arXiv subjects

Mathijs Wintraecken

Publications and source records attributed to Mathijs Wintraecken.

At least 19 recordsLinked to original sources

arXiv2026

Generalized altitudes and their bounds

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 position of the affine spans of such faces and yield a uniform formula for the angle between them. We also derive an equivalent algebraic expression in terms of generalized cross products and Gram determinants, linking the construction to standard determinant-based tools. Finally, we prove that every generalized altitude is bounded below by a quantity controlled by the ordinary height of the simplex. Thus, classical height or thickness assumptions imply control over this broader family of geometric quantities. The results provide a compact framework for studying simplex quality and are motivated by applications to triangulation criteria for Riemannian manifolds.

cs.CG↗

arXiv2026

Simplicial subdivision of simplices of arbitrary dimension in spaces of constant curvature with bounded quality

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 simplices in the subdivision is lower bounded. This answered a question of Brouwer. Recently, Brunck discussed the same problem for simplices in two-dimensional spaces of constant curvature and provided a closely related construction. In this paper we generalize Brunck's result to arbitrary dimensional spaces of constant curvature by combining Freudenthal's construction and radial projection. We contrast this approach with Brunck's construction.

cs.CG↗

arXiv2026

The Singular Source of Vineyard Monodromy

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 parameter varies. When the parameter traces a closed loop, a vineyard can exhibit monodromy: diagram points permute over the course of a full traversal, which obstructs feature tracking and can complicate downstream analysis of such data. Chambers et al. considered the periodic vineyards that arise from the radial persistence transform, which maps the manifold to a family of persistence diagrams, where each diagram fixes a base point and considers the filtration that is based on Euclidean distance to that point, and showed that monodromy and knotting can occur. Other recent work by Arya et al. considers geometric conditions that exclude monodromy in two dimensions, in an effort to better understand when this effect happens. That said, understanding when and why monodromy occurs is a fundamental open problem with direct practical consequences for many data analysis pipelines. In this work, we study this question for 1-manifolds in $\mathbb{R}^2$, using a surprising connection with tools from singularity theory, and provide a classification for the causes of monodromy in vineyards. More precisely, we prove that the vineyard of a sufficiently small loop $\gamma$ cannot exhibit monodromy unless it contains a specific singularity of the distance function. The central geometric object in our analysis is the symmetry set, which is the locus of centers of spheres tangent in more than one point to the manifold; this object classifies singularities of the distance function, and in our setting, dictates precisely when monodromy occurs. This characterization opens the door to the development of algorithmic criteria for detecting and utilizing (or avoiding) monodromy in TDA pipelines.

cs.CG↗

arXiv2025

Error estimates for the interpolation and approximation of gradients and vector fields on protected Delaunay meshes in $\mathbb{R}^d$

One frequently needs to interpolate or approximate gradients on simplicial meshes. Unfortunately, there are very few explicit mathematical results governing the interpolation or approximation of vector-valued functions on Delaunay meshes in more than two dimensions. Most of the existing results are tailored towards interpolation with piecewise linear polynomials. In contrast, interpolation with piecewise high-order polynomials is not well understood. In particular, the results in this area are sometimes difficult to immediately interpret, or to specialize to the Delaunay setting. In order to address this issue, we derive explicit error estimates for high-order, piecewise polynomial gradient interpolation and approximation on protected Delaunay meshes. In addition, we generalize our analysis beyond gradients, and obtain error estimates for sufficiently-smooth vector fields. Throughout the paper, we show that the quality of interpolation and approximation often depends (in part) on the minimum thickness of simplices in the mesh. Fortunately, the minimum thickness can be precisely controlled on protected Delaunay meshes in $\mathbb{R}^d$.

math.NA↗

arXiv2025

Braiding vineyards

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, topological data analysis (TDA) and knot theory. Given a function from a topological space to $\mathbb{R}$, TDA provides tools to simplify and study the importance of topological features: in particular, the $l^{th}$-dimensional persistence diagram encodes the $l$-homology in the sublevel set as the function value increases as a set of points in the plane. Given a continuous one-parameter family of such functions, we can combine the persistence diagrams into an object known as a vineyard, which track the evolution of points in the persistence diagram. If we further restrict that family of functions to be periodic, we identify the two ends of the vineyard, yielding a closed vineyard. This allows the study of monodromy, which in this context means that following the family of functions for a period permutes the set of points in a non-trivial way. In this work, given a link and value $l$, we construct a topological space and periodic family of functions such that the closed $l$-vineyard contains this link. This shows that vineyards are topologically as rich as one could possibly hope. Importantly, it has at least two immediate consequences: First, monodromy of any periodicity can occur in a $l$-vineyard, answering a variant of a question by [Arya et al 2024]. To exhibit this, we also reformulate monodromy in a more geometric way, which may be of interest in itself. Second, distinguishing vineyards is likely to be difficult given the known difficulty of knot and link recognition, which have strong connections to many NP-hard problems.

cs.CG↗

arXiv2024

A free lunch: manifolds of positive reach can be smoothed without decreasing the reach

Assumptions on the reach are crucial for ensuring the correctness of many geometric and topological algorithms, including triangulation, manifold reconstruction and learning, homotopy reconstruction, and methods for estimating curvature or reach. However, these assumptions are often coupled with the requirement that the manifold be smooth, typically at least C^2 .In this paper, we prove that any manifold with positive reach can be approximated arbitrarily well by a C^$\infty$ manifold without significantly reducing the reach, by employing techniques from differential topology -partitions of unity and smoothing using convolution kernels. This result implies that nearly all theorems established for C^2 manifolds with a certain reach naturally extend to manifolds with the same reach, even if they are not C^2 , for free!

cs.CG↗

arXiv2024

Manifolds of positive reach, differentiability, tangent variation, and attaining the reach

This 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 $\mathbb{R}^d$. 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.

cs.CG↗

arXiv2024

Quasi-optimal interpolation of gradients and vector-fields on protected Delaunay meshes in $\mathbb{R}^d$

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 techniques for proving optimal interpolation properties are often limited to triangular meshes. Furthermore, the results which do exist, are tailored towards interpolation with piecewise linear polynomials. In fact, we are unaware of any results which govern the high-order, piecewise polynomial interpolation of functions or their gradients on Delaunay meshes. In order to address this issue, we prove that quasi-optimal, high-order, piecewise polynomial gradient interpolation can be successfully achieved on protected Delaunay meshes. In addition, we generalize our analysis beyond gradient interpolation, and prove quasi-optimal interpolation properties for sufficiently-smooth vector fields. Throughout the paper, we use the words 'quasi-optimal', because the quality of interpolation depends (in part) on the minimum thickness of simplicies in the mesh. Fortunately, the minimum thickness can be precisely controlled on protected Delaunay meshes in $\mathbb{R}^d$. Furthermore, the current best mathematical estimates for minimum thickness have been obtained on such meshes. In this sense, the proposed interpolation is optimal, although, we acknowledge that future work may reveal an alternative Delaunay meshing strategy with better control over the minimum thickness. With this caveat in mind, we refer to our interpolation on protected Delaunay meshes as quasi-optimal.

math.NA↗

arXiv2024

Brillouin Zones of Integer Lattices and Their Perturbations

For a locally finite set, $A \subseteq \mathbb{R}^d$, the $k$-th Brillouin zone of $a \in A$ is the region of points $x \in \mathbb{R}^d$ for which $\|x-a\|$ is the $k$-th smallest among the Euclidean distances between $x$ and the points in $A$. If $A$ is a lattice, the $k$-th Brillouin zones of the points in $A$ are translates of each other, which tile space. Depending on the value of $k$, they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in $\mathbb{R}^2$, and the convergence of the maximum volume of a chamber to zero for the integer lattice.

math.CO↗

arXiv2024

Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds

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, Niyogi, Smale, and Weinberger studied samples of $C^2$ manifolds with positive reach embedded in $\mathbb{R}^d$. We extend their results in the following ways: In the first part of our paper we consider both manifolds of positive reach -- a more general setting than $C^2$ manifolds -- and sets of positive reach embedded in $\mathbb{R}^d$. The sample $P$ of such a set $\mathcal{S}$ does not have to lie directly on it. Instead, we assume that the two one-sided Hausdorff distances -- $\varepsilon$ and $δ$ -- between $P$ and $\mathcal{S}$ are bounded. We provide explicit bounds in terms of $\varepsilon$ and $ δ$, that guarantee that there exists a parameter $r$ such that the union of balls of radius $r$ centred at the sample $P$ deformation-retracts to $\mathcal{S}$. In the second part of our paper we study homotopy learning in a significantly more general setting -- we investigate sets of positive reach and submanifolds of positive reach embedded in a \emph{Riemannian manifold with bounded sectional curvature}. To this end we introduce a new version of the reach in the Riemannian setting inspired by the cut locus. Yet again, we provide tight bounds on $\varepsilon$ and $δ$ for both cases (submanifolds as well as sets of positive reach), exhibiting the tightness by an explicit construction.

cs.CG↗

arXiv2024

The medial axis of closed bounded sets is Lipschitz stable with respect to the Hausdorff distance under ambient diffeomorphisms

We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let $\mathcal{S} \subseteq \mathbb{R}^d$ be (fixed) closed set (that contains a bounding sphere). Consider the space of $C^{1,1}$ diffeomorphisms of $\mathbb{R}^d$ to itself, which keep the bounding sphere invariant. The map from this space of diffeomorphisms (endowed with some Banach norm) to the space of closed subsets of $\mathbb{R}^d$ (endowed with the Hausdorff distance), mapping a diffeomorphism $F$ to the closure of the medial axis of $F(\mathcal{S})$, is Lipschitz. This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of $C^2$ manifolds under $C^2$ ambient diffeomorphisms.

cs.CG↗

arXiv2023

Hausdorff and Gromov-Hausdorff stable subsets of the medial axis

In this paper we introduce a pruning of the medial axis called the $(λ,α)$-medial axis ($\textrm{ax}_λ^α$). 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 ($d_H$) sense then the $(λ,α)$-medial axes of $K$ and $K'$ are close as metric spaces, that is the Gromov-Hausdorff distance ($d_{GH}$) between the two is $\frac{1}{4}$-H{ö}lder in the sense that $d_{GH} (\textrm{ax}_λ^α(K),\textrm{ax}_λ^α(K')) \lesssim d_H(K,K')^{1/4}$. The Hausdorff distance between the two medial axes is also bounded, by $d_{H} (\textrm{ax}_λ^α(K),\textrm{ax}_λ^α(K')) \lesssim d_H(K,K')^{1/2}$. 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.

cs.CG↗

arXiv2023

Translation of "Simplizialzerlegungen von Beschrankter Flachheit'' by Hans Freudenthal, Annals of Mathematics, Second Series, Volume 43, Number 3, July 1942, Pages 580-583

Translation 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.''

math.CO↗

arXiv2021

The Density Fingerprint of a Periodic Point Set

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. 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.

cs.CG↗

arXiv2018

Local criteria for triangulation of manifolds

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 presented in local coordinate charts for M, and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on M. No Delaunay property of A is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily verified in such a setting, they are expected to be of general use.

cs.CG↗

arXiv2017

Optimal Triangulation of Saddle Surfaces

We consider the piecewise linear approximation of saddle functions of the form $f(x,y)=ax^2-by^2$ under the L-infinity error norm. We show that interpolating approximations are not optimal. One can get slightly smaller errors by allowing the vertices of the approximation to move away from the graph of the function.

math.MG↗

arXiv2017

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 $\mathbb{R}^2$ 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 $\cal P$ in a domain $Ω$ equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of $\cal P$ 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 $Ω$ 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.

cs.CG↗

arXiv2017

The Topology of the Cosmic Web in Terms of Persistent Betti Numbers

We introduce a multiscale topological description of the Megaparsec weblike 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 weblike 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 weblike 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.

astro-ph.CO↗