Non-Euclidean Cross-Ratios and Carnot's Theorem for Conics

Abstract

When considering geometry, one might think of working with lines and circles on a flat plane as in Euclidean geometry. However, doing geometry in other spaces is possible, as the existence of spherical and hyperbolic geometry demonstrates. Despite the differences between these three geometries, striking connections appear among the three. In this paper, we illuminate one such connection by generalizing the cross-ratio, a powerful invariant associating a number to four points on a line, into non-Euclidean geometry. Along the way, we see how projections between these geometries can allow us to directly export results from one geometry into the others. The paper culminates by generalizing Carnot's Theorem for Conics - a classical result relating when six points on a triangle lie on a conic - into spherical and hyperbolic geometry. These same techniques are then applied to Carnot's Theorem for higher degree curves.