Poincaré disk model

In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle.
The group of orientation preserving isometries of the disk model is given by the projective special unitary group, the quotient of the special unitary group SU by its center.
Along with the Klein model and the Poincaré half-space model, it was proposed by Eugenio Beltrami who used these models to show that hyperbolic geometry was equiconsistent with Euclidean geometry. It is named after Henri Poincaré, because his rediscovery of this representation fourteen years later became better known than the original work of Beltrami.
The Poincaré ball model is the similar model for 3 or n-dimensional hyperbolic geometry in which the points of the geometry are in the n-dimensional unit ball.

History

The disk model was first described by Bernhard Riemann in an 1854 lecture, which inspired an 1868 paper by Eugenio Beltrami. Henri Poincaré employed it in his 1882 treatment of hyperbolic, parabolic and elliptic functions, but it became widely known following Poincaré's presentation in his 1905 philosophical treatise, Science and Hypothesis. There he describes a world, now known as the Poincaré disk, in which space was Euclidean, but which appeared to its inhabitants to satisfy the axioms of hyperbolic geometry:

"Suppose, for example, a world enclosed in a large sphere and subject to the following laws: The temperature is not uniform; it is greatest at their centre, and gradually decreases as we move towards the circumference of the sphere, where it is absolute zero. The law of this temperature is as follows: If be the radius of the sphere, and the distance of the point considered from the centre, the absolute temperature will be proportional to. Further, I shall suppose that in this world all bodies have the same co-efficient of dilatation, so that the linear dilatation of any body is proportional to its absolute temperature. Finally, I shall assume that a body transported from one point to another of different temperature is instantaneously in thermal equilibrium with its new environment....
If they construct a geometry, it will not be like ours, which is the study of the movements of our invariable solids; it will be the study of the changes of position which they will have thus distinguished, and will be 'non-Euclidean displacements,' and this will be non-Euclidean geometry. So that beings like ourselves, educated in such a world, will not have the same geometry as ours."

Poincaré's disk was an important piece of evidence for the hypothesis that the choice of spatial geometry is conventional rather than factual, especially in the influential philosophical discussions of Rudolf Carnap and of Hans Reichenbach.

Lines and distance

Hyperbolic straight lines or geodesics consist of all arcs of Euclidean circles contained within the disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.
Distances in this model are Cayley–Klein metrics.
Given two distinct points p and q inside the disk, the unique hyperbolic line connecting them intersects the boundary at two ideal points, a and b. Label them so that the points are, in order, a, p, q, b, that is, so that and.
The hyperbolic distance between p and q is then
The vertical bars indicate Euclidean length of the line segment connecting the points between them in the model ; ln is the natural logarithm.
Equivalently, 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
where denotes the usual Euclidean norm. Then the distance function is
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.
Specializing to the case where one of the points is the origin and the Euclidean distance between the points is r, the hyperbolic distance is:
where is the inverse hyperbolic function of the hyperbolic tangent. If the two points lie on the same radius and point lies between the origin and point , their hyperbolic distance is
This reduces to the previous special case if.

Metric and curvature

The associated metric tensor of the Poincaré disk model is given by
where the x_i are the Cartesian coordinates of the ambient Euclidean space.
An orthonormal frame with respect to this Riemannian metric is given by
with dual coframe of 1-forms

In two dimensions

In two dimensions, with respect to these frames and the Levi-Civita connection, the connection forms are given by the unique skew-symmetric matrix of 1-forms that is torsion-free, i.e., that satisfies the matrix equation. Solving this equation for yields
where the curvature matrix is
Therefore, the curvature of the hyperbolic disk is

Construction of lines

By compass and straightedge

The unique hyperbolic line through two points and not on a diameter of the boundary circle can be constructed by:

let be the inversion in the boundary circle of point
let be the inversion in the boundary circle of point
let be the midpoint of segment
let be the midpoint of segment
Draw line through perpendicular to segment
Draw line through perpendicular to segment
let be where line and line intersect.
Draw circle with center and going through .
The part of circle that is inside the disk is the hyperbolic line.

If P and Q are on a diameter of the boundary circle that diameter is the hyperbolic line.
Another way is:

let be the midpoint of segment
Draw line m through perpendicular to segment
let be the inversion in the boundary circle of point
let be the midpoint of segment
Draw line through perpendicular to segment
let be where line and line intersect.
Draw circle with center and going through .
The part of circle that is inside the disk is the hyperbolic line.
By analytic geometry

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
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
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

Angles

We may compute the angle between the circular arc whose endpoints 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
If v = −u but not t = −s, the formula becomes, in terms of the wedge product,
where
If both chords are not diameters, the general formula obtains
where
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

Cycles

In the Euclidean plane the generalized circles are lines and circles. On the sphere, they are great and small circles. In the hyperbolic plane, there are 4 distinct types of generalized circles or cycles: circles, horocycles, hypercycles, and geodesics. In the Poincaré disk model, all of these are represented by straight lines or circles.
A Euclidean circle:

that is completely inside the disk is a hyperbolic circle;
that is inside the disk and tangent to the boundary is a horocycle;
that intersects the boundary orthogonally is a hyperbolic line; and
that intersects the boundary non-orthogonally is a hypercycle.

A Euclidean chord of the boundary circle:

that goes through the center is a hyperbolic line; and
that does not go through the center is a hypercycle.
Circles

A circle is a circle completely inside the disk not touching or intersecting its boundary. The hyperbolic center of the circle in the model does not in general correspond to the Euclidean center of the circle, but they are on the same radius of the Poincaré disk.

Hypercycles

A hypercycle is a Euclidean circle arc or chord of the boundary circle that intersects the boundary circle at a positive but non-right angle. Its axis is the hyperbolic line that shares the same two ideal points. This is also known as an equidistant curve.

Horocycles

A horocycle, is a circle inside the disk that is tangent to the boundary circle of the disk. The point where it touches the boundary circle is not part of the horocycle. It is an ideal point and is the hyperbolic center of the horocycle. It is also the point to which all the perpendicular geodesics converge.
In the Poincaré disk model, the Euclidean points representing opposite "ends" of a horocycle converge to its center on the boundary circle, but in the hyperbolic plane every point of a horocycle is infinitely far from its center, and opposite ends of the horocycle are not connected.