# Hyperboloid model

## Model of n-dimensional hyperbolic geometry / From Wikipedia, the free encyclopedia

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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.