Spherical geometry: geometry on the surface of a sphere

Basic concepts and elements

Key objects in spherical geometry are defined much like their flat counterparts but with important changes in meaning. A point is a location on the surface; a line is a great circle arc connecting two points. The shortest path between nearby points on the surface follows a great-circle arc and is called a geodesic. Two great circles always meet in two antipodal points, so the idea of parallel lines from Euclidean geometry does not exist in the same way on a sphere.

Principal properties and theorems

• Triangle angle sum: the sum of angles of a spherical triangle exceeds 180 degrees; the excess is proportional to the triangle’s area.
• No parallel lines: any two distinct great circles intersect in two points.
• Area and angles are closely related: spherical excess gives a direct measure of area on a unit sphere.
• Congruence and similarity behave differently: similar shapes of different size are not generally obtainable by scaling on a sphere as they are in a plane.

History and development

Spherical geometry has roots in ancient astronomy and navigation. Early practitioners used spherical constructions to predict celestial positions and to create sea charts. The subject evolved alongside advances in trigonometry and cartography, with further formalization when mathematicians contrasted it with Euclidean and other non-Euclidean geometries. For historical connections to mapmaking and star charts see cartography and Euclidean geometry.

Applications and examples

Practical uses include great-circle navigation for aircraft and ships (shortest long-distance routes), design and interpretation of global maps, and the modeling of the celestial sphere in astronomy. Modern technologies such as global positioning systems and geodesy rely on spherical or spheroidal models of Earth; see discussions of map projection trade-offs at map projection resources and basic geodesy references at geodesy.

Notable distinctions and further study

Compared with planar geometry, many familiar theorems must be modified when used on a sphere. Studying spherical geometry provides intuition for other curved spaces and for general relativity’s geometric viewpoint. For concise definitions of the core terms and advanced topics consult further reading.