article
Incircular nets and confocal conics
published
yes
Arseniy
Akopyan
author 430D2C90-F248-11E8-B48F-1D18A9856A870000-0002-2548-617X
Alexander
Bobenko
author
HeEd
department
International IST Postdoc Fellowship Programme
project
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.
American Mathematical Society2018
eng
Transactions of the American Mathematical Society10.1090/tran/7292
37042825 - 2854
Akopyan, Arseniy, and Alexander Bobenko. “Incircular Nets and Confocal Conics.” <i>Transactions of the American Mathematical Society</i>, vol. 370, no. 4, American Mathematical Society, 2018, pp. 2825–54, doi:<a href="https://doi.org/10.1090/tran/7292">10.1090/tran/7292</a>.
Akopyan, Arseniy, and Alexander Bobenko. “Incircular Nets and Confocal Conics.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2018. <a href="https://doi.org/10.1090/tran/7292">https://doi.org/10.1090/tran/7292</a>.
A. Akopyan, A. Bobenko, Transactions of the American Mathematical Society 370 (2018) 2825–2854.
Akopyan, A., & Bobenko, A. (2018). Incircular nets and confocal conics. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href="https://doi.org/10.1090/tran/7292">https://doi.org/10.1090/tran/7292</a>
Akopyan A, Bobenko A. 2018. Incircular nets and confocal conics. Transactions of the American Mathematical Society. 370(4), 2825–2854.
Akopyan A, Bobenko A. Incircular nets and confocal conics. <i>Transactions of the American Mathematical Society</i>. 2018;370(4):2825-2854. doi:<a href="https://doi.org/10.1090/tran/7292">10.1090/tran/7292</a>
A. Akopyan and A. Bobenko, “Incircular nets and confocal conics,” <i>Transactions of the American Mathematical Society</i>, vol. 370, no. 4. American Mathematical Society, pp. 2825–2854, 2018.
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