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Poincaré Hyperbolic Disk


HyperbolicTessellation

The Poincaré hyperbolic disk is a two-dimensional space having hyperbolic geometry defined as the disk {x in R^2:|x|<1}, with hyperbolic metric

 ds^2=(dx^2+dy^2)/((1-x^2-y^2)^2).
(1)

The Poincaré disk is a model for hyperbolic geometry in which a line is represented as an arc of a circle whose ends are perpendicular 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).

PoincareHyperbolicDiskPoincareDisk
PoincareDiskCons

The endpoints of any arc can be specified by two angles around the disk theta_1 and theta_2. Define

theta=1/2(theta_1+theta_2)
(2)
dtheta=1/2|theta_1-theta_2|.
(3)

Then trigonometry shows that in the above diagram,

r=tan(dtheta)
(4)
y=sin(dtheta)tan(dtheta),
(5)

so the radius of the circle forming the arc is r and its center is located at (Rcostheta,Rsintheta), where

 R=cos(dtheta)+y=sec(dtheta).
(6)

The half-angle subtended by the arc is then

 sinphi=(sin(dtheta))/(tan(dtheta))=cos(dtheta),
(7)

so

 phi=sin^(-1)[cos(dtheta)].
(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.

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).


See also

Elliptic Plane, Hyperbolic Geometry, Hyperbolic Metric, Klein-Beltrami Model, Poincaré Metric

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References

Anderson, J. W. "The Poincaré Disc Model." §4.1 in Hyperbolic Geometry. New York: Springer-Verlag, pp. 95-104, 1999.Escher, M. C. Circle Limit IV (Heaven and Hell). Woodcut in black and ocre. 1960. http://www.mcescher.com/Gallery/recogn-bmp/LW436.jpg.Goodman-Strauss, C. "Compass and Straightedge in the Poincaré Disk." Amer. Math. Monthly 108, 38-49, 2001.Segerman, H. "Autologlyphs." http://www.stanford.edu/~segerman/autologlyphs.html#Poincaredisk.Segerman, H. and Dehaye, P.-O. Cover of Math. Intell. 26, No. 2, 2004.Trott, M. Graphica 1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael Trott. Champaign, IL: Wolfram Media, pp. 10 and 83, 1999.Trott, M. The Mathematica GuideBook for Graphics. New York: Springer-Verlag, p. xxxvi, 2004. http://www.mathematicaguidebooks.org/.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 188-189, 1991.

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Poincaré Hyperbolic Disk

Cite this as:

Weisstein, Eric W. "Poincaré Hyperbolic Disk." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PoincareHyperbolicDisk.html

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