Klein-Beltrami Model

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

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

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

There is an isomorphism between the Poincaré Hyperbolic Disk model and theKlein-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 lowerHemisphere, they become arcs of Circles orthogonal to the equator. If we then stereographicallyproject the Sphere's lower Hemisphere back onto the plane of the Klein disk from the north pole, the equatorwill map onto a disk somewhat larger than the Klein disk, and the chords of the original Klein disk will now be arcs ofCircles orthogonal to this larger disk. That is, they will be Poincaré lines. Now we can say that twoKlein lines or angles are congruent Iff their corresponding Poincaré lines and angles under this isomorphism arecongruent in the sense of the Poincaré model.

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