Spherical geometry

Spherical geometry studies distances, angles, and figures on a sphere’s surface. Its shortest paths are arcs of great circles, and its triangles have angle sums greater than 180 degrees.

Great circle: A circle on a sphere whose center coincides with the sphere’s center. Great circles provide the straightest paths and the sides of spherical triangles.
Spherical law of cosines: A relation among the sides and angles of a triangle on a sphere. It computes spherical side lengths or angles where planar triangle formulas fail.
Euclidean geometry: The geometry of flat space described by Euclid’s postulates. Its triangle angles sum to 180 degrees, unlike spherical triangles.
Spherical trigonometry: The study of relationships among sides and angles of spherical triangles. It supplies calculation tools for navigation and celestial measurements on a sphere.
Geodesic: A locally shortest path between points in a curved space. On a sphere, geodesics are arcs of great circles.
Haversine formula: A trigonometric formula for the great-circle distance between two points on a sphere. It turns latitude and longitude differences into spherical distance.
Hyperbolic geometry: A non-Euclidean geometry with constant negative curvature and many parallels through an external point. It reverses spherical geometry’s angle-sum behavior: triangles have less than 180 degrees.
Celestial navigation: The determination of position using observed celestial bodies and their predicted locations. Spherical triangles connect an observer, the celestial pole, and a measured star.
Spherical triangle: A region on a sphere bounded by three arcs of great circles. Its angles sum to more than 180 degrees, revealing the sphere’s curvature.
Gaussian curvature: The intrinsic curvature of a surface, defined as the product of its principal curvatures. A sphere’s positive Gaussian curvature accounts for its non-Euclidean triangle angle sums.

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