De Tilly’s mechanical view on hyperbolic and spherical geometries
Dmitriy SlutskiyUniversité de Cergy-Pontoise, France
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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 from a geodesic (function ) and of the length of a circle of radius (function ) in the -plane of any constant curvature , . We study the rotation and the translation of a segment and of a triangle to obtain various formulae relating the functions and . As a corollary we give an elementary proof of the Laws of Sines and Cosines in hyperbolic and spherical spaces.