120 citations · 148 across the 18 of their papers we have counts for
21 papers
Simplicial subdivision of simplices of arbitrary dimension in spaces of constant curvature with bounded quality
Jean-Daniel Boissonnat, Hana Dal Poz Kourimska, Arijit Ghosh +1
In 1942, Freudenthal showed that a simplex in Euclidean space can be subdivided such that the quality (well-shapedness of the simplex, quantified in terms of e.g. fatness) of the s…
Generalized altitudes and their bounds
Hana Dal Poz Kourimska, Mathijs Wintraecken
We introduce generalized altitudes of a simplex, extending the usual vertex-to-opposite-face altitude to arbitrary pairs of opposite faces. These quantities encode the relative pos…
The Singular Source of Vineyard Monodromy
Erin W. Chambers, Christopher Fillmore, Shankha Shubhra Mukherjee +3
Vineyards, or time-varying families of persistence diagrams, are widely used in topological data analysis (TDA) pipelines to track how topological features change and evolve as a p…
Error estimates for the interpolation and approximation of gradients and vector fields on protected Delaunay meshes in
David M. Williams, Mathijs Wintraecken
One frequently needs to interpolate or approximate gradients on simplicial meshes. Unfortunately, there are very few explicit mathematical results governing the interpolation or ap…
Braiding vineyards
Erin Chambers, Christopher Fillmore, Elizabeth Stephenson +1
In this work, we introduce and study what we believe is an intriguing and, to the best of our knowledge, previously unknown connection between two areas in computational topology,…
Manifolds of positive reach, differentiability, tangent variation, and attaining the reach
André Lieutier, Mathijs Wintraecken
This paper contains three main results. Firstly, we give an elementary proof of the following statement: Let be a (closed, in both the geometrical and topological sense of the…
A free lunch: manifolds of positive reach can be smoothed without decreasing the reach
Hana Dal Poz Kouřimská, André Lieutier, Mathijs Wintraecken
Assumptions on the reach are crucial for ensuring the correctness of many geometric and topological algorithms, including triangulation, manifold reconstruction and learning, homot…
Quasi-optimal interpolation of gradients and vector-fields on protected Delaunay meshes in
David M. Williams, Mathijs Wintraecken
There are very few mathematical results governing the interpolation of functions or their gradients on Delaunay meshes in more than two dimensions. Unfortunately, the standard tech…
Hausdorff and Gromov-Hausdorff stable subsets of the medial axis
André Lieutier, Mathijs Wintraecken
In this paper we introduce a pruning of the medial axis called the -medial axis (). We prove that the -medial axis of a set is stable in a Gromov…
Translation of "Simplizialzerlegungen von Beschrankter Flachheit'' by Hans Freudenthal, Annals of Mathematics, Second Series, Volume 43, Number 3, July 1942, Pages 580-583
Mathijs Wintraecken
Translation of the paper ``Simplizialzerlegungen von Beschrankter Flachheit'' by Hans Freudenthal (https://doi.org/10.2307/1968813), in which Freudenthal answers ``a question by Br…
The medial axis of closed bounded sets is Lipschitz stable with respect to the Hausdorff distance under ambient diffeomorphisms
Hana Dal Poz Kouřimská, André Lieutier, Mathijs Wintraecken
We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let be (fixed) closed set (that contains a bounding sp…
Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds
Dominique Attali, Hana Dal Poz Kouřimská, Christopher Fillmore +4
In this article we extend and strengthen the seminal work by Niyogi, Smale, and Weinberger on the learning of the homotopy type from a sample of an underlying space. In their work,…
Brillouin Zones of Integer Lattices and Their Perturbations
Herbert Edelsbrunner, Alexey Garber, Mohadese Ghafari +3
For a locally finite set, , the -th Brillouin zone of is the region of points for which is the -th smallest…
The Density Fingerprint of a Periodic Point Set
Herbert Edelsbrunner, Teresa Heiss, Vitaliy Kurlin +2
Modeling 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…
Local criteria for triangulation of manifolds
Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh +1
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 pr…
Anisotropic triangulations via discrete Riemannian Voronoi diagrams
Jean-Daniel Boissonnat, Mael Rouxel-Labbé, Mathijs Wintraecken
The construction of anisotropic triangulations is desirable for various applications, such as the numerical solving of partial differential equations and the representation of surf…
The Topology of the Cosmic Web in Terms of Persistent Betti Numbers
Pratyush Pranav, Herbert Edelsbrunner, Rien van de Weygaert +4
We introduce a multiscale topological description of the Megaparsec weblike cosmic matter distribution. Betti numbers and topological persistence offer a powerful means of describi…
Barycentric coordinate neighbourhoods in Riemannian manifolds
Ramsay Dyer, Gert Vegter, Mathijs Wintraecken
We quantify conditions that ensure that a signed measure on a Riemannian manifold has a well defined centre of mass. We then use this result to quantify the extent of a neighbourho…
Optimal Triangulation of Saddle Surfaces
Dror Atariah, Günter Rote, Mathijs Wintraecken
We consider the piecewise linear approximation of saddle functions of the form under the L-infinity error norm. We show that interpolating approximations are not…
Riemannian simplices and triangulations
Ramsay Dyer, Gert Vegter, Mathijs Wintraecken
We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary dimension , and exploit these simplices to obtain criteria for triangulating…
Probing Dark Energy with Alpha Shapes and Betti Numbers
Rien van de Weygaert, Pratyush Pranav, Bernard J. T. Jones +8
We introduce a new descriptor of the weblike pattern in the distribution of galaxies and matter: the scale dependent Betti numbers which formalize the topological information conte…