Poincaré disk model

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In geometry, the Poincaré disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or unit ball, and the lines of the geometry are segments of circles contained in the disk orthogonal to the boundary of the disk, or else diameters of the disk. Along with the Klein model and the Poincaré half-space model, it was proposed by Eugenio Beltrami who used these models to show hyperbolic geometry was equiconsistent with Euclidean geometry.

Contents

1 Distance function
2 Metric form
3 Relation to the hyperboloid model
4 Analytic geometry constructions in the hyperbolic plane
5 Angles in the Poincaré disk model
6 See also
7 References

Distance function

If u and v are two vectors in real n-dimensional vector space Rⁿ with the usual Euclidean norm, both of which have norm less than one, then we may define an isometric invariant by

[\delta (u, v) = 2 \frac,]

where ||*|| denotes the usual Euclidean norm. Then the distance function is

[d(u, v) = \operatorname (1+\delta (u,v)).]

Such a distance function is defined for any two vectors of norm less than one, and makes the set of such vectors into a metric space which is a model of hyperbolic space of constant curvature -1. The model has the conformal property that the angle between two intersecting curves in hyperbolic space is the same as the angle in the model.

Metric form

The metric form of the Poincaré disk model is given by

[ds^2 = 4 \frac.]

Relation to the hyperboloid model

The Poincaré disk model, as well as the Klein model, are related to the hyperboloid model projectively. If we have a point [t, x₁, ... x_n] on the upper sheet of the hyperboloid of the hyperboloid model, thereby defining a point in the hyperboloid model, we may project it onto the hypersurface t=0 by intersecting it with a line drawn through [-1, 0, ..., 0]. The result is the corresponding point of the Poincaré disk model.

Analytic geometry constructions in the hyperbolic plane

A basic construction of analytic geometry is to find a line through two given points. In the Poincaré disk model, lines in the plane are defined by portions of circles having equations of the form

[x^2 + y^2 + a x + b y + 1 = 0,]

which is the general form of a circle orthogonal to the unit circle, or else by diameters. Given two points u and v in the disk which do not lie on a diameter, we can solve for the circle of this form passing through both points, and obtain

[x^2 + y^2 + \fracx +]

[\fracy + 1 = 0.]

If the points u and v are points on the boundary of the disk not lying at the endpoints of a diameter, the above simplifies to

[x^2+y^2+\fracx - \fracy + 1 = 0.]

Angles in the Poincaré disk model

We may compute the angle between the circular arc whose ideal points, which are its endpoints given by unit vectors u and v, and the arc 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 model lines 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 θ is

[\cos(\theta) = u \cdot s.]

If v=-u but not t=-s, the formula becomes, in terms of the wedge product,

[\cos^2(\theta) = \frac,]

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(\theta) = \frac,]

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 = (u \cdot v)(2 - u \cdot v),]

[R = (s \cdot t)(2 - s \cdot t).]

References

James W. Anderson, Hyperbolic Geometry, second edition, Springer, 2005

Eugenio Beltrami, Theoria fondamentale delgi spazil di curvatura constanta, Annali. di Mat., ser II 2 (1868), 232-255

Saul Stahl, The Poincaré Half-Plane, Jones and Bartlett, 1993

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