Checkerboard incircular nets. Laguerre geometry and parametrisation

TL;DR

This paper introduces an explicit integration procedure for checkerboard incircular nets, extending classical incircular nets, and explores their connections with elliptic billiards, confocal coordinate systems, and integrable systems within Laguerre geometry.

Contribution

It provides a novel parametrisation method for checkerboard IC-nets using Laguerre geometry, linking them to elliptic billiards and integrable systems.

Findings

Explicit integration procedure for checkerboard IC-nets.

Connection established with elliptic billiards and confocal coordinate systems.

Framework based on pencils of conics and quadrics in Laguerre geometry.

Abstract

We present a procedure which allows one to integrate explicitly the class of checkerboard IC-nets which has recently 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. Connections with discrete confocal coordinate systems and the fundamental QRT maps of integrable systems theory are made. The formalism developed in this paper is based on the existence of underlying pencils of conics and quadrics which is exploited in a Laguerre geometric setting.