Beltrami–Klein model

In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk (or n-dimensional unit ball) and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.

It is analogous to the gnomonic projection of spherical geometry, in that geodesics (great circles in spherical geometry) are mapped to straight lines.

This model is not conformal: angles are not faithfully represented, and circles become ellipses, increasingly flattened near the edge. This is in contrast to the Poincaré disk model, which is conformal. However, lines in the Poincaré model are not represented by straight line segments, but by arcs that meet the boundary orthogonally.

The Beltrami–Klein model is named after the Italian geometer Eugenio Beltrami and the German Felix Klein while "Cayley" in Cayley–Klein model refers to the English geometer Arthur Cayley.