In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S+ of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m-planes are represented by the intersections of (m+1)-planes passing through the origin in Minkowski space with S+ or by wedge products of m vectors. Hyperbolic space is embedded isometrically in Minkowski space; that is, the hyperbolic distance function is inherited from Minkowski space, analogous to the way spherical distance is inherited from Euclidean distance when the n-sphere is embedded in (n+1)-dimensional Euclidean space. Other models of hyperbolic space can be thought of as map projections of S+: the Beltrami–Klein model is the projection of S+ through the origin onto a plane perpendicular to a vector from the origin to specific point in S+ analogous to the gnomonic projection of the sphere; the Poincaré disk model is a projection of S+ through a point on the other sheet S− onto perpendicular plane, analogous to the stereographic projection of the sphere; the Gans model is the orthogonal projection of S+ onto a plane perpendicular to a specific point in S+, analogous to the orthographic projection; the band model of the hyperbolic plane is a conformal “cylindrical” projection analogous to the Mercator projection of the sphere; Lobachevsky coordinates are a cylindrical projection analogous to the equirectangular projection (longitude, latitude) of the sphere. Minkowski quadratic form If (x0, x1, ..., xn) is a vector in the -dimensional coordinate space Rn+1, the Minkowski quadratic form is defined to be The vectors such that 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, (This is sometimes also written using scalar product notation ) Explicitly, The hyperbolic distance between two points u and v of S+ is given by the formula where is the inverse function of hyperbolic cosine. Choice of metric signature The bilinear form also functions as the metric tensor over the space. In n+1 dimensional Minkowski space, there are two choices for