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A new method to obtain trigonometry for the real spaces of constant curvatureand metric of any (even degenerate) signature is presented. The methodencapsulates trigonometry for all these spaces into a single basictrigonometric group equation. This brings to its logical end the idea of anabsolute trigonometry, and provides equations which hold true for the ninetwo-dimensional spaces of constant curvature and any signature. This family ofspaces includes both relativistic and non-relativistic homogeneous spacetimes;therefore a complete discussion of trigonometry in the six de Sitter,minkowskian, Newton--Hooke and galilean spacetimes follow as particularinstances of the general approach. Any equation previously known for the threeclassical riemannian spaces also has a version for the remaining sixspacetimes; in most cases these equations are new. Distinctive traits of themethod are universality and self-duality: every equation is meaningful for thenine spaces at once, and displays explicitly invariance under a dualitytransformation relating the nine spaces. The derivation of the single basictrigonometric equation at group level, its translation to a set of equations(cosine, sine and dual cosine laws) and the natural apparition of angular andlateral excesses, area and coarea are explicitly discussed in detail. Theexposition also aims to introduce the main ideas of this direct grouptheoretical way to trigonometry, and may well provide a path to systematicallystudy trigonometry for any homogeneous symmetric space.Comment: 51 pages, LaTe

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