# De Tilly’s mechanical view on hyperbolic and spherical geometries

### Dmitriy Slutskiy

Université de Cergy-Pontoise, France

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## 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 $∘(r)$) in the $2$-plane of any constant curvature $K$, $K∈R$. We study the rotation and the translation of a segment and of a triangle to obtain various formulae relating the functions \( \eq(r) \) and $∘(r)$. As a corollary we give an elementary proof of the Laws of Sines and Cosines in hyperbolic and spherical spaces.