in geometric design
Subdivision Methods for Geometric Design: A Constructive Approach
Subdivision Methods for Geometric Design: A Constructive Approach
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We study the problem of convergence of geodesics on PL-surfaces and in particular on subdivision surfaces. More precisely, if a sequence (Tn)n∈N of PL-surfaces converges in distance and in normals to a smooth surface S and if Cn is a geodesic of Tn (i.e. it is locally a shortest path) such that (Cn)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 Cn 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.