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.