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 curvature and metric of any (even degenerate) signature is presented. The method could be described as `curvature/signature (in)dependent trigonometry' and encapsulates trigonometry for all these spaces into a single basic trigonometric group equation. This brings to its logical end the idea of an `absolute trigonometry', and provides equations which hold true for the nine two-dimensional spaces of constant curvature and any signature. This family of spaces includes both relativistic and non-relativistic spacetimes; therefore a complete discussion of trigonometry in the six de Sitter, Minkowskian, Newton-Hooke and Galilean spacetimes follow as particular instances of the general approach. Distinctive traits of the method are `universality' and `self-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 complete study of trigonometry and, in fact, any equation previously known for the three classical (Riemannian) spaces also has a version for the remaining six `spacetimes'; in most cases these equations are new.