Incircular and Checkerboard Incircular Nets

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Considered are 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.
The 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. It is shown 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 the results are valid in hyperbolic and spherical geometries as well. Presented are 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.

This leads to a procedure which allows one to integrate explicitly the class of checkerboard IC-nets which has been introduced as a generalisation of incircular (IC) nets. The latter class of privileged congruences of lines in the plane is known to admit a great variety of geometric properties which are also present in the case of checkerboard IC-nets. The parametrisation obtained in this manner is reminiscent of that associated with elliptic billiards. The formalism developed in the paper is based on the existence of underlying pencils of conics and quadrics which is exploited in a Laguerre geometric setting.

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