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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 ...
A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation. A hyperboloid is a quadric surface , that is, a surface defined as the zero set of a polynomial of degree two in three variables.
In 1966 David Gans proposed a flattened hyperboloid model in the journal American Mathematical Monthly. [35] It is an orthographic projection of the hyperboloid model onto the xy-plane. This model is not as widely used as other models but nevertheless is quite useful in the understanding of hyperbolic geometry.
The isometry to the half-space model can be realised by a homography sending a point of the unit sphere to infinity. Hyperboloid model : In contrast with the previous two models this realises hyperbolic n {\displaystyle n} -space as isometrically embedded inside the ( n + 1 ) {\displaystyle (n+1)} -dimensional Minkowski space (which is not a ...
The hyperboloid model can be represented as the equation t 2 = x 1 2 + x 2 2 + 1, t > 1. It can be used to construct a Poincaré disk model as a projection viewed from (t = −1, x 1 = 0, x 2 = 0), projecting the upper half hyperboloid onto the unit disk at t = 0. The red geodesic in the Poincaré disk model projects to the brown geodesic on ...
Combined projections from the Klein disk model (yellow) to the Poincaré disk model (red) via the hemisphere model (blue) The Beltrami–Klein model (K in the picture) is an orthographic projection from the hemispherical model and a gnomonic projection of the hyperboloid model (Hy) with the center of the hyperboloid (O) as its center.
The Poincaré half-plane model is closely related to a model of the hyperbolic plane in the quadrant Q = {(x,y): x > 0, y > 0}. For such a point the geometric mean = and the hyperbolic angle = / produce a point (u,v) in the upper half-plane. The hyperbolic metric in the quadrant depends on the Poincaré half-plane metric.
In the hyperboloid model horocycles are represented by intersections of the hyperboloid with planes that generate parabolas on the asymptotic cone (see conic sections" the cutting plane is parallel to exactly one generating line of the cone " ) The normal of the cutting plane is a null vector in three-dimensional Minkowski space.