the Euclidean norm. Then the distance between U and V in the modelled hyperbolic space is expressed as where the factor of one half is needed to make the curvature −1. The associated metric tensor is given by Relation to the hyperboloid model The hyperboloid model is a model of hyperbolic geometry within -dimensional Minkowski space. The Minkowski inner product is given by and the norm by . The hyperbolic plane is embedded in this space as the vectors x with and x0 (the "timelike component") positive. The intrinsic distance (in the embedding) between points u and v is then given by This may also be written in the homogeneous form which allows the vectors to be rescaled for convenience. The Beltrami–Klein model is obtained from the hyperboloid model by rescaling all vectors so that the timelike component is 1, that is, by projecting the hyperboloid embedding through the origin onto the plane . The distance function, in its homogeneous form, is unchanged. Since the intrinsic lines (geodesics) of the hyperboloid model are the intersection of the embedding with planes through the Minkowski origin, the intrinsic lines of the Beltrami–Klein model are the chords of the sphere. Relation to the Poincaré ball model Both the Poincaré ball model and the Beltrami–Klein model are models of the n-dimensional hyperbolic space in the n-dimensional unit ball in Rn. If is a vector of norm less than one representing a point of the Poincaré disk model, then the corresponding point of the Beltrami–Klein model is given by Conversely, from a vector of norm less than one representing a point of the Beltrami–Klein model, the corresponding point of the Poincaré disk model is given by Given two points on the boundary of the unit disk, which are traditionally called ideal points, the straight line connecting them in the Beltrami–Klein model is the chord between them, while in the corresponding Poincaré model the line is a circular arc on the two-dimensional subspace generated by the two boundary point vectors, meeting the boundary of the ball at right angles. The two models are related through a projection from the center of the disk; a ray from the center passing through a point of one model line passes through the corresponding point of the line in the other model. See also Poincaré half-plane model Poincaré disk model Poincaré metric Inversive geometry Notes References Luis Santaló (1961), ''Geometrias