Beltrami–Klein model

This model is not conformal: angles are not faithfully represented, and circles become ellipses, increasingly flattened as they are nearer to the edge.

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.

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: Later, Felix Klein realized that Cayley's ideas give rise to a projective model of the non-Euclidean plane.

The vertical bars indicate Euclidean distances between the points in the model, where ln is the natural logarithm and the factor of one half is needed to give the model the standard curvature of −1.

Lines in this model are represented by chords of the boundary circle (also called the absolute).

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.

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.

An advantage of the Poincaré disk model is that it is conformal (circles and angles are not distorted); a disadvantage is that lines of the geometry are circular arcs orthogonal to the boundary circle of the disk.

(the ideal points remain on the same spot) also the pole of the chord is the centre of the circle that contains the arc.

from the centre of the unit circle in the Beltrami–Klein model, then the corresponding point on the Poincaré disk model a distance of u on the same radius: Conversely, If P is a point a distance

from the centre of the unit circle in the Poincaré disk model, then the corresponding point of the Beltrami–Klein model is a distance of s on the same radius: The gnomonic projection of the sphere projects from the sphere's center onto a tangent plane.

Every great circle on the sphere is projected to a straight line, but it is not conformal.

Angles are not faithfully represented, and circles become ellipses, increasingly stretched as they get further from the tangent point.

Taking the centre of the unit ball of the model as the origin, and assigning position vectors u, v, a, b respectively to the points U, V, A, B, we have that that ‖a − v‖ > ‖a − u‖ and ‖u − b‖ > ‖v − b‖, where ‖ · ‖ denotes the Euclidean norm.

Then the distance between U and V in the modelled hyperbolic space is expressed as where the factor of one half is needed to make the curvature −1.

The hyperbolic plane is embedded in this space as the vectors x with ‖x‖ = 1 and x0 (the "timelike component") positive.

The intrinsic distance (in the embedding) between points u and v is then given by This may also be written in the homogeneous form which allows the vectors to be rescaled for convenience.

Since the intrinsic lines (geodesics) of the hyperboloid model are the intersection of the embedding with planes through the Minkowski origin, the intrinsic lines of the Beltrami–Klein model are the chords of the sphere.