TY - JOUR
AB - Coxeter triangulations are triangulations of Euclidean space based on a single simplex. By this we mean that given an individual simplex we can recover the entire triangulation of Euclidean space by inductively reflecting in the faces of the simplex. In this paper we establish that the quality of the simplices in all Coxeter triangulations is O(1/d−−√) of the quality of regular simplex. We further investigate the Delaunay property for these triangulations. Moreover, we consider an extension of the Delaunay property, namely protection, which is a measure of non-degeneracy of a Delaunay triangulation. In particular, one family of Coxeter triangulations achieves the protection O(1/d2). We conjecture that both bounds are optimal for triangulations in Euclidean space.
AU - Choudhary, Aruni
AU - Kachanovich, Siargey
AU - Wintraecken, Mathijs
ID - 7567
JF - Mathematics in Computer Science
SN - 1661-8270
TI - Coxeter triangulations have good quality
ER -
TY - GEN
AB - Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e.manifolds defined as the zero set of some multivariate multivalued functionf:Rd→Rd−n.A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear(PL) approximation based on a triangulationTof the ambient spaceRd. In this paper, we giveconditions under which the PL-approximation of an isomanifold is topologically equivalent to theisomanifold. The conditions can always be met by taking a sufficiently fine triangulationT.
AU - Boissonnat, Jean-Daniel
AU - Wintraecken, Mathijs
ID - 7568
T2 - EUROCG 2020
TI - The topological correctness of the PL-approximation of isomanifolds
ER -
TY - CONF
AB - 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: 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.
AU - Boissonnat, Jean-Daniel
AU - Wintraecken, Mathijs
ID - 7952
SN - 1868-8969
T2 - 36th International Symposium on Computational Geometry
TI - The topological correctness of PL-approximations of isomanifolds
VL - 164
ER -
TY - JOUR
AB - Fejes Tóth [3] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the square of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.
AU - Vegter, Gert
AU - Wintraecken, Mathijs
ID - 8163
IS - 2
JF - Studia Scientiarum Mathematicarum Hungarica
SN - 0081-6906
TI - Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes
VL - 57
ER -
TY - JOUR
AB - We consider the following setting: suppose that we are given a manifold M in Rd with positive reach. Moreover assume that we have an embedded simplical complex A without boundary, whose vertex set lies on the manifold, is sufficiently dense and such that all simplices in A have sufficient quality. We prove that if, locally, interiors of the projection of the simplices onto the tangent space do not intersect, then A is a triangulation of the manifold, that is, they are homeomorphic.
AU - Boissonnat, Jean-Daniel
AU - Dyer, Ramsay
AU - Ghosh, Arijit
AU - Lieutier, Andre
AU - Wintraecken, Mathijs
ID - 8248
JF - Discrete & Computational Geometry
SN - 0179-5376
TI - Local conditions for triangulating submanifolds of Euclidean space
ER -
TY - JOUR
AB - 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.
AU - Dyer, Ramsay
AU - Vegter, Gert
AU - Wintraecken, Mathijs
ID - 6515
IS - 1
JF - Journal of Computational Geometry
SN - 1920-180X
TI - Simplices modelled on spaces of constant curvature
VL - 10
ER -
TY - CONF
AB - 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.
AU - Vegter, Gert
AU - Wintraecken, Mathijs
ID - 6628
T2 - The 31st Canadian Conference in Computational Geometry
TI - The extrinsic nature of the Hausdorff distance of optimal triangulations of manifolds
ER -
TY - JOUR
AB - In this paper we discuss three results. The first two concern general sets of positive reach: we first characterize the reach of a closed set by means of a bound on the metric distortion between the distance measured in the ambient Euclidean space and the shortest path distance measured in the set. Secondly, we prove that the intersection of a ball with radius less than the reach with the set is geodesically convex, meaning that the shortest path between any two points in the intersection lies itself in the intersection. For our third result we focus on manifolds with positive reach and give a bound on the angle between tangent spaces at two different points in terms of the reach and the distance between the two points.
AU - Boissonnat, Jean-Daniel
AU - Lieutier, André
AU - Wintraecken, Mathijs
ID - 6671
IS - 1-2
JF - Journal of Applied and Computational Topology
SN - 2367-1726
TI - The reach, metric distortion, geodesic convexity and the variation of tangent spaces
VL - 3
ER -
TY - JOUR
AB - 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 $\mathbb{R}^2$ and on surfaces embedded in $\mathbb{R}^3$ as detailed in our experimental companion paper. In this paper, we study theoretical aspects of our structure. Given a finite set of points $\mathcal{P}$ in a domain $\Omega$ equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of $\mathcal{P}$ 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.
AU - Boissonnat, Jean-Daniel
AU - Rouxel-Labbé, Mael
AU - Wintraecken, Mathijs
ID - 6672
IS - 3
JF - SIAM Journal on Computing
SN - 0097-5397
TI - Anisotropic triangulations via discrete Riemannian Voronoi diagrams
VL - 48
ER -
TY - JOUR
AB - We introduce a multiscale topological description of the Megaparsec web-like cosmic matter distribution. Betti numbers and topological persistence offer a powerful means of describing the rich connectivity structure of the cosmic web and of its multiscale arrangement of matter and galaxies. Emanating from algebraic topology and Morse theory, Betti numbers and persistence diagrams represent an extension and deepening of the cosmologically familiar topological genus measure and the related geometric Minkowski functionals. In addition to a description of the mathematical background, this study presents the computational procedure for computing Betti numbers and persistence diagrams for density field filtrations. The field may be computed starting from a discrete spatial distribution of galaxies or simulation particles. The main emphasis of this study concerns an extensive and systematic exploration of the imprint of different web-like morphologies and different levels of multiscale clustering in the corresponding computed Betti numbers and persistence diagrams. To this end, we use Voronoi clustering models as templates for a rich variety of web-like configurations and the fractal-like Soneira-Peebles models exemplify a range of multiscale configurations. We have identified the clear imprint of cluster nodes, filaments, walls, and voids in persistence diagrams, along with that of the nested hierarchy of structures in multiscale point distributions. We conclude by outlining the potential of persistent topology for understanding the connectivity structure of the cosmic web, in large simulations of cosmic structure formation and in the challenging context of the observed galaxy distribution in large galaxy surveys.
AU - Pranav, Pratyush
AU - Edelsbrunner, Herbert
AU - Van De Weygaert, Rien
AU - Vegter, Gert
AU - Kerber, Michael
AU - Jones, Bernard
AU - Wintraecken, Mathijs
ID - 1022
IS - 4
JF - Monthly Notices of the Royal Astronomical Society
SN - 00358711
TI - The topology of the cosmic web in terms of persistent Betti numbers
VL - 465
ER -