Spherical geometry

geometry of the two-dimensional surface of a sphere

Spherical geometry is the use of geometry on a sphere. It was started for cartography, as well as for making maps of stars. It is different from Euclidean geometry (which is always on a plane), and Non-Euclidean geometry. Points are defined in the same way as they are in Euclidean geometry: A point is at a defined location on the sphere. A "staight line" is different though: It is the shortest path between two points, which stays on the surface of the plane. Some theorems of Euclidean geometry cannot be used on the sphere, many of them have been adapted though.

On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180. The surface of a sphere can be represented by a collection of two dimensional maps. Therefore it is a two dimensional manifold.