Beltrami–Klein model

Model of hyperbolic geometry

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. 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. The Beltrami–Klein model 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, meaning that angles and circles are distorted, whereas the Poincaré disk model preserves these. In this model, lines and segments are straight Euclidean segments, whereas in the Poincaré disk model, lines are arcs that meet the boundary orthogonally.

This model made its first appearance for hyperbolic geometry in two memoirs of Eugenio Beltrami published in 1868, first for dimension 1=n = 2 and then for general n, these essays proved the equiconsistency of hyperbolic geometry with ordinary Euclidean geometry. The papers of Beltrami remained little noticed until recently and the model was named after Klein ("The Klein disk model"). This happened as follows. In 1859 Arthur Cayley used the cross ratio definition of angle due to Laguerre to show how Euclidean geometry could be defined using projective geometry. His definition of distance later became known as the Cayley metric. In 1869, the young (twenty year old) Felix Klein became acquainted with Cayley's work. He recalled that in 1870 he gave a talk on the work of Cayley at the seminar of Weierstrass and he wrote:

"I finished with a question whether there might exist a connection between the ideas of Cayley and Lobachevsky. I was given the answer that these two systems were conceptually widely separated." Later, Felix Klein realized that Cayley's ideas give rise to a projective model of the non Euclidean plane. As Klein puts it, "I allowed myself to be convinced by these objections and put aside this already mature idea." However, in 1871, he returned to this idea, formulated it mathematically, and published it.

Chords that meet on the boundary circle are limiting parallel lines. Two chords are perpendicular if, when extended outside the disk, each goes through the pole of the other. (The pole of a chord is an ultra ideal point: the point outside the disk where the tangents to the disk at the endpoints of the chord meet.) Chords that go through the centre of the disk have their pole at infinity, orthogonal to the direction of the chord (this implies that right angles on diameters are not distorted).

Here is how one can use compass and straightedge constructions in the model to achieve the effect of the basic constructions in the hyperbolic plane. The pole of a line. While the pole is not a point in the hyperbolic plane (it is an ultra ideal point) most constructions will use the pole of a line in one or more ways. For a line: construct the tangents to the boundary circle through the ideal (end) points of the line. the point where these tangents intersect is the pole. For diameters of the disk: the pole is at infinity perpendicular to the diameter. To construct a perpendicular to a given line through a given point draw the ray from the pole of the line through the given point. The part of the ray that is inside the disk is the perpendicular. When the line is a diameter of the disk then the perpendicular is the chord that is (Euclidean) perpendicular to that diameter and going through the given point. To find the midpoint of given segment : Draw the lines through A and B that are perpendicular to (see above) Draw the lines connecting the ideal points of these lines, two of these lines will intersect the segment and will do this at the same point. This point is the (hyperbolic) midpoint of. To bisect a given angle : Draw the rays AB and AC. Draw tangents to the circle where the rays intersect the boundary circle. Draw a line from A to the point where the tangents intersect. The part of this line between A and the boundary circle is the bisector. The common perpendicular of two lines is the chord that when extended goes through both poles of the chords. When one of the chords is a diameter of the boundary circle then the common perpendicular is the chord that is perpendicular to the diameter and that when lengthened goes through the pole of the other chord. To reflect a point P in a line l: From a point R on the line l draw the ray through P. Let X be the ideal point where the ray intersects the absolute. Draw the ray from the pole of line l through X, let Y be another ideal point that intersects the ray. Draw the segment RY. The reflection of point P is the point where the ray from the pole of line l through P intersects RY.

While lines in the hyperbolic plane are easy to draw in the Klein disk model, it is not the same with circles, hypercycles and horocycles. Circles (the set of all points in a plane that are at a given distance from a given point, its center) in the model become ellipses increasingly flattened as they are nearer to the edge. Also angles in the Klein disk model are deformed. For constructions in the hyperbolic plane that contain circles, hypercycles, horocycles or non right angles it is better to use the Poincaré disk model or the Poincaré half plane model.

Both the hyperboloid model and the Klein disk model are models of the hyperbolic plane. The Klein disk (K, in the picture) is a gnomonic projection of the hyperboloid model (Hy) with as center the center of the hyperboloid (O) and the projection plane tangent to the nearest point of the hyperboloid.