Poincaré disk model

It is named after Henri Poincaré, because his rediscovery of this representation fourteen years later became better known than the original work of Beltrami.

[1] 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.

The disk model was first described by Bernhard Riemann in an 1854 lecture (published 1868), which inspired an 1868 paper by Eugenio Beltrami.

[2] Henri Poincaré employed it in his 1882 treatment of hyperbolic, parabolic and elliptic functions,[3] but it became widely known following Poincaré's presentation in his 1905 philosophical treatise, Science and Hypothesis.

[4] 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 distance of the point considered from the centre, the absolute temperature will be proportional to

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.

"[4] (pp.65-68)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[5] and of Hans Reichenbach.

The vertical bars indicate Euclidean length of the line segment connecting the points between them in the model (not along the circle arc); ln is the natural logarithm.

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 associated metric tensor of the Poincaré disk model is given by[8] where the xi 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, with respect to these frames and the Levi-Civita connection, the connection forms are given by the unique skew-symmetric matrix of 1-forms

Another way is: A basic construction of analytic geometry is to find a line through two given points.

), 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 In the Euclidean plane the generalized circles (curves of constant curvature) are lines and circles.

In the Poincaré disk model, all of these are represented by straight lines or circles.

Its axis is the hyperbolic line that shares the same two ideal points.

A horocycle (a curve whose normal or perpendicular geodesics are limiting parallels, all converging asymptotically to the same ideal point), 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.

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.

(Euclidean intuition can be misleading because the scale of the model increases to infinity at the boundary circle.)

A disadvantage is that the Klein disk model is not conformal (circles and angles are distorted).

(the ideal points remain on the same spot) also the pole of the chord in the Klein disk model is the center of the circle that contains the arc in the Poincaré disk model.

of the upper half plane is given by the inverse of the Cayley transform

In terms of real coordinates, a point (x,y) in the disk model maps to

The result is the corresponding point of the Poincaré disk model.

For Cartesian coordinates (t, xi) on the hyperboloid and (yi) on the plane, the conversion formulas are:

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

M. C. Escher explored the concept of representing infinity on a two-dimensional 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.

HyperRogue, a roguelike game, uses the hyperbolic plane for its world geometry, and also uses the Poincaré disk model.