Incircular nets and confocal conics

TL;DR

This paper explores incircular nets in the plane with square grid combinatorics, revealing their vertices lie on confocal conics, and introduces checkerboard IC-nets with new incidence theorems in Laguerre geometry, extending to higher dimensions.

Contribution

It introduces checkerboard IC-nets and their natural occurrence in Laguerre geometry, along with a new 9 inspheres incidence theorem, extending classical results to hyperbolic, spherical, and higher-dimensional geometries.

Findings

Vertices of IC-nets lie on confocal conics.

Checkerboard IC-nets relate to Laguerre geometry and incidence theorems.

Generalizations to higher dimensions called checkerboard IS-nets.

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.