Klein-Beltrami Model

The Klein-Beltrami model of Hyperbolic Geometry consists of an Open Disk in the Euclidean plane whose open chords correspond to hyperbolic lines. Two lines $\l$ and $m$ are then considered parallel if their chords fail to intersect and are Perpendicular under the following conditions,

1. If at least one of $l$ and $m$ is a diameter of the Disk, they are hyperbolically perpendicular Iff they are perpendicular in the Euclidean sense.

2. If neither is a diameter, $l$ is perpendicular to $m$ Iff the Euclidean line extending $l$ passes through the pole of $m$ (defined as the point of intersection of the tangents to the disk at the ``endpoints'' of $m$).

There is an isomorphism between the Poincaré Hyperbolic Disk model and the Klein-Beltrami model. Consider a Klein disk in Euclidean 3-space with a Sphere of the same radius seated atop it, tangent at the Origin. If we now project chords on the disk orthogonally upward onto the Sphere's lower Hemisphere, they become arcs of Circles orthogonal to the equator. If we then stereographically project the Sphere's lower Hemisphere back onto the plane of the Klein disk from the north pole, the equator will map onto a disk somewhat larger than the Klein disk, and the chords of the original Klein disk will now be arcs of Circles orthogonal to this larger disk. That is, they will be Poincaré lines. Now we can say that two Klein lines or angles are congruent Iff their corresponding Poincaré lines and angles under this isomorphism are congruent in the sense of the Poincaré model.

See also Hyperbolic Geometry, Poincaré Hyperbolic Disk