@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}, } @article{8338, abstract = {Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines that are diagonally related to lines of curvature is proved theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory.}, author = {Akopyan, Arseniy and Bobenko, Alexander I. and Schief, Wolfgang K. and Techter, Jan}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {938--976}, publisher = {Springer Nature}, title = {{On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs}}, doi = {10.1007/s00454-020-00240-w}, volume = {66}, year = {2021}, } @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{7905, abstract = {We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1355–1370, ACM, New York, 2012), and the cohomology stratification algorithm given in Nanda (Found. Comput. Math. 20(2), 195–222, 2020). Additionally, we give examples of stratifications based on the geometric techniques of Breiding et al. (Rev. Mat. Complut. 31(3), 545–593, 2018), illustrating how the sheaf-theoretic approach can be used to study stratifications from both topological and geometric perspectives. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.}, author = {Brown, Adam and Wang, Bei}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {1166--1198}, publisher = {Springer Nature}, title = {{Sheaf-theoretic stratification learning from geometric and topological perspectives}}, doi = {10.1007/s00454-020-00206-y}, volume = {65}, year = {2021}, } @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}, } @inproceedings{8135, abstract = {Discrete Morse theory has recently lead to new developments in the theory of random geometric complexes. This article surveys the methods and results obtained with this new approach, and discusses some of its shortcomings. It uses simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics.}, author = {Edelsbrunner, Herbert and Nikitenko, Anton and Ölsböck, Katharina and Synak, Peter}, booktitle = {Topological Data Analysis}, isbn = {9783030434076}, issn = {21978549}, pages = {181--218}, publisher = {Springer Nature}, title = {{Radius functions on Poisson–Delaunay mosaics and related complexes experimentally}}, doi = {10.1007/978-3-030-43408-3_8}, volume = {15}, year = {2020}, } @article{9249, abstract = {Rhombic dodecahedron is a space filling polyhedron which represents the close packing of spheres in 3D space and the Voronoi structures of the face centered cubic (FCC) lattice. In this paper, we describe a new coordinate system where every 3-integer coordinates grid point corresponds to a rhombic dodecahedron centroid. In order to illustrate the interest of the new coordinate system, we propose the characterization of 3D digital plane with its topological features, such as the interrelation between the thickness of the digital plane and the separability constraint we aim to obtain. We also present the characterization of 3D digital lines and study it as the intersection of multiple digital planes. Characterization of 3D digital sphere with relevant topological features is proposed as well along with the 48-symmetry appearing in the new coordinate system.}, author = {Biswas, Ranita and Largeteau-Skapin, Gaëlle and Zrour, Rita and Andres, Eric}, issn = {2353-3390}, journal = {Mathematical Morphology - Theory and Applications}, number = {1}, pages = {143--158}, publisher = {De Gruyter}, title = {{Digital objects in rhombic dodecahedron grid}}, doi = {10.1515/mathm-2020-0106}, volume = {4}, year = {2020}, } @inproceedings{9299, abstract = {We call a multigraph non-homotopic if it can be drawn in the plane in such a way that no two edges connecting the same pair of vertices can be continuously transformed into each other without passing through a vertex, and no loop can be shrunk to its end-vertex in the same way. It is easy to see that a non-homotopic multigraph on n>1 vertices can have arbitrarily many edges. We prove that the number of crossings between the edges of a non-homotopic multigraph with n vertices and m>4n edges is larger than cm2n for some constant c>0 , and that this bound is tight up to a polylogarithmic factor. We also show that the lower bound is not asymptotically sharp as n is fixed and m⟶∞ .}, author = {Pach, János and Tardos, Gábor and Tóth, Géza}, booktitle = {28th International Symposium on Graph Drawing and Network Visualization}, isbn = {9783030687656}, issn = {1611-3349}, location = {Virtual, Online}, pages = {359--371}, publisher = {Springer Nature}, title = {{Crossings between non-homotopic edges}}, doi = {10.1007/978-3-030-68766-3_28}, volume = {12590}, year = {2020}, } @article{9630, 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}, issn = {1920180X}, journal = {Journal of Computational Geometry}, number = {2}, pages = {162--182}, publisher = {Carleton University}, title = {{Topological data analysis in information space}}, doi = {10.20382/jocg.v11i2a7}, volume = {11}, year = {2020}, } @article{8538, abstract = {We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.}, author = {Akopyan, Arseniy and Schwartz, Richard and Tabachnikov, Serge}, issn = {2199-6768}, journal = {European Journal of Mathematics}, publisher = {Springer Nature}, title = {{Billiards in ellipses revisited}}, doi = {10.1007/s40879-020-00426-9}, year = {2020}, }