De Tilly’s mechanical view on hyperbolic and spherical geometries

Abstract

In this chapter, we describe a kinematic approach developed by J.-M. de Tilly for the computation of the length of a curve at distance $r$ from a geodesic (function $\eq(r)$) and of the length of a circle of radius $r$ (function ${\circ}(r)$) in the $2$-plane of any constant curvature $K$, $K\in\mathbb{R}$. We study the rotation and the translation of a segment and of a triangle to obtain various formulae relating the functions $\eq(r)$ and ${\circ}(r)$. As a corollary we give an elementary proof of the Laws of Sines and Cosines in hyperbolic and spherical spaces.