Dictionary:Spherical excess

The amount by which the sum of the angles of a triangle on the surface of a sphere exceeds 180 degrees. For a sphere, this excess is the area of the triangle. For a spherical triangle on an ellipsoid, such as the Earth approximates, the spherical excess $\varepsilon$ is approximately:

$\varepsilon =mbc{\text{sin}}\alpha$,

where b and c are two adjacent sides of the triangle which intersect at the angle $\alpha$, m = latitude function=${\frac {\rho }{2RN}}$, $\rho$=number of seconds of arc/radian=206 264.8, R=radius of curvature in the meridian, and N=radius of curvature in the prime vertical. Values of m for various ellipsoids are obtained from tables; the value for the center of the triangle to the nearest half degree is usually used. For greater precision a correction factor is often applied in iterative fashion.

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