Riemannian simplices and triangulations

Abstract

We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary dimension n, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on the manifold, the point that minimises the weighted sum of squared distances to the vertices is the Karcher mean relative to the weights. Using barycentric coordinates as the weights, we obtain a smooth map from the standard Euclidean simplex to the manifold. A Riemannian simplex is defined as the image of this barycentric coordinate map. In this work we articulate criteria that guarantee that the barycentric coordinate map is a smooth embedding. If it is not, we say the Riemannian simplex is degenerate. Quality measures for the "thickness" or "fatness" of Euclidean simplices can be adapted to apply to these Riemannian simplices. For manifolds of dimension 2, the simplex is non-degenerate if it has a positive quality measure, as in the Euclidean case. However, when the dimension is greater than two, non-degeneracy can be guaranteed only when the quality exceeds a positive bound that depends on the size of the simplex and local bounds on the absolute values of the sectional curvatures of the manifold. An analysis of the geometry of non-degenerate Riemannian simplices leads to conditions which guarantee that a simplicial complex is homeomorphic to the manifold.

Delaunay Triangulation of Manifolds

“…If M is a Riemannian manifold, then Del(P ) is a Delaunay triangulation of M and is equipped with a piecewise flat metric that is a good approximation of d M . This follows from recent results [DVW14] that guarantee a homeomorphism in this setting.…”

“…We use the exponential map to define the coordinate charts. Proposition 17 and Lemma 11 of [DVW14] directly imply that if…”

Distributed mean curvature on a discrete manifold for Regge calculus

“…This line is perpendicular to both λ h and l. The triplet (l, m l , λ h ) thus form an orthonormal frame within H h . been made in perturbing given triangulations to form Delaunay triangulations on non-Euclidean manifolds however, irrespective of any embedding [17,18]. Again, the intuition in the following relies on circumcenters lying in the interior of n-simplices, however reasonable deviations should not cause problems.…”

Geometric and Topological Inference

“…Variants of Theorem 7.16 have been proved by Cheng and al. [56], Boissonnat and Ghosh [18] and Dyer et al [65]. The proof presented here is based on [16].…”