BooksirmaCollected Volumepp. 93–111

De Tilly’s mechanical view on hyperbolic and spherical geometries

  • Dmitriy Slutskiy

    Université de Cergy-Pontoise, France
De Tilly’s mechanical view on hyperbolic and spherical geometries cover
Download Chapter PDF

A subscription is required to access this book chapter.

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 rr from a geodesic (function \eq(r)\eq(r)) and of the length of a circle of radius rr (function (r){\circ}(r)) in the 22-plane of any constant curvature KK, KRK\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)\eq(r) and (r){\circ}(r). As a corollary we give an elementary proof of the Laws of Sines and Cosines in hyperbolic and spherical spaces.