@unpublished{7568,
abstract = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e.manifolds defined as the zero set of some multivariate multivalued functionf:Rd→Rd−n.A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear(PL) approximation based on a triangulationTof the ambient spaceRd. In this paper, we giveconditions under which the PL-approximation of an isomanifold is topologically equivalent to theisomanifold. The conditions can always be met by taking a sufficiently fine triangulationT.},
author = {Boissonnat, Jean-Daniel and Wintraecken, Mathijs},
booktitle = {EUROCG 2020},
pages = {8},
title = {{The topological correctness of the PL-approximation of isomanifolds}},
year = {2020},
}
@article{7791,
abstract = {Extending a result of Milena Radnovic and Serge Tabachnikov, we establish conditionsfor two different non-symmetric norms to define the same billiard reflection law.},
author = {Akopyan, Arseniy and Karasev, Roman},
issn = {21996768},
journal = {European Journal of Mathematics},
publisher = {Springer Nature},
title = {{When different norms lead to same billiard trajectories?}},
doi = {10.1007/s40879-020-00405-0},
year = {2020},
}
@article{7666,
abstract = {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.},
author = {Edelsbrunner, Herbert and Ölsböck, Katharina},
issn = {14320444},
journal = {Discrete and Computational Geometry},
publisher = {Springer Nature},
title = {{Tri-partitions and bases of an ordered complex}},
doi = {10.1007/s00454-020-00188-x},
year = {2020},
}
@phdthesis{7460,
abstract = {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.},
author = {Ölsböck, Katharina},
issn = {2663-337X},
keyword = {shape reconstruction, hole manipulation, ordered complexes, Alpha complex, Wrap complex, computational topology, Bregman geometry},
pages = {155},
publisher = {IST Austria},
title = {{The hole system of triangulated shapes}},
doi = {10.15479/AT:ISTA:7460},
year = {2020},
}
@article{7554,
abstract = {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 ${R}^n$.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton},
issn = {10957219},
journal = {Theory of Probability and its Applications},
number = {4},
pages = {595--614},
publisher = {SIAM},
title = {{Weighted Poisson–Delaunay mosaics}},
doi = {10.1137/S0040585X97T989726},
volume = {64},
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},
publisher = {Springer Nature},
title = {{Coxeter triangulations have good quality}},
doi = {10.1007/s11786-020-00461-5},
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},
}
@article{6793,
abstract = {The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here, we give a simple geometric proof of Regge symmetries in Euclidean, spherical, and hyperbolic geometry.},
author = {Akopyan, Arseniy and Izmestiev, Ivan},
issn = {14692120},
journal = {Bulletin of the London Mathematical Society},
number = {5},
pages = {765--775},
publisher = {London Mathematical Society},
title = {{The Regge symmetry, confocal conics, and the Schläfli formula}},
doi = {10.1112/blms.12276},
volume = {51},
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},
}