@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 = {14320444},
journal = {Discrete and Computational Geometry},
publisher = {Springer Nature},
title = {{On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs}},
doi = {10.1007/s00454-020-00240-w},
year = {2020},
}
@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{6050,
abstract = {We answer a question of David Hilbert: given two circles it is not possible in general to construct their centers using only a straightedge. On the other hand, we give infinitely many families of pairs of circles for which such construction is possible. },
author = {Akopyan, Arseniy and Fedorov, Roman},
journal = {Proceedings of the American Mathematical Society},
pages = {91--102},
publisher = {AMS},
title = {{Two circles and only a straightedge}},
doi = {10.1090/proc/14240},
volume = {147},
year = {2019},
}
@article{5678,
abstract = {The order-k Voronoi tessellation of a locally finite set 𝑋⊆ℝ𝑛 decomposes ℝ𝑛 into convex domains whose points have the same k nearest neighbors in X. Assuming X is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a relaxed version of discrete Morse theory and generalize by counting only faces, for which the k nearest points in X are within a given distance threshold.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton},
issn = {14320444},
journal = {Discrete and Computational Geometry},
number = {4},
pages = {865–878},
publisher = {Springer},
title = {{Poisson–Delaunay Mosaics of Order k}},
doi = {10.1007/s00454-018-0049-2},
volume = {62},
year = {2019},
}
@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},
}