geometry - What is the relevance of hyperbolic sine and cosine? What is ...
triangles - A notion of similarity in hyperbolic geometry - Mathematics ...
Any manifold is the quotient of its universal cover by its fundamental group, so this statement is a special case of a general principle. So what you are looking for is the statement that a complete simply connected manifold of sectional curvature $-1$ is isometric to the hyperbolic space. This is a basic result in Riemannian geometry and can be found for instance in do Carmo, Manfredo ...
While $\mathbb H^n$ is not really an affine space, the general equation for hyperbolic hyperplanes is just a manifestation of this broad correspondence between affine spaces (inhomogeneous) and vector spaces with one more dimension (homogeneous), which also manifests itself in algebra and algebraic geometry as homogenization of polynomials.
The Conversation: Corals, crochet and the cosmos: how hyperbolic geometry pervades the universe
Hyperbolic space is a Pringle-like alternative to flat, Euclidean geometry where the normal rules don’t apply: angles of a triangle add up to less than 180 degrees and Euclid’s parallel postulate, ...
Mathematicians often comment on the beauty of their chosen discipline. For the non-mathematicians among us, that can be hard to visualise. But in Prof Caroline Series’s field of hyperbolic geometry, ...
I also know that the hyperbolic functions parametrize the unit hyperbola $ (\cosh a, \sinh a)$ and that $\frac {a} {2}$ corresponds to the area of the region bounded by the hyperbola, the rays from the origin, and the point $ (\cosh a, \sinh a)$.