Poincaré Hyperbolic Disk

The Poincaré hyperbolic disk is a two-dimensional space having hyperbolic geometry defined as the disk , with hyperbolic metric

(1)

The Poincaré disk is a model for hyperbolic geometry in which a geodesic is an arc of a circle orthogonal to the disk's boundary (and diameters are also permitted). Two arcs which do not meet correspond to parallel rays, arcs which meet orthogonally correspond to perpendicular lines, and arcs which meet on the boundary are a pair of limits rays. The illustration above shows a hyperbolic tessellation similar to M. C. Escher's Circle Limit IV (Heaven and Hell) (Trott 1999, pp. 10 and 83).

The endpoints of any arc can be specified by two angles around the disk and . Define

			(2)
			(3)

Then trigonometry shows that in the above diagram,

			(4)
			(5)

so the radius of the circle forming the arc is and its center is located at , where

(6)

The half-angle subtended by the arc is then

(7)

so

(8)

The Poincaré hyperbolic disk represents a conformal mapping, so angles between rays can be measured directly. There is an isomorphism between the Poincaré disk model and the Klein-Beltrami model.

The modular group Gamma acts on the disk by conjugating its action on the upper half-plane with the Möbius transformation

(9)

The generators and then act as

			(10)
			(11)

The standard fundamental region , in the upper half-plane maps to a pair of reflected hyperbolic triangles (Milne 2017, pp. 32-33). One has vertices , , and the ideal vertex on the unit circle, with angles , , and 0. Its sides are geodesics, so their drawn boundaries are arcs orthogonal to the unit circle or diameters, rather than straight chords. The left panel shows this pair, and the right panel shows 20 of its modular transforms.

A tiling of the Poincaré disk using the words "Poincaré disk" with five pentagons around each vertex appears on the cover of a 2004 volume of the Mathematical Intelligencer (Segerman and Dehaye 2004).