Incircular nets and confocal conics

A. Akopyan, A. Bobenko, Transactions of the American Mathematical Society 370 (2018) 2825–2854.

Journal Article | Published | English
Author
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Department
Abstract
We consider congruences of straight lines in a plane with the combinatorics of the square grid, with all elementary quadrilaterals possessing an incircle. It is shown that all the vertices of such nets (we call them incircular or IC-nets) lie on confocal conics. Our main new results are on checkerboard IC-nets in the plane. These are congruences of straight lines in the plane with the combinatorics of the square grid, combinatorially colored as a checkerboard, such that all black coordinate quadrilaterals possess inscribed circles. We show how this larger class of IC-nets appears quite naturally in Laguerre geometry of oriented planes and spheres and leads to new remarkable incidence theorems. Most of our results are valid in hyperbolic and spherical geometries as well. We present also generalizations in spaces of higher dimension, called checkerboard IS-nets. The construction of these nets is based on a new 9 inspheres incidence theorem.
Publishing Year
Date Published
2018-04-01
Journal Title
Transactions of the American Mathematical Society
Acknowledgement
DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”; People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) REA grant agreement n◦[291734]
Volume
370
Issue
4
Page
2825 - 2854
IST-REx-ID

Cite this

Akopyan A, Bobenko A. Incircular nets and confocal conics. Transactions of the American Mathematical Society. 2018;370(4):2825-2854. doi:10.1090/tran/7292
Akopyan, A., & Bobenko, A. (2018). Incircular nets and confocal conics. Transactions of the American Mathematical Society, 370(4), 2825–2854. https://doi.org/10.1090/tran/7292
Akopyan, Arseniy, and Alexander Bobenko. “Incircular Nets and Confocal Conics.” Transactions of the American Mathematical Society 370, no. 4 (2018): 2825–54. https://doi.org/10.1090/tran/7292.
A. Akopyan and A. Bobenko, “Incircular nets and confocal conics,” Transactions of the American Mathematical Society, vol. 370, no. 4, pp. 2825–2854, 2018.
Akopyan A, Bobenko A. 2018. Incircular nets and confocal conics. Transactions of the American Mathematical Society. 370(4), 2825–2854.
Akopyan, Arseniy, and Alexander Bobenko. “Incircular Nets and Confocal Conics.” Transactions of the American Mathematical Society, vol. 370, no. 4, American Mathematical Society, 2018, pp. 2825–54, doi:10.1090/tran/7292.

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