Triangulating stratified manifolds I: a reach comparison theorem

Jean-Daniel Boissonnat (1) , Mathijs Wintraecken (1)

1 DATASHAPE - Understanding the Shape of Data

In this paper, we define the reach for submanifolds of Riemannian manifolds, in a way that is similar to the Euclidean case. Given a d-dimensional submanifold S of a smooth Riemannian manifold M and a point p ∈ M that is not too far from S we want to give bounds on local feature size of exp −1 p (S). Here exp −1 p is the inverse exponential map, a canonical map from the manifold to the tangent space. Bounds on the local feature size of exp −1 p (S) can be reduced to giving bounds on the reach of exp −1 p (B), where B is a geodesic ball, centred at c with radius equal to the reach of S. Equivalently we can give bounds on the reach of exp −1 p • exp c (B c), where now B c is a ball in the tangent space T c M, with the same radius. To establish bounds on the reach of exp −1 p • exp c (B c) we use bounds on the metric and on its derivative in Riemann normal coordinates. This result is a first step towards answering the important question of how to triangulate stratified manifolds.

Mots clés: and phrases Reach; Manifolds; surfaces; comparison theory

Domaines: Géométrie algorithmique [cs.CG]; Algorithme et structure de données [cs.DS]; Topologie algébrique [math.AT]; Topologie géométrique [math.GT]

Jean-Daniel Boissonnat, Mathijs Wintraecken. Triangulating stratified manifolds I: a reach comparison theorem. 2017. ⟨hal-01661233⟩