Nets of lines with the combinatorics of the square grid and with touching inscribed conics

TL;DR

This paper introduces a new class of line nets in the projective plane with touching inscribed conics, generalizing incircular nets and revealing their geometric properties and connections to Koenigs nets and billiards.

Contribution

It defines and analyzes nets of lines with touching inscribed conics, showing they are planar Koenigs nets and characterizing Koenigs nets by inscribed conics, extending the theory of incircular nets.

Findings

Nets are planar Koenigs nets.

Existence of a 1-parameter family of touching inscribed conics characterizes Koenigs nets.

Lines of the grid are tangent to a common conic.

Abstract

In the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially neighbouring quadrilaterals have the same touching point on their common edge-line. We suggest that these nets are a natural projective generalisation of incircular nets. It is shown that these nets are planar Koenigs nets. Moreover, we show that general Koenigs nets are characterised by the existence of a 1-parameter family of touching inscribed conics. It is shown that the lines of any grid of quadrilaterals with touching inscribed conics are tangent to a common conic. These grids can be constructed via polygonal chains that are inscribed in conics. The special case of billiards in conics corresponds to incircular nets.