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 (n + 1)-dimensional coordinate space Rn+1, the Minkowski quadratic form is defined to be
Q ( x 0 , x 1 , … , x n ) = − x 0 2 + x 1 2 + … + x n 2 . {\displaystyle Q(x_{0},x_{1},\ldots ,x_{n})=-x_{0}^{2}+x_{1}^{2}+\ldots +x_{n}^{2}.}
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 metric on the hyperboloid is d s 2 = Q ( d x 0 , d x 1 , … , d x n ) = − d x 0 2 + d x 1 2 + … + d x n 2 . {\displaystyle ds^{2}=Q(dx_{0},dx_{1},\ldots ,dx_{n})=-dx_{0}^{2}+dx_{1}^{2}+\ldots +dx_{n}^{2}.} The Minkowski bilinear form B is the polarization of the Minkowski quadratic form Q,
B ( u , v ) = ( Q ( u + v ) − Q ( u ) − Q ( v ) ) / 2. {\displaystyle B(\mathbf {u} ,\mathbf {v} )=(Q(\mathbf {u} +\mathbf {v} )-Q(\mathbf {u} )-Q(\mathbf {v} ))/2.}
(This is sometimes also written using scalar product notation u ⋅ v . {\displaystyle \mathbf {u} \cdot \mathbf {v} .} ) Explicitly,
B ( ( x 0 , x 1 , … , x n ) , ( y 0 , y 1 , … , y n ) ) = − x 0 y 0 + x 1 y 1 + … + x n y n . {\displaystyle B((x_{0},x_{1},\ldots ,x_{n}),(y_{0},y_{1},\ldots ,y_{n}))=-x_{0}y_{0}+x_{1}y_{1}+\ldots +x_{n}y_{n}.}
The hyperbolic distance between two points u and v of S+ is given by the formula
d ( u , v ) = arcosh ( − B ( u , v ) ) , {\displaystyle d(\mathbf {u} ,\mathbf {v} )=\operatorname {arcosh} (-B(\mathbf {u} ,\mathbf {v} )),}
where arcosh is the inverse function of hyperbolic cosine.
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