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Transactions of the American Mathematical Society

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Incircular nets and confocal conics


Authors: Arseniy V. Akopyan and Alexander I. Bobenko
Journal: Trans. Amer. Math. Soc. 370 (2018), 2825-2854
MSC (2010): Primary 51A05, 51B15, 52C35; Secondary 51K10, 51F10, 52C26
DOI: https://doi.org/10.1090/tran/7292
Published electronically: November 16, 2017
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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.


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Additional Information

Arseniy V. Akopyan
Affiliation: Institute of Science and Technology Austria (IST Austria), Am Campus 1, A - 3400, Klosterneuburg, Austria
Email: akopjan@gmail.com

Alexander I. Bobenko
Affiliation: Institut für Mathematik, Technische Universität Berlin, Strasse des 17 June 136, 10623 Berlin, Germany
Email: bobenko@math.tu-berlin.de

DOI: https://doi.org/10.1090/tran/7292
Received by editor(s): February 15, 2016
Received by editor(s) in revised form: January 10, 2017
Published electronically: November 16, 2017
Additional Notes: This research was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”. The first author was also supported by People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n$^{∘}$[291734].
Article copyright: © Copyright 2017 American Mathematical Society

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