Talk:Hyperboloid model/Draft

Red circular arc is geodesic in Poincaré disk model; it projects to the brown geodesic on the green hyperboloid.

Animation of partial {7,3} hyperbolic tiling of the hyperboloid rotated into the Poincare perspective.

In hyperbolic geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of the hyperbolic plain in which points are represented by points on the positive sheet S⁺ of a two-sheeted hyperboloid in (2+1)-dimensional Minkowski space.

Lines geodetics are represented by the intersections of planes passing through the origin of Minkowski space. The hyperbolic plane 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 a sphere is embedded in 3-dimensional Euclidean space.

the hyperboloid is mostlty used for investigations in space time. some students think that the hyperboloid is the only model of hyperbolic geometry.

Advantages of this model are:

• as there is no boundary on this model there is no place where the scale of the model gets to 0. (less rounding errors in coordinates this is one of the reasons why hyperrouge and other hyperbolic games use this model.

Disadvantages of this model are

• 3 dimensionalty doesn't fit easy on a sheet of paper or screen

history

see https://hsm.stackexchange.com/questions/3230/what-was-the-motivation-for-minkowski-spacetime-before-special-relativity/3232#3232

and parts of history as given before be carefull some confuse hyperbolic geometry with the hyperboloid model ( and an hyperboloid like isometric model of part of the hyperbolic plane)

geometry of the hyperboloid , and cone

distance formula ed

relation to other models of hyperbolic geometry

Other models of hyperbolic space can be thought of as map projections of this model:

The Beltrami–Klein model is the projection of points on the positive sheet S⁺ trough the origin onto a plane perpendicular to the normal vector from the origin to specific point on this model analogous to the gnomonic projection of the sphere. ?????

The Poincaré disk model is a projection of the sheet 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.

in higher dimensional geometry

mostly rest of the article