Spherical and hyperbolic conics

Abstract

This is a survey of metric properties of spherical and hyperbolic conics, mainly based on works of Chasles and Story. A spherical conic is the intersection of the sphere with a quadratic cone; similarly, a hyperbolic conic is the intersection of the Beltrami–Cayley–Klein disk with an affine conic. Non-Euclidean conics have numerous metric properties similar to those of Euclidean conics. The presence of absolute polarity makes the non-Euclidean case richer than the Euclidean one.