Hyperbolic trigonometry in two-dimensional space-time geometry

Details Hyperbolic trigonometry in two-dimensional space-time geometry Hyperbolic trigonometry in two-dimensional space-time geometry

2005-08-03

arxiv.org/abs/math-ph/0508011v1

Nuovo Cimento B, 118 (2003) 475

Physics

Mathematical Physics

16 pages, 3 figures

Scientific paper

10.1393/ncb/i2003-10012-9

By analogy with complex numbers, a system of hyperbolic numbers can be introduced in the same way: z=x+h*y with h*h=1 and x,y real numbers. As complex numbers are linked to the Euclidean geometry, so this system of numbers is linked to the pseudo-Euclidean plane geometry (space-time geometry). In this paper we will show how this system of numbers allows, by means of a Cartesian representation, an operative definition of hyperbolic functions using the invariance respect to special relativity Lorentz group. From this definition, by using elementary mathematics and an Euclidean approach, it is straightforward to formalize the pseudo-Euclidean trigonometry in the Cartesian plane with the same coherence as the Euclidean trigonometry.

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