@inproceedings{8703, abstract = {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. }, author = {Osang, Georg F and Rouxel-Labbé, Mael and Teillaud, Monique}, booktitle = {28th Annual European Symposium on Algorithms}, isbn = {9783959771627}, issn = {18688969}, location = {Virtual, Online; Pisa, Italy}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Generalizing CGAL periodic Delaunay triangulations}}, doi = {10.4230/LIPIcs.ESA.2020.75}, volume = {173}, 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{9157, abstract = {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.}, author = {Akopyan, Arseniy and Edelsbrunner, Herbert}, issn = {2544-7297}, journal = {Computational and Mathematical Biophysics}, number = {1}, pages = {51--67}, publisher = {De Gruyter}, title = {{The weighted mean curvature derivative of a space-filling diagram}}, doi = {10.1515/cmb-2020-0100}, volume = {8}, year = {2020}, } @article{9156, abstract = {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.}, author = {Akopyan, Arseniy and Edelsbrunner, Herbert}, issn = {2544-7297}, journal = {Computational and Mathematical Biophysics}, number = {1}, pages = {74--88}, publisher = {De Gruyter}, title = {{The weighted Gaussian curvature derivative of a space-filling diagram}}, doi = {10.1515/cmb-2020-0101}, volume = {8}, year = {2020}, } @article{15064, abstract = {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.}, author = {Bauer, U. and Edelsbrunner, Herbert and Jablonski, Grzegorz and Mrozek, M.}, issn = {2367-1734}, journal = {Journal of Applied and Computational Topology}, number = {4}, pages = {455--480}, publisher = {Springer Nature}, title = {{Čech-Delaunay gradient flow and homology inference for self-maps}}, doi = {10.1007/s41468-020-00058-8}, volume = {4}, year = {2020}, } @article{6515, abstract = {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.}, author = {Dyer, Ramsay and Vegter, Gert and Wintraecken, Mathijs}, issn = {1920-180X}, journal = {Journal of Computational Geometry }, number = {1}, pages = {223–256}, publisher = {Carleton University}, title = {{Simplices modelled on spaces of constant curvature}}, doi = {10.20382/jocg.v10i1a9}, volume = {10}, year = {2019}, } @inproceedings{6628, abstract = {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.}, author = {Vegter, Gert and Wintraecken, Mathijs}, booktitle = {The 31st Canadian Conference in Computational Geometry}, location = {Edmonton, Canada}, pages = {275--279}, title = {{The extrinsic nature of the Hausdorff distance of optimal triangulations of manifolds}}, year = {2019}, } @inproceedings{6648, abstract = {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.}, author = {Edelsbrunner, Herbert and Virk, Ziga and Wagner, Hubert}, booktitle = {35th International Symposium on Computational Geometry}, isbn = {9783959771047}, location = {Portland, OR, United States}, pages = {31:1--31:14}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Topological data analysis in information space}}, doi = {10.4230/LIPICS.SOCG.2019.31}, volume = {129}, year = {2019}, } @inproceedings{6989, abstract = {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. }, author = {Aichholzer, Oswin and Akitaya, Hugo A and Cheung, Kenneth C and Demaine, Erik D and Demaine, Martin L and Fekete, Sandor P and Kleist, Linda and Kostitsyna, Irina and Löffler, Maarten and Masárová, Zuzana and Mundilova, Klara and Schmidt, Christiane}, booktitle = {Proceedings of the 31st Canadian Conference on Computational Geometry}, location = {Edmonton, Canada}, pages = {164--170}, publisher = {Canadian Conference on Computational Geometry}, title = {{Folding polyominoes with holes into a cube}}, year = {2019}, } @article{6671, abstract = {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.}, author = {Boissonnat, Jean-Daniel and Lieutier, André and Wintraecken, Mathijs}, issn = {2367-1734}, journal = {Journal of Applied and Computational Topology}, number = {1-2}, pages = {29–58}, publisher = {Springer Nature}, title = {{The reach, metric distortion, geodesic convexity and the variation of tangent spaces}}, doi = {10.1007/s41468-019-00029-8}, volume = {3}, year = {2019}, }