Hyperbolic Geometry
In hyperbolic geometry, a hyperbolic triangle is a figure in the hyperbolic plane, analogous to a triangle in Euclidean geometry, consisting of three sides and three angles. The relations among the angles and sides are analogous to those of spherical trigonometry; they are most conveniently stated if the lengths are measured in terms of a special unit of length analogous to a radian. In terms of the Gaussian curvature K of the plane this unit is given by
In all the trig formulas stated below the sides a, b, and c must be measured in this unit. In a hyperbolic triangle the sum of the angles A, B, C (respectively opposite to the side with the corresponding letter) is strictly less than a straight angle. The difference is often called the defect of the triangle. The area of a hyperbolic triangle is equal to its defect multiplied by the square of R:
The corresponding theorem in spherical geometry is Girard's theorem first proven by Johann Heinrich Lambert.
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