Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere, where the primary objects are points, great circles, and arcs of great circles. A great circle is the intersection of the sphere with a plane through its center; it is the spherical analogue of a straight line. The distance between two points is the length of the shorter arc of the great circle joining them, and the angle between two curves is the angle between their tangent lines at the intersection point.

Unlike Euclidean geometry, spherical geometry violates the parallel postulate: any two great circles intersect in two antipodal points, so there are no parallel lines. The sum of the angles of a spherical triangle is always greater than 180° and less than 540°, with the excess proportional to the triangle's area (Girard's theorem). The geometry is locally Euclidean but globally curved, with constant positive Gaussian curvature equal to 1/R² for a sphere of radius R.

The origins of spherical geometry lie in ancient Greek astronomy and geography. Theodosius of Bithynia (c. 100 BC) wrote Spherics, a systematic treatise on the geometry of the sphere, which was later translated into Arabic and Latin. Menelaus of Alexandria (c. 100 AD) introduced the spherical triangle and proved the spherical version of the Menelaus theorem, laying the groundwork for spherical trigonometry.

In the Islamic Golden Age, scholars such as Al-Battani and Ibn al-Haytham refined spherical methods for determining the qibla (direction to Mecca) and for astronomical calculations. The modern formulation of spherical geometry as a non-Euclidean geometry emerged in the 19th century with the work of Nikolai Lobachevsky and János Bolyai, who recognized that spherical geometry is a model of elliptic geometry, and later with Bernhard Riemann's concept of manifolds of constant positive curvature.

Spherical geometry is indispensable in navigation and geodesy: the shortest path between two points on Earth is a great-circle arc, and navigators use spherical trigonometry to compute courses and distances. In astronomy, spherical coordinates and spherical triangles are used to map the celestial sphere and to calculate positions of stars and planets. Cartography relies on spherical geometry to project the Earth's surface onto maps, with distortions managed by various map projections.

In modern physics, spherical geometry appears in general relativity, where the spatial geometry of a closed universe can be spherical, and in the study of black hole horizons. In computer graphics and robotics, spherical interpolation (slerp) is used for smooth rotations. Spherical geometry also underpins the mathematics of the 3-sphere and higher-dimensional spheres, which are central to topology and string theory.

Spherical geometry has several surprising features. For example, a spherical triangle can have two right angles, and its area is given by the spherical excess (sum of angles minus π) times the square of the radius. The polar triangle of a spherical triangle has angles that are supplementary to the original sides, and vice versa. Spherical geometry also includes the concept of lunes, which are regions bounded by two great circles; the area of a lune is proportional to its angle.

An often-overlooked fact is that spherical geometry is not the only elliptic geometry: the real projective plane, obtained by identifying antipodal points, gives a model of elliptic geometry where lines intersect in exactly one point. This model, called the elliptic plane, has no antipodal points and is the basis for the geometry of the projective plane. Spherical geometry also appears in the study of polyhedra: the spherical excess of a spherical polygon relates to the curvature at a vertex of a polyhedron, a connection used in the Gauss–Bonnet theorem.

Great circle

A circle on a sphere whose plane passes through the sphere's center; the shortest path between two points on the sphere.

Spherical triangle

A triangle formed by arcs of three great circles on a sphere.

Spherical excess

The amount by which the sum of the angles of a spherical triangle exceeds 180°.

Lune

A region on a sphere bounded by two great circles.

Antipodal points

Pairs of points on a sphere that are directly opposite each other, such as the North and South poles.