Editing Hyperbolic geometry (section)

=== The hyperboloid model ===
{{main article|hyperboloid model}}

The [[hyperboloid model]] or Lorentz model employs a 2-dimensional [[hyperboloid]] of revolution (of two sheets, but using one) embedded in 3-dimensional [[Minkowski space]]. This model is generally credited to Poincaré, but Reynolds<ref>{{aut|Reynolds, William F.}}, (1993) ''Hyperbolic Geometry on a Hyperboloid'', [[American Mathematical Monthly]] 100:442–455.</ref> says that [[Wilhelm Killing]] used this model in 1885
* This model has direct application to [[special relativity]], as Minkowski 3-space is a model for [[spacetime]], suppressing one spatial dimension. One can take the hyperboloid to represent the events (positions in spacetime) that various [[Inertial frame of reference|inertially]] moving observers, starting from a common event, will reach in a fixed [[proper time]].
* The hyperbolic distance between two points on the hyperboloid can then be identified with the relative [[rapidity]] between the two corresponding observers.
* The model generalizes directly to an additional dimension: a hyperbolic 3-space  three-dimensional hyperbolic geometry relates to Minkowski 4-space.