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The Poincaré disk is a model of hyperbolic space. Geodesics in the Poincaré disk model are of two types:

1. The intersection of a line through the origin and the open disk.

2. The intersection of a circle not through the origin that intersects the boundary of the disk orthogonally.

Given any two points in the Poincaré disk, there is a unique geodesic that passes through both of them. Hence, if two points are given and a third point is chosen that is not contained in the geodesic determined by the first two, these three points determine a unique hyperbolic triangle.

Hyperbolic space has a metric that confers a constant curvature of . In the Poincaré disk model the metric tensor is

.

The Gauss-Bonnet theorem applied to this metric on the boundary of the triangle gives a formula for the area of the hyperbolic triangle, namely

,

where , , and are the measures of the interior angles of the triangle. It is worth pointing out that the sum of the interior angles of a hyperbolic triangle must be less than (which is quite different from Euclidean geometry, in which that sum must equal ).

References:

J. W. Anderson, Hyperbolic Geometry, New York: Springer-Verlag, 1999 pp. 95–104.

S. Katok, Fuchsian Groups, Chicago: University of Chicago Press, 1992.