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.
History
This model made its first appearance for hyperbolic geometry in two memoirs of Eugenio Beltrami published in 1868, first for dimension 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.
Distance formula
The distance function for the Beltrami–Klein model is a Cayley–Klein metric. Given two distinct points p and q in the open unit ball, the unique straight line connecting them intersects the boundary at two ideal points, a and b, label them so that the points are, in order, a, p, q, b , so that and .
The hyperbolic distance between p and q is then:
The vertical bars indicate Euclidean distances between the points in the model, ln is the natural logarithm and the factor of one half is needed to give the model the standard curvature of −1.
When one of the points is the origin and Euclidean distance between the points is r then the hyperbolic distance is:
where artanh is the inverse hyperbolic function of the hyperbolic tangent.
The Klein disk model
In two dimensions the Beltrami–Klein model is called the Klein disk model. It is a disk and the inside of the disk is a model of the entire hyperbolic plane. Lines in this model are represented by chords of the boundary circle (also called the absolute). The points on the boundary circle are called ideal points; although well defined, they do not belong to the hyperbolic plane. Neither do points outside the disk, which are sometimes called ultra ideal points.
The model is not conformal, meaning that angles
are distorted, and circles on the hyperbolic plane are in general not circular in the model. Only circles that have their centre at the centre of the boundary circle are not distorted. All other circles are distorted, as are horocycles and hypercycles
Properties
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).
Compass and straightedge constructions
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.