On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs

A. Akopyan, A.I. Bobenko, W.K. Schief, J. Techter, Discrete and Computational Geometry (2020).


Journal Article | Epub ahead of print | English

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Author
Akopyan, ArseniyIST Austria; Bobenko, Alexander I.; Schief, Wolfgang K.; Techter, Jan
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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.
Publishing Year
Date Published
2020-08-31
Journal Title
Discrete and Computational Geometry
Acknowledgement
This research was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”. W.K.S. was also supported by the Australian Research Council (DP1401000851). A.V.A. was also supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha).
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Cite this

Akopyan A, Bobenko AI, Schief WK, Techter J. On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. Discrete and Computational Geometry. 2020. doi:10.1007/s00454-020-00240-w
Akopyan, A., Bobenko, A. I., Schief, W. K., & Techter, J. (2020). On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. Discrete and Computational Geometry. https://doi.org/10.1007/s00454-020-00240-w
Akopyan, Arseniy, Alexander I. Bobenko, Wolfgang K. Schief, and Jan Techter. “On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs.” Discrete and Computational Geometry, 2020. https://doi.org/10.1007/s00454-020-00240-w.
A. Akopyan, A. I. Bobenko, W. K. Schief, and J. Techter, “On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs,” Discrete and Computational Geometry, 2020.
Akopyan A, Bobenko AI, Schief WK, Techter J. 2020. On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. Discrete and Computational Geometry.
Akopyan, Arseniy, et al. “On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs.” Discrete and Computational Geometry, Springer Nature, 2020, doi:10.1007/s00454-020-00240-w.
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