Trigonometry in extended hyperbolic space and extended de Sitter space

We study the hyperbolic cosine and sine laws in the extended hyperbolic space which contains hyperbolic space as a subset and is an analytic continuation of the hyperbolic space. And we also study the spherical cosine and sine laws in the extended de Sitter space which contains de Sitter Space $S^n_1$ as a subset and is also an analytic continuation of de Sitter space. In fact, the extended hyperbolic space and extended de Sitter space are the same space only differ by -1 multiple in the metric. Hence these two extended spaces clearly show and apparently explain that why many corresponding formulas in hyperbolic and spherical space are very similar each other. From these extended trigonometry laws, we can give a coherent and geometrically simple explanation for the various relations between the lengths and angles of hyperbolic polygons and relations on de Sitter polygons which lie on $S^2_1$.

💡 Research Summary

The paper introduces a unified geometric framework that simultaneously encompasses hyperbolic space ( \mathbb H^n ) and de Sitter space ( S^n_1 ) by means of analytic continuation. The authors construct two “extended” spaces: the extended hyperbolic space ( \widehat{\mathbb H}^n ) and the extended de Sitter space ( \widehat{S}^n_1 ). Both are defined on the same underlying manifold ( \mathcal M \subset \mathbb C^{n+1} ), but they carry metrics that differ only by an overall sign, i.e. ( g_{\widehat{S}} = -,g_{\widehat{\mathbb H}} ). This simple sign reversal makes the two spaces analytically continuations of each other: replacing a real distance parameter by an imaginary one transforms hyperbolic distances into de Sitter distances, and vice‑versa.

Within this common setting the classical cosine and sine laws are derived in a single, compact form. For a triangle with side‑lengths (a,b,c) and opposite angles ( \alpha,\beta,\gamma ) the authors obtain the “master” cosine law

\