Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry

A new method to obtain trigonometry for the real spaces of constant curvatu re and metric of any (even degenerate) signature is presented. The method c ould be described as 'curvature/signature (in)dependent trigonometry' and e ncapsulates trigonometry for all these spaces into a single basic trigonome tric group equation. This brings to its logical end the idea of an 'absolut e trigonometry', and provides equations which hold true for the nine two-di mensional spaces of constant curvature and any signature. This family of sp aces includes both relativistic and non-relativistic spacetimes; therefore a complete discussion of trigonometry in the six de Sitter, Minkowskian, Ne wton-Hooke and Galilean spacetimes follow as particular instances of the ge neral approach. Distinctive traits of the method are 'universality' and 'se lf-duality': every equation is meaningful for the nine spaces at once, and displays invariance explicitly under a duality transformation relating the nine spaces amongst themselves. These basic structural properties allow a c omplete study of trigonometry and, in fact, any equation previously known f or the three classical (Riemannian) spaces also has a version for the remai ning six 'spacetimes'; in most cases these equations are new.