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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Incircular nets and confocal conics
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by Arseniy V. Akopyan and Alexander I. Bobenko PDF
Trans. Amer. Math. Soc. 370 (2018), 2825-2854 Request permission


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
  • MR Author ID: 824468
  • Email:
  • Alexander I. Bobenko
  • Affiliation: Institut für Mathematik, Technische Universität Berlin, Strasse des 17 June 136, 10623 Berlin, Germany
  • MR Author ID: 191410
  • Email:
  • 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$^\circ$[291734].
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 2825-2854
  • MSC (2010): Primary 51A05, 51B15, 52C35; Secondary 51K10, 51F10, 52C26
  • DOI:
  • MathSciNet review: 3748586