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**1 - 7**of**7**### FR +E N

, 2013

"... Delaunay stability via perturbations Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh ha l-0 ..."

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Delaunay stability via perturbations Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh ha l-0

### Only distances are required to reconstruct submanifolds

, 2014

"... In this paper, we give the first algorithm that outputs a faithful reconstruction of a subman-ifold of Euclidean space without maintaining or even constructing complicated data structures such as Voronoi diagrams or Delaunay complexes. Our algorithm uses the witness complex and relies on the stabili ..."

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In this paper, we give the first algorithm that outputs a faithful reconstruction of a subman-ifold of Euclidean space without maintaining or even constructing complicated data structures such as Voronoi diagrams or Delaunay complexes. Our algorithm uses the witness complex and relies on the stability of power protection, a notion introduced in this paper. The complexity of the algorithm depends exponentially on the intrinsic dimension of the manifold, rather than the dimension of ambient space, and linearly on the dimension of the ambient space. Another interesting feature of this work is that no explicit coordinates of the points in the point sample is needed. The algorithm only needs the distance matrix as input, i.e., only distance between points in the point sample as input.

### +E

, 2013

"... Delaunay stability via perturbations Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh ar X iv ..."

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Delaunay stability via perturbations Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh ar X iv

### Riemannian simplices and triangulations

, 2014

"... 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 com-pact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices i ..."

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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 com-pact 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 sim-plices 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.