205 research outputs found
Riemannian simplices and triangulations
We study a natural intrinsic definition of geometric simplices in Riemannian
manifolds of arbitrary dimension , 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
Constructing Intrinsic Delaunay Triangulations of Submanifolds
We describe an algorithm to construct an intrinsic Delaunay triangulation of
a smooth closed submanifold of Euclidean space. Using results established in a
companion paper on the stability of Delaunay triangulations on -generic
point sets, we establish sampling criteria which ensure that the intrinsic
Delaunay complex coincides with the restricted Delaunay complex and also with
the recently introduced tangential Delaunay complex. The algorithm generates a
point set that meets the required criteria while the tangential complex is
being constructed. In this way the computation of geodesic distances is
avoided, the runtime is only linearly dependent on the ambient dimension, and
the Delaunay complexes are guaranteed to be triangulations of the manifold
A probabilistic approach to reducing the algebraic complexity of computing Delaunay triangulations
Computing Delaunay triangulations in involves evaluating the
so-called in\_sphere predicate that determines if a point lies inside, on
or outside the sphere circumscribing points . This
predicate reduces to evaluating the sign of a multivariate polynomial of degree
in the coordinates of the points . Despite
much progress on exact geometric computing, the fact that the degree of the
polynomial increases with makes the evaluation of the sign of such a
polynomial problematic except in very low dimensions. In this paper, we propose
a new approach that is based on the witness complex, a weak form of the
Delaunay complex introduced by Carlsson and de Silva. The witness complex
is defined from two sets and in some metric space
: a finite set of points on which the complex is built, and a set of
witnesses that serves as an approximation of . A fundamental result of de
Silva states that if .
In this paper, we give conditions on that ensure that the witness complex
and the Delaunay triangulation coincide when is a finite set, and we
introduce a new perturbation scheme to compute a perturbed set close to
such that . Our perturbation
algorithm is a geometric application of the Moser-Tardos constructive proof of
the Lov\'asz local lemma. The only numerical operations we use are (squared)
distance comparisons (i.e., predicates of degree 2). The time-complexity of the
algorithm is sublinear in . Interestingly, although the algorithm does not
compute any measure of simplex quality, a lower bound on the thickness of the
output simplices can be guaranteed.Comment: 24 page
An obstruction to Delaunay triangulations in Riemannian manifolds
Delaunay has shown that the Delaunay complex of a finite set of points of
Euclidean space triangulates the convex hull of , provided
that satisfies a mild genericity property. Voronoi diagrams and Delaunay
complexes can be defined for arbitrary Riemannian manifolds. However,
Delaunay's genericity assumption no longer guarantees that the Delaunay complex
will yield a triangulation; stronger assumptions on are required. A natural
one is to assume that is sufficiently dense. Although results in this
direction have been claimed, we show that sample density alone is insufficient
to ensure that the Delaunay complex triangulates a manifold of dimension
greater than 2.Comment: This is a revision and extension of a note that appeared as an
appendix in the (otherwise unpublished) report arXiv:1303.649
Only distances are required to reconstruct submanifolds
In this paper, we give the first algorithm that outputs a faithful
reconstruction of a submanifold 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.Comment: Major revision, 16 figures, 47 page
Simplices modelled on spaces of constant curvature
We give non-degeneracy criteria for Riemannian simplices based on simplices in spaces of constant sectional curvature. It extends previous work on Riemannian simplices, where we developed Riemannian simplices with respect to Euclidean reference simplices. The criteria we give in this article are in terms of quality measures for spaces of constant curvature that we develop here. We see that simplices in spaces that have nearly constant curvature, are already non-degenerate under very weak quality demands. This is of importance because it allows for sampling of Riemannian manifolds based on anisotropy of the manifold and not (absolute) curvature
Local Criteria for Triangulation of Manifolds
We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex A, and a map H from the underlying space of A to M, our criteria are presented in local coordinate charts for M, and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on M. No Delaunay property of A is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily verified in such a setting, they are expected to be of general use
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