Singular Frégier Conics in Non-Euclidean Geometry

Abstract

The hypotenuses of all right triangles inscribed into a fixed conic $C$ with fixed right-angle vertex $p$ are incident with the Frégier point $f$ to $p$ and $C$. As $p$ varies on the conic, the locus of the Frégier point is, in general, a conic as well. We study conics $C$ whose Frégier 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 Frégier locus.

arXiv commentAccepted for publication in the Proceedings of ICGG 2016, The 17th International Conference on Geometry and Graphics