@inbook{21056, abstract = {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 fundamental areas in computational topology, namely topological data analysis (TDA) and knot theory. Given a function from a topological space to ℝ, TDA provides tools to simplify and study the importance of topological features: in particular, the 𝑙^𝑡⁢ℎ-dimensional persistence diagram encodes the topological changes (or 𝑙-homology) in the sublevel set as the function value increases into 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 tracks the evolution of points in the persistence diagram as the function changes. 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. Recent work has studied monodromy in the directional persistent homology transform, demonstrating some interesting connections between an input shape and monodromy in the persistent homology transform for 0-dimensional homology embedded in ℝ^2. In this work, given a link and a value 𝑙, we construct a topological space (based on the given link) and periodic family of functions on this space (based on the Euclidean distance function), such that the closed 𝑙-vineyard contains this link. This shows that vineyards are topologically as rich as one could possibly hope, suggesting many future directions of work. Importantly, it has at least two immediate consequences we explicitly point out: 1. Monodromy of any periodicity can occur in a 𝑙-vineyard for any 𝑙. This answers a variant of a question by Arya and collaborators. To exhibit this as a consequence of our first main result we also reformulate monodromy in a more geometric way, which may be of interest in itself. 2. Topologically distinguishing closed vineyards is likely to be difficult (from a complexity theory as well as from a practical perspective) because of the difficulty of knot and link recognition, which have strong connections to many NP-hard problems.}, author = {Chambers, Erin W. and Fillmore, Christopher D and Stephenson, Elizabeth R and Wintraecken, Mathijs}, booktitle = {Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms}, editor = {Green Larsen, Kasper and Saha, Barna}, pages = {6240--6263}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Braiding Vineyards}}, doi = {10.1137/1.9781611978971.225}, year = {2026}, } @unpublished{21051, abstract = {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.}, author = { Chambers, Erin and Fillmore, Christopher D and Stephenson, Elizabeth R and Wintraecken, Mathijs}, booktitle = {arXiv}, title = {{Braiding vineyards}}, doi = {10.48550/ARXIV.2504.11203}, year = {2026}, } @article{20260, abstract = {The medial axis of a set consists of the points in the ambient space without a unique closest point in the original set. Since its introduction, the medial axis has been used extensively in many applications as a method of computing a skeleton topologically equivalent to the original set. Unfortunately, one limiting factor in the use of the medial axis of a smooth manifold is that it is not necessarily topologically stable under small perturbations of the manifold. To counter these instabilities, various prunings of the medial axis have been proposed in the computational geometry community. Here, we examine one type of pruning, called burning. Because of the good experimental results it was hoped that the burning method of simplifying the medial axis would be stable. In this work, we show a simple example that dashes such hopes. Based on Bing’s house with two rooms, we demonstrate an isotopy of a shape where the medial axis goes from collapsible to non-collapsible. More precisely, we consider the standard deformation retract from the closed ball to Bing’s house with two rooms, but stop just short of the point where Bing’s house becomes two dimensional. This way we obtain an isotopy from the 3-ball to a thickened version of Bing’s house. Under this isotopy, the medial axis goes from collapsible to non-collapsible. We stress that this isotopy can be made generic, in the sense of singularity theory, as developed by Arnol’d and Thom.}, author = {Chambers, Erin Wolf and Fillmore, Christopher D and Stephenson, Elizabeth R and Wintraecken, Mathijs}, issn = {2730-9657}, journal = {La Matematica}, pages = {811--828}, publisher = {Springer Nature}, title = {{Burning or collapsing the medial axis is unstable}}, doi = {10.1007/s44007-025-00170-0}, volume = {4}, year = {2025}, } @inproceedings{18097, abstract = {In our companion paper "Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds" we gave optimal bounds (in terms of the two one-sided Hausdorff distances) on a sample P of an input shape 𝒮 (either manifold or general set with positive reach) such that one can infer the homotopy of 𝒮 from the union of balls with some radius centred at P, both in Euclidean space and in a Riemannian manifold of bounded curvature. The construction showing the optimality of the bounds is not straightforward. The purpose of this video is to visualize and thus elucidate said construction in the Euclidean setting.}, author = {Attali, Dominique and Kourimska, Hana and Fillmore, Christopher D and Ghosh, Ishika and Lieutier, Andre and Stephenson, Elizabeth R and Wintraecken, Mathijs}, booktitle = {40th International Symposium on Computational Geometry}, isbn = {9783959773164}, location = {Athens, Greece}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The ultimate frontier: An optimality construction for homotopy inference (media exposition)}}, doi = {10.4230/LIPIcs.SoCG.2024.87}, volume = {293}, year = {2024}, } @inproceedings{17144, abstract = {We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let 𝒮 ⊆ ℝ^d be a fixed closed set that contains a bounding sphere. That is, the bounding sphere is part of the set 𝒮. Consider the space of C^{1,1} diffeomorphisms of ℝ^d to itself, which keep the bounding sphere invariant. The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of ℝ^d (endowed with the Hausdorff distance), mapping a diffeomorphism F to the closure of the medial axis of F(𝒮), is Lipschitz. This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of C² manifolds under C² ambient diffeomorphisms.}, author = {Kourimska, Hana and Lieutier, André and Wintraecken, Mathijs}, booktitle = {40th International Symposium on Computational Geometry}, isbn = {9783959773164}, issn = {1868-8969}, location = {Athens, Greece}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms}}, doi = {10.4230/LIPIcs.SoCG.2024.69}, volume = {293}, year = {2024}, } @inproceedings{17170, abstract = {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 C² manifolds with positive reach embedded in ℝ^d. We extend their results in the following ways: - As the ambient space we consider both ℝ^d and Riemannian manifolds with lower bounded sectional curvature. - In both types of ambient spaces, we study sets of positive reach - a significantly more general setting than C² manifolds - as well as general manifolds of positive reach. - The sample P of a set (or a manifold) 𝒮 of positive reach may be noisy. We work with two one-sided Hausdorff distances - ε and δ - between P and 𝒮. We provide tight bounds in terms of ε 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 𝒮. We exhibit their tightness by an explicit construction. We carefully distinguish the roles of δ and ε. This is not only essential to achieve tight bounds, but also sensible in practical situations, since it allows one to adapt the bound according to sample density and the amount of noise present in the sample separately.}, author = {Attali, Dominique and Kourimska, Hana and Fillmore, Christopher D and Ghosh, Ishika and Lieutier, André and Stephenson, Elizabeth R and Wintraecken, Mathijs}, booktitle = {40th International Symposium on Computational Geometry}, isbn = {9783959773164}, issn = {1868-8969}, location = {Athens, Greece}, pages = {11:1--11:19}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds}}, doi = {10.4230/LIPIcs.SoCG.2024.11}, volume = {293}, year = {2024}, } @article{17190, abstract = {For a locally finite set, 𝐴⊆ℝ𝑑 , the 𝑘 th Brillouin zone of 𝑎∈𝐴 is the region of points 𝑥∈ℝ𝑑 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, and together they 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 ℝ2 , and the convergence of the maximum volume of a chamber to zero for the integer lattice.}, author = {Edelsbrunner, Herbert and Garber, Alexey and Ghafaris, Mohadese and Heiss, Teresa and Saghafiant, Morteza and Wintraecken, Mathijs}, issn = {0895-4801}, journal = {SIAM Journal on Discrete Mathematics}, number = {2}, pages = {1784--1807}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Brillouin zones of integer lattices and their perturbations}}, doi = {10.1137/22M1489071}, volume = {38}, year = {2024}, } @article{12287, abstract = {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.}, author = {Boissonnat, Jean-Daniel and Dyer, Ramsay and Ghosh, Arijit and Wintraecken, Mathijs}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, keywords = {Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology, Theoretical Computer Science}, pages = {156--191}, publisher = {Springer Nature}, title = {{Local criteria for triangulating general manifolds}}, doi = {10.1007/s00454-022-00431-7}, volume = {69}, year = {2023}, } @article{12763, abstract = {Kleinjohann (Archiv der Mathematik 35(1):574–582, 1980; Mathematische Zeitschrift 176(3), 327–344, 1981) and Bangert (Archiv der Mathematik 38(1):54–57, 1982) extended the reach rch(S) from subsets S of Euclidean space to the reach rchM(S) of subsets S of Riemannian manifolds M, where M is smooth (we’ll assume at least C3). Bangert showed that sets of positive reach in Euclidean space and Riemannian manifolds are very similar. In this paper we introduce a slight variant of Kleinjohann’s and Bangert’s extension and quantify the similarity between sets of positive reach in Euclidean space and Riemannian manifolds in a new way: Given p∈M and q∈S, we bound the local feature size (a local version of the reach) of its lifting to the tangent space via the inverse exponential map (exp−1p(S)) at q, assuming that rchM(S) and the geodesic distance dM(p,q) are bounded. These bounds are motivated by the importance of the reach and local feature size to manifold learning, topological inference, and triangulating manifolds and the fact that intrinsic approaches circumvent the curse of dimensionality.}, author = {Boissonnat, Jean Daniel and Wintraecken, Mathijs}, issn = {2367-1734}, journal = {Journal of Applied and Computational Topology}, pages = {619--641}, publisher = {Springer Nature}, title = {{The reach of subsets of manifolds}}, doi = {10.1007/s41468-023-00116-x}, volume = {7}, year = {2023}, } @article{12960, abstract = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e., submanifolds of Rd defined as the zero set of some multivariate multivalued smooth function f:Rd→Rd−n, where n is the intrinsic dimension of the manifold. A natural way to approximate a smooth isomanifold M=f−1(0) is to consider its piecewise linear (PL) approximation M^ based on a triangulation T of the ambient space Rd. In this paper, we describe a simple algorithm to trace isomanifolds from a given starting point. The algorithm works for arbitrary dimensions n and d, and any precision D. Our main result is that, when f (or M) has bounded complexity, the complexity of the algorithm is polynomial in d and δ=1/D (and unavoidably exponential in n). Since it is known that for δ=Ω(d2.5), M^ is O(D2)-close and isotopic to M , our algorithm produces a faithful PL-approximation of isomanifolds of bounded complexity in time polynomial in d. Combining this algorithm with dimensionality reduction techniques, the dependency on d in the size of M^ can be completely removed with high probability. We also show that the algorithm can handle isomanifolds with boundary and, more generally, isostratifolds. The algorithm for isomanifolds with boundary has been implemented and experimental results are reported, showing that it is practical and can handle cases that are far ahead of the state-of-the-art. }, author = {Boissonnat, Jean Daniel and Kachanovich, Siargey and Wintraecken, Mathijs}, issn = {1095-7111}, journal = {SIAM Journal on Computing}, number = {2}, pages = {452--486}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Tracing isomanifolds in Rd in time polynomial in d using Coxeter–Freudenthal–Kuhn triangulations}}, doi = {10.1137/21M1412918}, volume = {52}, year = {2023}, } @inproceedings{13048, abstract = {In this paper we introduce a pruning of the medial axis called the (λ,α)-medial axis (axλα). 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 1/4-Hölder in the sense that dGH (axλα(K),axλα(K′)) ≲ dH(K,K′)1/4. The Hausdorff distance between the two medial axes is also bounded, by dH (axλα(K),λα(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.}, author = {Lieutier, André and Wintraecken, Mathijs}, booktitle = {Proceedings of the 55th Annual ACM Symposium on Theory of Computing}, isbn = {9781450399135}, location = {Orlando, FL, United States}, pages = {1768--1776}, publisher = {Association for Computing Machinery}, title = {{Hausdorff and Gromov-Hausdorff stable subsets of the medial axis}}, doi = {10.1145/3564246.3585113}, year = {2023}, } @inproceedings{11428, abstract = {The medial axis of a set consists of the points in the ambient space without a unique closest point on the original set. Since its introduction, the medial axis has been used extensively in many applications as a method of computing a topologically equivalent skeleton. Unfortunately, one limiting factor in the use of the medial axis of a smooth manifold is that it is not necessarily topologically stable under small perturbations of the manifold. To counter these instabilities various prunings of the medial axis have been proposed. Here, we examine one type of pruning, called burning. Because of the good experimental results, it was hoped that the burning method of simplifying the medial axis would be stable. In this work we show a simple example that dashes such hopes based on Bing’s house with two rooms, demonstrating an isotopy of a shape where the medial axis goes from collapsible to non-collapsible.}, author = {Chambers, Erin and Fillmore, Christopher D and Stephenson, Elizabeth R and Wintraecken, Mathijs}, booktitle = {38th International Symposium on Computational Geometry}, editor = {Goaoc, Xavier and Kerber, Michael}, isbn = {978-3-95977-227-3}, issn = {1868-8969}, location = {Berlin, Germany}, pages = {66:1--66:9}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{A cautionary tale: Burning the medial axis is unstable}}, doi = {10.4230/LIPIcs.SoCG.2022.66}, volume = {224}, year = {2022}, } @article{9649, abstract = {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.}, author = {Boissonnat, Jean-Daniel and Wintraecken, Mathijs}, issn = {1615-3383}, journal = {Foundations of Computational Mathematics }, pages = {967--1012}, publisher = {Springer Nature}, title = {{The topological correctness of PL approximations of isomanifolds}}, doi = {10.1007/s10208-021-09520-0}, volume = {22}, year = {2022}, } @article{8248, abstract = {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.}, author = {Boissonnat, Jean-Daniel and Dyer, Ramsay and Ghosh, Arijit and Lieutier, Andre and Wintraecken, Mathijs}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {666--686}, publisher = {Springer Nature}, title = {{Local conditions for triangulating submanifolds of Euclidean space}}, doi = {10.1007/s00454-020-00233-9}, volume = {66}, year = {2021}, } @article{8940, abstract = {We quantise Whitney’s construction to prove the existence of a triangulation for any C^2 manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric.}, author = {Boissonnat, Jean-Daniel and Kachanovich, Siargey and Wintraecken, Mathijs}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, keywords = {Theoretical Computer Science, Computational Theory and Mathematics, Geometry and Topology, Discrete Mathematics and Combinatorics}, number = {1}, pages = {386--434}, publisher = {Springer Nature}, title = {{Triangulating submanifolds: An elementary and quantified version of Whitney’s method}}, doi = {10.1007/s00454-020-00250-8}, volume = {66}, year = {2021}, } @inproceedings{9441, abstract = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. submanifolds of ℝ^d defined as the zero set of some multivariate multivalued smooth function f: ℝ^d → ℝ^{d-n}, where n is the intrinsic dimension of the manifold. A natural way to approximate a smooth isomanifold M is to consider its Piecewise-Linear (PL) approximation M̂ based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we describe a simple algorithm to trace isomanifolds from a given starting point. The algorithm works for arbitrary dimensions n and d, and any precision D. Our main result is that, when f (or M) has bounded complexity, the complexity of the algorithm is polynomial in d and δ = 1/D (and unavoidably exponential in n). Since it is known that for δ = Ω (d^{2.5}), M̂ is O(D²)-close and isotopic to M, our algorithm produces a faithful PL-approximation of isomanifolds of bounded complexity in time polynomial in d. Combining this algorithm with dimensionality reduction techniques, the dependency on d in the size of M̂ can be completely removed with high probability. We also show that the algorithm can handle isomanifolds with boundary and, more generally, isostratifolds. The algorithm for isomanifolds with boundary has been implemented and experimental results are reported, showing that it is practical and can handle cases that are far ahead of the state-of-the-art. }, author = {Boissonnat, Jean-Daniel and Kachanovich, Siargey and Wintraecken, Mathijs}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, isbn = {978-3-95977-184-9}, issn = {1868-8969}, location = {Virtual}, pages = {17:1--17:16}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Tracing isomanifolds in Rd in time polynomial in d using Coxeter-Freudenthal-Kuhn triangulations}}, doi = {10.4230/LIPIcs.SoCG.2021.17}, volume = {189}, year = {2021}, } @inproceedings{9345, abstract = {Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functionsthat facilitates the efficient search for new materials and material properties. We prove invarianceunder isometries, continuity, and completeness in the generic case, which are necessary featuresfor the reliable comparison of crystals. The proof of continuity integrates methods from discretegeometry and lattice theory, while the proof of generic completeness combines techniques fromgeometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and relatedinclusion-exclusion formulae. We have implemented the algorithm and describe its application tocrystal structure prediction.}, author = {Edelsbrunner, Herbert and Heiss, Teresa and Kurlin , Vitaliy and Smith, Philip and Wintraecken, Mathijs}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, issn = {1868-8969}, location = {Virtual}, pages = {32:1--32:16}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The density fingerprint of a periodic point set}}, doi = {10.4230/LIPIcs.SoCG.2021.32}, volume = {189}, year = {2021}, } @inproceedings{7952, abstract = {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. }, author = {Boissonnat, Jean-Daniel and Wintraecken, Mathijs}, booktitle = {36th International Symposium on Computational Geometry}, isbn = {978-3-95977-143-6}, issn = {1868-8969}, location = {Zürich, Switzerland}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The topological correctness of PL-approximations of isomanifolds}}, doi = {10.4230/LIPIcs.SoCG.2020.20}, volume = {164}, year = {2020}, } @article{8163, abstract = {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.}, author = {Vegter, Gert and Wintraecken, Mathijs}, issn = {1588-2896}, journal = {Studia Scientiarum Mathematicarum Hungarica}, number = {2}, pages = {193--199}, publisher = {Akadémiai Kiadó}, title = {{Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes}}, doi = {10.1556/012.2020.57.2.1454}, volume = {57}, year = {2020}, } @article{7567, abstract = {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.}, author = {Choudhary, Aruni and Kachanovich, Siargey and Wintraecken, Mathijs}, issn = {1661-8289}, journal = {Mathematics in Computer Science}, pages = {141--176}, publisher = {Springer Nature}, title = {{Coxeter triangulations have good quality}}, doi = {10.1007/s11786-020-00461-5}, volume = {14}, year = {2020}, }