If (x0, x1, …, xn) 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
Read more about this topic: Hyperboloid Model
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