Hyperbolic law of haversines


In hyperbolic geometry, the law of cosines is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry.

Take a hyperbolic plane whose Gaussian curvature is -1/k2 . Then given a hyperbolic triangle ABC with angles α, β, γ, and side lengths BC = a, AC = b, and AB = c, the two rules hold.

In cases where a/k is small, and being solved for, the numerical precision of the standard form of the hyperbolic law of cosines will drop due to rounding errors, for the exact same reason it does in the Spherical law of cosines. The hyperbolic version of the law of haversines can prove useful in this case, as shown here.

Related formulas


alength (dimensionless)
kcarvature (dimensionless)
blength (dimensionless)
clength (dimensionless)
αangle (dimensionless)