Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a non-Euclidean geometry.

In plane geometry the basic concepts are points and lines.

On the sphere, points are defined in the usual sense. The equivalents of lines are not defined in the usual sense of "straight line" but in the sense of "the shortest paths between points" which is called a geodesic. On the sphere the geodesics are the great circles, so the other geometric concepts are defined like in plane geometry but with lines replaced by great circles. Thus, in spherical geometry angles are defined between great circles.

The spherical geometry is the simplest model of the elliptic or Riemannian geometry, in which a line has no parallels through a given point, and it is opposite to Lobachevskian or hyperbolic geometry, in which a line has at least two parallels through a given point.

Spherical geometry has important practical uses in celestial navigation and astronomy.

There are two basic versions of this theory; one in which opposite points of the sphere are not identified, and the other in which they are identified.