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 who used these models to show hyperbolic geometry was equiconsistent with Euclidean geometry. The distance function was originated first by Arthur Cayley and interpreted geometrically in hyperbolic geometry by Felix Klein.
Relation to the hyperboloid model
The hyperboloid model is a model of hyperbolic geometry within Minkowski space. If <math>[x_0, x_1, \cdots, x_n]</math> is a vector in real <math>(n+1)</math>-space, we may define the Minkowski quadratic form to be
- <math>Q([x_0, x_1, \cdots, x_n]) = x_0^2 - x_1^2 - \cdots - x_n^2.</math>
Corresponding to the Minkowski quadratic form <math>Q</math> there is a Minkowski bilinear form <math>B</math>, defined by
- <math>B(u, v) = (Q(u+v)-Q(u)-Q(v))/2.</math>
If
- <math>u = [x_0, x_1, \cdots, x_n], v = [y_0, y_1, \cdots, y_n]</math>
then we may write this as
- <math>B(u, v) = x_0 y_0-x_1 y_1 - \cdots - x_n y_n =
x_0 y_0 - \mathbf{x} \cdot \mathbf{y}.</math>
We may use this to put a hyperbolic metric on certain of the points of Minkowski projective space, which is to say, of lines through the origin which are rays defined by a vector <math>u</math> such that <math>Q(u)>0</math>. If <math>u</math> and <math>v</math> are two such vectors, then we may define a distance between
them by
- <math>d(u, v) = \operatorname{arccosh}(\frac{B(u,v)}{\sqrt{Q(u)Q(v)}}).</math>
This is a homogenous function, and so defines a distance between projective points. We can obtain either the hyperboloid model or the Klein model by normalizing these projective points. If we normalize <math>u</math> and <math>v</math> by changing sign if needed to make the first coordinate positive, and then dividing <math>u</math> and <math>v</math> to obtain
<math>u’ = \frac{u}{\sqrt{Q(u)}}, v’ = \frac{v}{\sqrt{Q(v)}}</math>
respectively, so that the points satisfy <math>Q(u’) = Q(v’) = 1</math>, we obtain
the hyperboloid model. If instead we normalize <math>u</math> and <math>v</math> by dividing through by the first coordinate, which since <math>Q(u)</math> and <math>Q(v)</math> are greater than zero cannot be zero, we obtain a subset of the projective plane, which are points in the interior of a unit disk. We may also view this as intersecting the lines through the origin with the hypersurface <math>t=1</math>.
Distance formula
From the projective hyperbolic distance function we may derive a distance function for the points in the unit disk. If <math>s</math> and <math>t</math> are two vectors with norm less than one, then we may define <math>u</math> as the vector in Minkowski space whose t coordinate is 1 followed by the coordinates for <math>s</math>, and <math>v</math> as the same for <math>t</math>. Then
- <math>d(s, t) = \operatorname{arccosh}(\frac{B(u,v)}{\sqrt{Q(u)Q(v)}})</math>
defines a distance function on the unit disk; this is the distance function of
the Klein model. In terms of the original vectors <math>s</math> and <math>t</math>, we may now rewrite this as
- <math>d(s, t) = \operatorname{arccosh}(\frac{1 - s \cdot t}{\sqrt{(1-s \cdot s)(1-t \cdot t)}}).</math>
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 <math>u</math> 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
- <math>s = \frac{2u}{1+u \cdot u}.</math>
Conversely, from a vector <math>s</math> of norm less than one representing a point of the Klein model, the corresponding point of the Poincaré disk model is given by
- <math>u = \frac{s}{1+\sqrt{1-s \cdot s}} =
\frac{(1-\sqrt{1-s \cdot s})s}{s \cdot s}.</math>
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 in the Klein model
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 <math>u</math> and <math>v</math>, and the chord whose endpoints are <math>s</math> and <math>t</math>, 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 <math>v=-u</math> and <math>t=-s</math>, then we are merely finding the angle between two unit vectors, and the formula for the angle <math>\theta</math> is
- <math>\cos(\theta) = u \cdot s.</math>
If <math>v=-u</math> but not <math>t=-s</math>, the formula becomes, in terms of the wedge product,
- <math>\cos^2(\theta) = \frac{P^2}{QR},</math>
where
- <math>P = u \cdot (s-t),</math>
- <math>Q = u \cdot u,</math>
- <math>R = (s-t) \cdot (s-t) - (s \wedge t) \cdot (s \wedge t)</math>
If both chords are not diameters, the general formula obtains
- <math>\cos^2(\theta) = \frac{P^2}{QR},</math>
where
- <math>P = (u-v) \cdot (s-t) - (u \wedge v) \cdot (s \wedge t),</math>
- <math>Q = (u-v) \cdot (u-v) - (u \wedge v) \cdot (u \wedge v),</math>
- <math>R = (s-t) \cdot (s-t) - (s \wedge t) \cdot (s \wedge t).</math>
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
- <math>P = (u-v) \cdot (s-t) + (u \cdot t)(v \cdot s) - (u \cdot s)(v \cdot t),</math>
- <math>Q = (1 - u \cdot v)^2,</math>
- <math>R = (1 - s \cdot t)^2.</math>
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 <math>B</math> intersecting a given chord <math>A</math> of the Klein model, which when extended to a line passes through the pole of the chord <math>A</math>, is perpendicular to <math>A</math>. This fact can be used to give an easy proof of the ultraparallel theorem.
See also
- Poincaré half-plane model
- Poincaré metric
- Inversive geometry
References
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