Klein model

In geometry, the Klein model, also called the projective model, the Beltrami–Klein model, the Klein–Beltrami model and the Cayley–Klein model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or ball, and the lines of the geometry are line segments contained in the disk; that is, with endpoints on the boundary of the disk. Along with the Poincaré half-plane model and the Poincaré disk model, it was first proposed by Eugenio Beltrami [{|mathbf{u} cdot frac{mathbf{v{|mathbf{v}\right)which gives us the freedom to rescale the vectors as we see fit.

The Klein model is obtained from the hyperboloid model by rescaling all vectors so that the timelike component is 1, that is, by projecting the hyperboloid embedding through the origin onto the plane "x"₀ = 1. This maps the hyperbolic plane into a ball of radius 1, with the spherical boundary of the ball corresponding to the conformal infinity of the hyperbolic plane. The distance function, in its homogeneous form, is unchanged. 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 Klein model are the chords of the sphere.

Distance formula

Cayley introduced the cross-ratio to measure distances in this model.Given two points in the unit ball, "p", "q", draw the line connecting them, which intersects the sphere in two points, "a" and "b", so the points are, in order, $a,p,q,b$.Then the cross ratio is defined by::$frac\{1\}\{2\}\; log\; frac\; cdot\; frac\{mathbf\{t\{|mathbf\{t\}\backslash right)\; =\; cosh^\{-1\}left(frac\{mathbf\{s\}\; cdot\; mathbf\{t\{sqrt\{(mathbf\{s\}cdotmathbf\{s\})(mathbf\{t\}cdotmathbf\{t\})\; ight)\; =\; cosh^\{-1\}left(frac\{1\; -\; s\; cdot\; t\}\{sqrt\{(1\; -\; s\; cdot\; s)(1\; -\; t\; cdot\; t)\; ight)$.

Relation to the Poincaré disk model

Both the Poincaré disk model and the Klein model are models of hyperbolic space on the unit n-disk. If $u$ is a vector of norm less than one representing a point of the Poincaré disk model, then the corresponding point of the Klein model is given by:$s\; =\; frac\{2u\}\{1+u\; cdot\; u\}.$Conversely, from a vector $s$ of norm less than one representing a point of the Klein model, the corresponding point of the Poincaré disk model is given by:$u\; =\; frac\{s\}\{1+sqrt\{1-s\; cdot\; s\; =\; frac\{(1-sqrt\{1-s\; cdot\; s\})s\}\{s\; cdot\; s\}.$

Given two points on the boundary of the unit disk, which are called "ideal points", the Klein model line is the chord between them, and the corresponding Poincaré model line is a circular arc on the two dimensional subspace generated by the two boundary point vectors, orthogonal to the boundary of the disk. The relationship between the two is simply a projection from the center of the disk; a ray from the center passing through a point of one model line passes through the corresponding point of the other model line.

Angles and perpendicularity

Given two intersecting lines in the Klein model, which are intersecting chords in the unit disk, we can find the angle between the lines by mapping the chords, expressed as parametric equations for a line, to parametric functions in the Poincaré disk model, finding unit tangent vectors, and using this to determine the angle.

We may also compute the angle between the chord whose ideal point endpoints are $u$ and $v$, and the chord whose endpoints are $s$ and $t$, by means of a formula. Since the ideal points are the same in the Klein model and the Poincaré disk model, the formulas are identical for each model.

If both chords are diameters, so that $v=-u$ and $t=-s$, then we are merely finding the angle between two unit vectors, and the formula for the angle $heta$ is:$cos(\; heta)\; =\; u\; cdot\; s.$If $v=-u$ but not $t=-s$, the formula becomes, in terms of the wedge product,:$cos^2(\; heta)\; =\; frac\{P^2\}\{QR\},$where :$P\; =\; u\; cdot\; (s-t),$:$Q\; =\; u\; cdot\; u,$:$R\; =\; (s-t)\; cdot\; (s-t)\; -\; (s\; wedge\; t)\; cdot\; (s\; wedge\; t)$If both chords are not diameters, the general formula obtains:$cos^2(\; heta)\; =\; frac\{P^2\}\{QR\},$where:$P\; =\; (u-v)\; cdot\; (s-t)\; -\; (u\; wedge\; v)\; cdot\; (s\; wedge\; t),$:$Q\; =\; (u-v)\; cdot\; (u-v)\; -\; (u\; wedge\; v)\; cdot\; (u\; wedge\; v),$:$R\; =\; (s-t)\; cdot\; (s-t)\; -\; (s\; wedge\; t)\; cdot\; (s\; wedge\; t).$Using the Binet–Cauchy identity and the fact that these are unit vectors we may rewrite the above expressions purely in terms of the dot product, as:$P\; =\; (u-v)\; cdot\; (s-t)\; +\; (u\; cdot\; t)(v\; cdot\; s)\; -\; (u\; cdot\; s)(v\; cdot\; t),$:$Q\; =\; (1\; -\; u\; cdot\; v)^2,$:$R\; =\; (1\; -\; s\; cdot\; t)^2.$

Determining angles is greatly simplified when the question is to determine or construct right angles in the hyperbolic plane. A line in the Poincaré disk model corresponds to a circle orthogonal to the unit disk boundary, with the corresponding Klein model line being the chord between the two points where this intersects the boundary. The tangents to the intersection at the two endpoints intersect in a point called the pole of the chord. Any line drawn through the pole, which is the center of the Poincaré model circle, will intersect the Poincaré model circle orthogonally, and hence the line segments intersect the chord in the Klein model, which corresponds to the circle, as perpendicular lines.

Restating this, a chord $B$ intersecting a given chord $A$ of the Klein model, which when extended to a line passes through the pole of the chord $A$, is perpendicular to $A$. This fact can be used to give an easy proof of the ultraparallel theorem.

ee also

*Poincaré half-plane model
*Poincaré metric
*Inversive geometry

References

* Luis Santaló (1961), "Geometrias no Euclidianas, EUDEBA.
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