The hyperbolic law of cosines is an identity in hyperbolic geometry relating the measure of an angle of a triangle in the hyperbolic plane with the lengths of its three sides.
In particular, for the figure at right with vertices (and angle measures) \(A\), \(B\), and \(C\) and opposite side lengths \(a\), \(b\), and \(c\), the hyperbolic law of cosines is—
\[\cosh c = \cosh a \cosh b - \sinh a \sinh b \cos C\]
There is also a dual law relating a single side length with the three angle measures:
\[\cos C = -\cos A \cos B + \sin A \sin B \cosh c\]
There are also analogous identities for more general types of triangles in the hyperbolic plane. We discuss these in the next section.
Laws for generalized hyperbolic triangles
By viewing the triangle used in the law of cosines as a triangle in the real projective plane \(\mathbf{RP}^{2}\) instead of the hyperbolic plane, and then permitting the triangle's vertices to range outside the hyperbolic plane, the law takes other forms. This yields similar identities for certain other types of polygons in the hyperbolic plane. We call these polygons generalized hyperbolic triangle. They fall into five types (see the linked article for more details):
- Type I: a triangle,
- Type II: a quadrilateral with exactly one pair of adjacent right angles,
- Type III: a quadrilateral with one pair of opposite right angles,
- Type IV: a pentagon with at least four right angles, and
- Type V: a hexagon with all right angles.
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References
- [Th] William P. Thurston, Three-dimensional Geometry and Topology, Vol. 1, edited by Silvio Levy, Princeton Mathematical Series, 35, 1997.
