Incircular nets and confocal conics

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 ICnets 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.

Ivory’s theorem revisited

“…One of our main inspirations in Section 2 were recent results of A. Akopyan and A. Bobenko [2] on the nets of lines whose quadrilaterals admit inscribed circles. In this direction, our billiard approach gives a proof of a theorem of Reye and Chasles (Theorem 2) and provides a configuration of circles associated with a periodic billiard trajectory in an ellipse ( Figure 12).…”

Fun Problems in Geometry and Beyond

“…We shall deduce Ivory's lemma from integrability of billiards in ellipses. See [2,3,13,16] for these and related topics, also including the material of the next section.…”

3-Webs generated by confocal conics and circles

“…Quadrilaterals formed by lines of this net can be circumscribed around circles and points of intersection of these lines can be split into families lying on confocal conics. This construction was rediscovered and generalized by the author and Bobenko in [1], where also it was noticed that Böhm's F 1 F 2 P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P…”