Legendre's theorem

The set of Legendre describes how small spherical triangles verebnet may be so in their calculation as in the flat trigonometry can be performed. It was set up in 1787 by Adrien-Marie Legendre .

statement

The sentence says that

a small spherical triangle according to sides and angles can be calculated approximately like a flat triangle with the same sides, if one takes as the angle of the flat triangle the angle of the spherical triangle reduced by one third of the spherical excess . ${\displaystyle \epsilon }$

With this simplification, the sides and angles can be calculated according to the sine or cosine law of plane trigonometry.

application

The theorem was used, for example, in geodesy for triangulations with large triangular meshes. Using the approximation, triangles on the earth's surface with a side length of up to 200 km can be calculated with millimeter accuracy. In these calculations, the spherical excess can be determined from the triangular area. If the spherical excess is not taken into account, the curvature of the earth would become noticeable in precision measurements from a few 10 km. For small triangles with a side length of a few kilometers, the application does not increase the accuracy in practice, since the influence of the angle measurement inaccuracy is greater than the influence of the earth's curvature.