57 research outputs found
Ambient and intrinsic triangulations and topological methods in cosmology
The 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
The 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
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
The extrinsic nature of the Hausdorff distance of optimal triangulations of manifolds
Fejes Tóth [5] and Schneider [9] studied approximations of smooth convex hypersurfaces in Euclidean space by piecewise flat triangular meshes with a given number of vertices on the hypersurface that are optimal with respect to Hausdorff distance. They proved that this Hausdorff distance decreases inversely proportional with m 2/(d−1), where m is the number of vertices and d is the dimension of Euclidean space. Moreover the pro-portionality constant can be expressed in terms of the Gaussian curvature, an intrinsic quantity. In this short note, we prove the extrinsic nature of this constant for manifolds of sufficiently high codimension. We do so by constructing an family of isometric embeddings of the flat torus in Euclidean space
Anisotropic Triangulations via Discrete Riemannian Voronoi Diagrams
The construction of anisotropic triangulations is desirable for various applications, such as the numerical solving of partial differential equations and the representation of surfaces in graphics. To solve this notoriously difficult problem in a practical way, we introduce the discrete Riemannian Voronoi diagram, a discrete structure that approximates the Riemannian Voronoi diagram. This structure has been implemented and was shown to lead to good triangulations in R^2 and on surfaces embedded in R^3 as detailed in our experimental companion paper.
In this paper, we study theoretical aspects of our structure. Given a finite set of points P in a domain Omega equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of P to its Riemannian Voronoi diagram. Both diagrams have dual structures called the discrete Riemannian Delaunay and the Riemannian Delaunay complex. We provide conditions that guarantee that these dual structures are identical. It then follows from previous results that the discrete Riemannian Delaunay complex can be embedded in Omega under sufficient conditions, leading to an anisotropic triangulation with curved simplices. Furthermore, we show that, under similar conditions, the simplices of this triangulation can be straightened
The topological correctness of PL approximations of isomanifolds
Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f : Rd → Rd−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 Rd. 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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