Hyperboloid model

The project uses the hyperboloid model of hyperbolic 3-space. Hyperbolic space then consists of all tuples (x,y,z,w) such that w^2=1+x^2+y^2+z^2.

Isometries

The isometries of hyperbolic space all extend to linear transformations of Euclidean 4-space. This means we can represent them using (4x4)-matrices (yay!).

By a translation of hyperbolic space we mean the map obtained by composing the following three maps:

• the inverse of the exponential map at a point;
• parallel transport along a geodesic;
• the exponential map at the new point.

The composition of two translations is not necessarily a translation.

By a rotation of hyperbolic space we mean the map obtained by composing the following three maps:

• the inverse of the exponential map at a point;
• rotation of the tangent space at the point (w.r.t. the inner product on that tangent space);
• the exponential map at the point.