Hyperboloid model

Model of n dimensional hyperbolic geometry

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.

The bilinear form also functions as the metric tensor over the space. In n+1 dimensional Minkowski space, there are two choices for the metric with opposite signature, in the 3 dimensional case either (+, −, −) or (−, +, +). If the signature (−, +, +) is chosen, then the scalar square of chords between distinct points on the same sheet of the hyperboloid will be positive, which more closely aligns with conventional definitions and expectations in mathematics. Then n dimensional hyperbolic space is a Riemannian space and distance or length can be defined as the square root of the scalar square. If the signature (+, −, −) is chosen, scalar square between distinct points on the hyperboloid will be negative, so various definitions of basic terms must be adjusted, which can be inconvenient. Nonetheless, the signature (+, −, −, −) is also common for describing spacetime in physics. (Cf. Sign convention#Metric signature.)