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 - CHAP AB - We study the Gromov waist in the sense of t-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov’s original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no t-neighborhood waist property, including a rather wide class of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2. We use a simpler form of Gromov’s pancake argument to produce some estimates of t-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. In the appendix of this paper we provide for reader’s convenience a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian measures. AU - Akopyan, Arseniy AU - Karasev, Roman ED - Klartag, Bo'az ED - Milman, Emanuel ID - 74 SN - 00758434 T2 - Geometric Aspects of Functional Analysis TI - Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures VL - 2256 ER - TY - JOUR AB - Slicing a Voronoi tessellation in ${R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as a power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in ${R}^n$, we study the expected number of simplices in the $k$-dimensional weighted Delaunay mosaic as well as the expected number of intervals of the Morse function, both as functions of a radius threshold. As a by-product, we obtain a new proof for the expected number of connected components (clumps) in a line section of a circular Boolean model in ${R}^n$. AU - Edelsbrunner, Herbert AU - Nikitenko, Anton ID - 7554 IS - 4 JF - Theory of Probability and its Applications SN - 0040585X TI - Weighted Poisson–Delaunay mosaics VL - 64 ER - TY - JOUR AB - Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups. AU - Edelsbrunner, Herbert AU - Ölsböck, Katharina ID - 7666 JF - Discrete and Computational Geometry SN - 01795376 TI - Tri-partitions and bases of an ordered complex VL - 64 ER - TY - JOUR AB - A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair of them is not connected by any edge (n→∞). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that almost all string graphs on n vertices are intersection graphs of plane convex sets. AU - Pach, János AU - Reed, Bruce AU - Yuditsky, Yelena ID - 7962 IS - 4 JF - Discrete and Computational Geometry SN - 01795376 TI - Almost all string graphs are intersection graphs of plane convex sets VL - 63 ER - TY - JOUR AU - Pach, János ID - 8323 JF - Discrete and Computational Geometry SN - 01795376 TI - A farewell to Ricky Pollack VL - 64 ER - TY - CONF AB - We evaluate the usefulness of persistent homology in the analysis of heart rate variability. In our approach we extract several topological descriptors characterising datasets of RR-intervals, which are later used in classical machine learning algorithms. By this method we are able to differentiate the group of patients with the history of transient ischemic attack and the group of hypertensive patients. AU - Graff, Grzegorz AU - Graff, Beata AU - Jablonski, Grzegorz AU - Narkiewicz, Krzysztof ID - 8580 SN - 9781728157511 T2 - 11th Conference of the European Study Group on Cardiovascular Oscillations: Computation and Modelling in Physiology: New Challenges and Opportunities, TI - The application of persistent homology in the analysis of heart rate variability ER - TY - JOUR AB - In this paper we find a tight estimate for Gromov’s waist of the balls in spaces of constant curvature, deduce the estimates for the balls in Riemannian manifolds with upper bounds on the curvature (CAT(ϰ)-spaces), and establish similar result for normed spaces. AU - Akopyan, Arseniy AU - Karasev, Roman ID - 10867 IS - 3 JF - International Mathematics Research Notices KW - General Mathematics SN - 1073-7928 TI - Waist of balls in hyperbolic and spherical spaces VL - 2020 ER - TY - THES AB - Many methods for the reconstruction of shapes from sets of points produce ordered simplicial complexes, which are collections of vertices, edges, triangles, and their higher-dimensional analogues, called simplices, in which every simplex gets assigned a real value measuring its size. This thesis studies ordered simplicial complexes, with a focus on their topology, which reflects the connectedness of the represented shapes and the presence of holes. We are interested both in understanding better the structure of these complexes, as well as in developing algorithms for applications. For the Delaunay triangulation, the most popular measure for a simplex is the radius of the smallest empty circumsphere. Based on it, we revisit Alpha and Wrap complexes and experimentally determine their probabilistic properties for random data. Also, we prove the existence of tri-partitions, propose algorithms to open and close holes, and extend the concepts from Euclidean to Bregman geometries. AU - Ölsböck, Katharina ID - 7460 KW - shape reconstruction KW - hole manipulation KW - ordered complexes KW - Alpha complex KW - Wrap complex KW - computational topology KW - Bregman geometry SN - 2663-337X TI - The hole system of triangulated shapes ER - TY - THES AB - This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph. For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton. In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars. AU - Masárová, Zuzana ID - 7944 KW - reconfiguration KW - reconfiguration graph KW - triangulations KW - flip KW - constrained triangulations KW - shellability KW - piecewise-linear balls KW - token swapping KW - trees KW - coloured weighted token swapping SN - 2663-337X TI - Reconfiguration problems ER - TY - CONF AB - Even though Delaunay originally introduced his famous triangulations in the case of infinite point sets with translational periodicity, a software that computes such triangulations in the general case is not yet available, to the best of our knowledge. Combining and generalizing previous work, we present a practical algorithm for computing such triangulations. The algorithm has been implemented and experiments show that its performance is as good as the one of the CGAL package, which is restricted to cubic periodicity. AU - Osang, Georg F AU - Rouxel-Labbé, Mael AU - Teillaud, Monique ID - 8703 SN - 18688969 T2 - 28th Annual European Symposium on Algorithms TI - Generalizing CGAL periodic Delaunay triangulations VL - 173 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 - Representing an atom by a solid sphere in 3-dimensional Euclidean space, we get the space-filling diagram of a molecule by taking the union. Molecular dynamics simulates its motion subject to bonds and other forces, including the solvation free energy. The morphometric approach [12, 17] writes the latter as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted mean curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [3], and the weighted Gaussian curvature [1], this yields the derivative of the morphometric expression of the solvation free energy. AU - Akopyan, Arseniy AU - Edelsbrunner, Herbert ID - 9157 IS - 1 JF - Computational and Mathematical Biophysics SN - 2544-7297 TI - The weighted mean curvature derivative of a space-filling diagram VL - 8 ER - TY - JOUR AB - The morphometric approach [11, 14] writes the solvation free energy as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted Gaussian curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [4], and the weighted mean curvature in [1], this yields the derivative of the morphometric expression of solvation free energy. AU - Akopyan, Arseniy AU - Edelsbrunner, Herbert ID - 9156 IS - 1 JF - Computational and Mathematical Biophysics SN - 2544-7297 TI - The weighted Gaussian curvature derivative of a space-filling diagram VL - 8 ER - TY - JOUR AB - We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior. We establish a sampling theorem to recover the eigenspaces of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for Čech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive Čech complexes to the computationally efficient Delaunay triangulations. The fast chain map algorithm has applications beyond dynamical systems. AU - Bauer, U. AU - Edelsbrunner, Herbert AU - Jablonski, Grzegorz AU - Mrozek, M. ID - 15064 IS - 4 JF - Journal of Applied and Computational Topology SN - 2367-1726 TI - Čech-Delaunay gradient flow and homology inference for self-maps VL - 4 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 - CONF AB - Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context. AU - Edelsbrunner, Herbert AU - Virk, Ziga AU - Wagner, Hubert ID - 6648 SN - 9783959771047 T2 - 35th International Symposium on Computational Geometry TI - Topological data analysis in information space VL - 129 ER - TY - CONF AB - When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with hole(s) to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special simple holes guarantee foldability. AU - Aichholzer, Oswin AU - Akitaya, Hugo A AU - Cheung, Kenneth C AU - Demaine, Erik D AU - Demaine, Martin L AU - Fekete, Sandor P AU - Kleist, Linda AU - Kostitsyna, Irina AU - Löffler, Maarten AU - Masárová, Zuzana AU - Mundilova, Klara AU - Schmidt, Christiane ID - 6989 T2 - Proceedings of the 31st Canadian Conference on Computational Geometry TI - Folding polyominoes with holes into a cube 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 -