Incircular nets and confocal conics Journal Article


Author(s): Akopyan, Arseniy V; Bobenko, Alexander I
Article Title: Incircular nets and confocal conics
Affiliation IST Austria
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.
Journal Title: Transactions of the American Mathematical Society
Volume: 370
Issue 4
ISSN: 1088-6850
Publisher: American Mathematical Society  
Date Published: 2018-04-01
Start Page: 2825
End Page: 2854
Sponsor: 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]
URL:
DOI: 10.1090/tran/7292
Open access: yes (repository)
IST Austria Authors
  1. Arseniy V Akopyan
    11 Akopyan
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