Geodesic as Limit of Geodesics on PL-Surfaces

In this paper, 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 want to know if the limit curve C is a geodesic of S. Hildebrandt et al. [12] have already shown that if Cn is a shortest path, then C is also a shortest path. The result does not hold anymore for geodesics that are not (global) shortest paths. In this paper, we first provide a counter example for geodesics: we build a sequence (Tn)n∈N of PL-surfaces that converges in distance and in normals to the plane. On each Tn, we build a geodesic Cn, such that (Cn)n∈N converges to a planar curve which is not a line-segment (and thus not a geodesic of the plane). In a second step, we give a positive result of convergence for geodesics that needs additional assumptions concerning the rate of convergence of the normals and of the lengths of the edges of the PL-surfaces. Finally, we apply this result to different subdivisions surfaces (following schemes for bicubic B-splines, or Catmull-Clark schemes, or schemes for Bezier surfaces). In particular, these results validate an algorithm of PhamTrong et al. [20] that builds geodesics on subdivision surfaces.