Poincaré disk model

Poincaré disc model of the truncated triheptagonal tiling.

Poincaré 'ball' model view of the hyperbolic regular icosahedral honeycomb, {3,5,3}

In geometry, the Poincaré disk model or Poincaré ball 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 straight lines consist of all segments of circles contained within the disk that are orthogonal to the boundary of the disk, plus all 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.

Metric

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 1, then we may define an isometric invariant by

\delta (u, v) = 2 \frac{\lVert u-v \rVert^2}{(1-\lVert u \rVert^2)(1-\lVert v \rVert^2)} \,,

where $\lVert \cdot \rVert$ denotes the usual Euclidean norm. Then the distance function is

d(u, v) = \operatorname{arcosh} (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.

The associated metric tensor of the Poincaré disk model is given by

ds^2 = 4 \frac{\sum_i dx_i^2}{(1-\sum_i x_i^2)^2}

where the x_i are the Cartesian coordinates of the ambient Euclidean space. The geodesics of the disk model are circles perpendicular to the boundary sphere Sⁿ⁻¹.

Relation to the hyperboloid model

The hyperboloid model can be seen as the equation of t²=x²+y²+1. It can be used to construct a Poincaré disk model as a perspective projection viewed from (t=-1,x=0,y=0), projecting the upper half hyperboloid onto an (x,y) unit disk at t=0. Planes passing through the origin represents geodesics on the hyperbolic plane. The red circular arc is geodesic in Poincaré disk model; it projects to the brown geodesic on the green hyperboloid.

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.

For Cartesian coordinates (t, x_i) on the hyperboloid and (y_i) on the plane, the conversion formulae are:

y_i = \frac{x_i}{1 + t}

(t, x_i) = \frac {\left( 1+\sum{y_i^2},\, 2 y_i \right)} {1-\sum{y_i^2}} \,.

Compare the formulae for stereographic projection between a sphere and a plane.

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

\begin{align} & {} x^2 + y^2 + \frac{u_2(v_1^2+v_2^2)-v_2(u_1^2+u_2^2)+u_2-v_2}{u_1v_2-u_2v_1}x \\[8pt] & {} \quad + \frac{v_1(u_1^2+u_2^2)-u_1(v_1^2+v_2^2)+v_1-u_1}{u_1v_2-u_2v_1}y + 1 = 0 \,. \end{align}

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+\frac{2(u_2-v_2)}{u_1v_2-u_2v_1}x - \frac{2(u_1-v_1)}{u_1v_2-u_2v_1}y + 1 = 0 \,.

Angles

We may compute the angle between the circular arc whose endpoints (ideal points) are 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 models' 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 ( $\wedge$ ),

\cos^2(\theta) = \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(\theta) = \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 \,.

Isometric Transformations

The analog of a reflection about a line in hyperbolic space is a reflection about a geodesic, which can be represented in the model as a circle inversion about the circle that represents the geodesic. Rotations and translations can be represented as a combination of two reflections about different geodesics. In the case of rotations, the two geodesics intersect, while in the case of translations, they do not.

One result of this is that if hyperbolic space is translated such that the origin in the unit Poincaré disk is translated to $\mathbf{v}$ , $\mathbf{x}$ is translated to

\frac{(1+2\mathbf{v}\cdot\mathbf{x}+|\mathbf{x}|^2)\mathbf{v}+(1-|\mathbf{v}|^2)\mathbf{x}}{1+2\mathbf{v}\cdot\mathbf{x}+|\mathbf{v}|^2|\mathbf{x}|^2}

This also applies for higher dimensions.

Artistic realizations

M. C. Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter around 1956 inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher's wood engravings Circle Limit I–IV demonstrate this concept between 1958 and 1960, the final one being Circle Limit IV: Heaven and Hell in 1960.^[1] According to Bruno Ernst, the best of them is Circle Limit III.

References

↑ Escher's Circle Limit Exploration