Hyperboloid model

The hyperboloid model of the curvature-minus-one hyperbolic plane isIts metric is the restriction of the Minkowski bilinear form to tangent planes, and its geodesics are intersections with two-dimensional linear subspaces through the origin.

The three-dimensional hyperboloid model isIt is a regular level set with

This indefinite bilinear form supplies the metric of Minkowski spacetime. On ${\mathbb{R}}^{n+1}$ the Minkowski inner product with mostly-positive convention is