Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry that is not Euclidean. Two practical applications of the principles of spherical geometry are navigation and astronomy. Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry that is not Euclidean. Two practical applications of the principles of spherical geometry are navigation and astronomy. In plane (Euclidean) geometry, the basic concepts are points and (straight) lines. On a sphere, points are defined in the usual sense. The equivalents of lines are not defined in the usual sense of 'straight line' in Euclidean geometry, but in the sense of 'the shortest paths between points', which are called geodesics. On a sphere, the geodesics are the great circles; other geometric concepts are defined as in plane geometry, but with straight lines replaced by great circles. Thus, in spherical geometry, angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects; for example, the sum of the interior angles of a triangle exceeds 180 degrees. Spherical geometry is not elliptic geometry, but is rather a subset of elliptic geometry. For example, it shares with that geometry the property that a line has no parallels through a given point. Contrast this with Euclidean geometry, in which a line has one parallel through a given point, and hyperbolic geometry, in which a line has two parallels and an infinite number of ultraparallels through a given point. An important geometry related to that of the sphere is that of the real projective plane; it is obtained by identifying antipodal points (pairs of opposite points) on the sphere. Locally, the projective plane has all the properties of spherical geometry, but it has different global properties. In particular, it is non-orientable, or one-sided.