Convergence of geodesics on triangulations

Is it possible to approximate a geodesic on a smooth surface S by geodesics on nearby triangulations (i.e. on piecewise linear surfaces)? In other words, given a sequence (Tn)n∈N of triangulations whose points and normals converge to those of a smooth surface S, if Cn is a geodesic of Tn (i.e. it is locally a shortest path) and if (Cn)n∈N converges to a curve C, we want to know if the limit curve C is a geodesic of S. It is already known 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 triangulations whose points and normals converge to those of a 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 triangulations. Finally, we apply this result to different subdivision surfaces (following schemes for B-splines, Bézier surfaces, or Catmull–Clark schemes assuming that geodesics avoid extraordinary vertices). In particular, these results validate an existing algorithm that builds geodesics on subdivision surfaces.