Hyperboloid Model - Minkowski Quadratic Form

If (x₀, x₁, …, x_n) is a vector in the (n+1)-dimensional coordinate space Rn+1, the Minkowski quadratic form is defined to be

The vectors v∈ Rn+1 such that Q(v) = 1 form an n-dimensional hyperboloid S consisting of two connected components, or sheets: the forward, or future, sheet S+, where x₀>0 and the backward, or past, sheet S−, where x₀<0. The points of the n-dimensional hyperboloid model are the points on the forward sheet S+.

The Minkowski bilinear form B is the polarization of the Minkowski quadratic form Q,

Explicitly,

.

The hyperbolic distance between two points u and v of S+ is given by the formula