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.

If (x₀, x₁, ..., x_n) is a vector in the (n + 1)-dimensional coordinate space Rⁿ⁺¹, the Minkowski quadratic form is defined to be

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$${\displaystyle ds^{2}=Q(dx_{0},dx_{1},\ldots ,dx_{n})=-dx_{0}^{2}+dx_{1}^{2}+\ldots +dx_{n}^{2}.}$$The Minkowski bilinear form B is the polarization of the Minkowski quadratic form Q,

(This is sometimes also written using scalar product notation ${\displaystyle \mathbf {u} \cdot \mathbf {v} .}$) Explicitly,

The hyperbolic distance between two points u and v of S⁺ is given by the formula

where arcosh is the inverse function of hyperbolic cosine.