ABSTRACT
A number of central concepts and methods for computations on the sphere are presented, for use both on a spherical approximation of the Earth, and on the celestial sphere. Firstly, the spherical excess of a spherical triangle is discussed, and its connection with the surface area of the triangle. Equations for computing the surface area are derived. Trigonometric identities for a rectangular spherical triangle are derived, and then for a general spherical triangle. The results obtained are the known sine and cosine rules on the sphere. Then, the same sine and cosine rules are derived by representing the triangle vertices by three-dimensional vectors on the unit sphere. Then, the concept of polarization is presented, allowing the derivation of a second type of spherical cosine rule. After shortly presenting historical methods for computing “small” spherical triangles, i.e., those with small spherical excesses, including a useful half-angle cosine rule that preserves numerical accuracy, the forward and reverse geodetic problems on the sphere are discussed.
