The Poincaré hyperbolic disk is a two-dimensional space having hyperbolic geometry defined as the disk {x element R^2 : left bracketing bar x right bracketing bar <1}, with hyperbolic metric d s^2 = (d x^2 + d y^2)/(1 - x^2 - y^2)^2. 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).
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