277 research outputs found
Riemannian Simplices and Triangulations
Get PDFWe study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary finite 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 the standard simplex under 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
A Cautionary Tale: Burning the Medial Axis Is Unstable (Media Exposition)
No full textThe medial axis of a set consists of the points in the ambient space without a unique closest point on the original set. Since its introduction, the medial axis has been used extensively in many applications as a method of computing a topologically equivalent skeleton. Unfortunately, one limiting factor in the use of the medial axis of a smooth manifold is that it is not necessarily topologically stable under small perturbations of the manifold. To counter these instabilities various prunings of the medial axis have been proposed. Here, we examine one type of pruning, called burning. Because of the good experimental results, it was hoped that the burning method of simplifying the medial axis would be stable. In this work we show a simple example that dashes such hopes based on Bing’s house with two rooms, demonstrating an isotopy of a shape where the medial axis goes from collapsible to non-collapsible
The density fingerprint of a periodic point set
Get PDFModeling a crystal as a periodic point set, we present a fingerprint consisting of density functions that facilitates the efficient search for new materials and material properties. We prove invariance under isometries, continuity, and completeness in the generic case, which are necessary features for the reliable comparison of crystals. The proof of continuity integrates methods from discrete geometry and lattice theory, while the proof of generic completeness combines techniques from geometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and related inclusion-exclusion formulae. We have implemented the algorithm and describe its application to crystal structure prediction
The topological correctness of PL-approximations of isomanifolds
Get PDFIsomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate multivalued function f : R^d → R^{d−n}. A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation T of the ambient space R^d. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine triangulation T. This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary
LIPIcs
Get PDFWe prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let ⊆ ℝ^d be a fixed closed set that contains a bounding sphere. That is, the bounding sphere is part of the set . Consider the space of C^{1,1} diffeomorphisms of ℝ^d to itself, which keep the bounding sphere invariant. The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of ℝ^d (endowed with the Hausdorff distance), mapping a diffeomorphism F to the closure of the medial axis of F(), is Lipschitz. This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of C² manifolds under C² ambient diffeomorphisms
Translation of "Simplizialzerlegungen von Beschrankter Flachheit'' by Hans Freudenthal, Annals of Mathematics, Second Series, Volume 43, Number 3, July 1942, Pages 580-583
No full text"Simplizialzerlegungen von Beschrankter Flachheit'' by Hans Freudenthal, Annals of Mathematics, Second Series, Volume 43, Number 3, July 1942, Pages 580-583Translation of the paper ``Simplizialzerlegungen von Beschrankter Flachheit'' by Hans Freudenthal (https://doi.org/10.2307/1968813), in which Freudenthal answers ``a question by Brouwer about the construction of an infinite series of subdivisions of a polytope, such that the next element in the sequence is a subdivision of the previous one and such that the subsimplices that arise do not become arbitrarily flat.'
Ambient and intrinsic triangulations and topological methods in cosmology
Get PDFThe thesis consist of two parts, one part concerns triangulations the other the structure of the universe. 1Images in films such as Shrek or Frozen and in computer games are often made using small triangles. Subdividing a figure (such as Shrek) into small triangles is called triangulating. This may be done in two different ways.The first method makes use of straight triangles and is used most often. Because computer power is limited, we want to use as few triangles as possible, while maintaining the quality of the image. This means that one has to choose the triangles in a clever manner. Much is known about the choice of triangles if the surface is convex (egg-shaped). This thesis contributes to our understanding of non-convex surfaces.The second and new method uses curved triangles that follow the surface. The triangles we use are determined by the intrinsic geometry of the surface and are called intrinsic triangles.2Shortly after the Big Bang the universe was very hot and dense. Quantum mechanical effects introduced structure into the matter distribution in the early universe. The universe expanded according the laws of General Relativity and the matter cooled down. After the matter in the universe had cooled down, clusters of galaxies formed out of the densest regions. These clusters of galaxies are connected by stringy structures consisting of galaxies. This thesis contributes to the understanding of this intricate structure
Ambient and intrinsic triangulations and topological methods in cosmology
Get PDFThe thesis consist of two parts, one part concerns triangulations the other the structure of the universe. 1Images in films such as Shrek or Frozen and in computer games are often made using small triangles. Subdividing a figure (such as Shrek) into small triangles is called triangulating. This may be done in two different ways.The first method makes use of straight triangles and is used most often. Because computer power is limited, we want to use as few triangles as possible, while maintaining the quality of the image. This means that one has to choose the triangles in a clever manner. Much is known about the choice of triangles if the surface is convex (egg-shaped). This thesis contributes to our understanding of non-convex surfaces.The second and new method uses curved triangles that follow the surface. The triangles we use are determined by the intrinsic geometry of the surface and are called intrinsic triangles.2Shortly after the Big Bang the universe was very hot and dense. Quantum mechanical effects introduced structure into the matter distribution in the early universe. The universe expanded according the laws of General Relativity and the matter cooled down. After the matter in the universe had cooled down, clusters of galaxies formed out of the densest regions. These clusters of galaxies are connected by stringy structures consisting of galaxies. This thesis contributes to the understanding of this intricate structure
Ambient and intrinsic triangulations and topological methods in cosmology
Get PDFThe thesis consist of two parts, one part concerns triangulations the other the structure of the universe. 1 Images in films such as Shrek or Frozen and in computer games are often made using small triangles. Subdividing a figure (such as Shrek) into small triangles is called triangulating. This may be done in two different ways. The first method makes use of straight triangles and is used most often. Because computer power is limited, we want to use as few triangles as possible, while maintaining the quality of the image. This means that one has to choose the triangles in a clever manner. Much is known about the choice of triangles if the surface is convex (egg-shaped). This thesis contributes to our understanding of non-convex surfaces. The second and new method uses curved triangles that follow the surface. The triangles we use are determined by the intrinsic geometry of the surface and are called intrinsic triangles. 2 Shortly after the Big Bang the universe was very hot and dense. Quantum mechanical effects introduced structure into the matter distribution in the early universe. The universe expanded according the laws of General Relativity and the matter cooled down. After the matter in the universe had cooled down, clusters of galaxies formed out of the densest regions. These clusters of galaxies are connected by stringy structures consisting of galaxies. This thesis contributes to the understanding of this intricate structure
Ambient and intrinsic triangulations and topological methods in cosmology
Get PDFThe thesis consist of two parts, one part concerns triangulations the other the structure of the universe. 1Images in films such as Shrek or Frozen and in computer games are often made using small triangles. Subdividing a figure (such as Shrek) into small triangles is called triangulating. This may be done in two different ways.The first method makes use of straight triangles and is used most often. Because computer power is limited, we want to use as few triangles as possible, while maintaining the quality of the image. This means that one has to choose the triangles in a clever manner. Much is known about the choice of triangles if the surface is convex (egg-shaped). This thesis contributes to our understanding of non-convex surfaces.The second and new method uses curved triangles that follow the surface. The triangles we use are determined by the intrinsic geometry of the surface and are called intrinsic triangles.2Shortly after the Big Bang the universe was very hot and dense. Quantum mechanical effects introduced structure into the matter distribution in the early universe. The universe expanded according the laws of General Relativity and the matter cooled down. After the matter in the universe had cooled down, clusters of galaxies formed out of the densest regions. These clusters of galaxies are connected by stringy structures consisting of galaxies. This thesis contributes to the understanding of this intricate structure
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