# Hyperbolic Geometry¶

Physics students are in fact quite familiar with the properties of hyperbolic geometry, even though some of the terms are not usually used in physics.

## Visualizations of Hyperbolic Space¶

A lot of models are used to describe hyperbolic space

Numerious models are developed to describe the hyperbolic space.

- Klein model;
- Poincare model;
- Gans model;
- Weierstrass model or hyperboloid mode.

One of the useful visualizations of hyperbolic space is the the hyerboloid model, a.k.a. Weierstrass model. As the name indicates, hyperbolic space is embeded in Euclid space as a hyperboloid.

Hyperboloid on Two Sheets

A unit hyerboloid is described by the equation

where \(x,y,z\) are the coordinates.

Lines on a hyperboloid is defined by the intersection of a plane with the hyperboloid.

We would also imagine that the so called light cone is basically

For simplicity, we consider two dimensions, space \(x\) and time \(t\). To build a theory of special relativity, we have to first specify the distance on hyperbola geometry. It’s straightforward as it seems to be, we just extract distance from the definition of hyperbola since it is a conserved quantity,

For a standard hyperbola \(x^2-t^2=1\), we can parameterize the coordinates using a single parameter,

since

Hyperbolic Trig Identities

As a review, the hyperbolic functions are defined as

Hyperbolic trig functions have several identities that could help us with the understanding of the geometry.

To visualize such relations, we draw an triangle, Fig. 19. We use this triangle to illustrate \(\sin\beta=b/a=3/4\)

We choose the edge \(b=3\) and \(a=4\). Given the condition that

we have to set

which is not so intuitive from Fig. 19.