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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
Geodesics of length less than πR in a set of reach R are unique and continuous with respect to the end points
No full textPositive reach underpins many results in computational geometry and topology. It is used for triangulation criteria, topological inference, and manifold learning. The geometric properties of these sets have therefore been studied intensely. Here we focus on the shortest paths or geodesics in these sets. Our main result states that geodesics of length strictly less than πR in a set of reach R are unique. This in turn implies that such geodesics are continuous with respect to the end points
Hausdorff and Gromov-Hausdorff Stable Subsets of the Medial Axis
No full textConference paper, the full version can be found as hal-04083167, it is advised to consult the full paper.International audienceIn this paper we introduce a pruning of the medial axis called the (λ,α)-medial axis (axαλ). We prove that the (λ,α)-medial axis of a set K is stable in a Gromov-Hausdorff sense under weak assumptions. More formally we prove that if K and K′ are close in the Hausdorff (dH) sense then the (λ,α)-medial axes of K and K′ are close as metric spaces, that is the Gromov-Hausdorff distance (dGH) between the two is 14-H{ö}lder in the sense that dGH(axαλ(K),axαλ(K′))≲dH(K,K′)1/4. The Hausdorff distance between the two medial axes is also bounded, by dH(axαλ(K),axαλ(K′))≲dH(K,K′)1/2. These quantified stability results provide guarantees for practical computations of medial axes from approximations. Moreover, they provide key ingredients for studying the computability of the medial axis in the context of computable analysis
Manifolds of positive reach, differentiability, tangent variation, and attaining the reach
No full textThis paper contains three main results.Firstly, we give an elementary proof of the following statement: Let M be a (closed, in both the geometrical and topological sense of the word) topological manifold embedded in R^d . If M has positive reach, then M can locally be written as the graph of a C^{1,1} from the tangent space to the normal space. Conversely if M can locally written as the graph of a C^{1,1} function from the tangent space to the normal space, then M has positive reach. The result was hinted at by Federer when he introduced the reach, and proved by Lytchak. Lytchak's proof relies heavily CAT(k)-theory. The proof presented here uses only basic results on homology.Secondly, we give optimal Lipschitz-constants for the derivative, in other words we give an optimal bound for the angle between tangent spaces in term of the distance between the points. This improves earlier results, that were either suboptimal or assumed that the manifold was C^2 .Thirdly, we generalize a result by Aamari et al. which explains the how the reach is attained from the smooth setting to general sets of positive reach.</p
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
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