Geodesics as limits of geodesics on PL-surfaces

André Lieutier (1) , Boris Thibert (1)

1 MGMI - Modélisation Géométrique & Multirésolution pour l'Image

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 (T_n) 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 (C_n) converges to a curve C, we want to know if the limit curve C is a geodesic of S. Hildebrandt and his coauthors have already shown that if C_n 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 (T_n) of PL-surfaces that converges in distance and in normals to the plane. On each T_n, we build a geodesic C_n, such that (C_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 Pham-Trong and her coauthors that builds geodesics on subdivision surfaces.

Mots clés: subdivision surfaces; triangulations; PL-surfaces; geodesics; shortest paths; convergence

Domaines: Géométrie algorithmique [cs.CG]; Géométrie différentielle [math.DG]

André Lieutier, Boris Thibert. Geodesics as limits of geodesics on PL-surfaces. 2007. ⟨hal-00160820v2⟩