Singular Fr\'egier Conics in Non-Euclidean Geometry

TL;DR

This paper investigates conditions under which the Frégier locus associated with conics in non-Euclidean geometries becomes singular, revealing diverse behaviors across Euclidean, elliptic, and hyperbolic planes.

Contribution

It characterizes conics with singular Frégier loci in non-Euclidean geometries, highlighting the complexity in hyperbolic and elliptic cases.

Findings

Hyperbolic plane admits the richest variety of such conics.

In elliptic geometry, only three families of conics have singular Frégier loci.

The Frégier locus is generally a conic, but can be singular in specific cases.

Abstract

The hypotenuses of all right triangles inscribed into a fixed conic $C$ with fixed right-angle vertex $p$ are incident with the Fr\'egier point $f$ to $p$ and $C$. As $p$ varies on the conic, the locus of the Fr\'egier point is, in general, a conic as well. We study conics $C$ whose Fr\'egier locus is singular in Euclidean, elliptic and hyperbolic geometry. The richest variety of conics with this property is obtained in hyperbolic plane while in elliptic geometry only three families of conics have a singular Fr\'egier locus.