Spherical circle

In spherical geometry, a spherical circle (often shortened to circle) is the locus of points on a sphere at constant spherical distance (the spherical radius) from a given point on the sphere (the pole or spherical center). It is a curve of constant geodesic curvature relative to the sphere, analogous to a line or circle in the Euclidean plane; the curves analogous to straight lines are called great circles, and the curves analogous to planar circles are called small circles or lesser circles. If the sphere is embedded in three-dimensional Euclidean space, its circles are the intersections of the sphere with planes, and the great circles are intersections with planes passing through the center of the sphere.

Fundamental concepts

Intrinsic characterization

A spherical circle with zero geodesic curvature is called a great circle, and is a geodesic analogous to a straight line in the plane. A great circle separates the sphere into two equal hemispheres, each with the great circle as its boundary. If a great circle passes through a point on the sphere, it also passes through the antipodal point (the unique furthest other point on the sphere). For any pair of distinct non-antipodal points, a unique great circle passes through both. Any two points on a great circle separate it into two arcs analogous to line segments in the plane; the shorter is called the minor arc and is the shortest path between the points, and the longer is called the major arc.

A circle with non-zero geodesic curvature is called a small circle, and is analogous to a circle in the plane. A small circle separates the sphere into two spherical disks or spherical caps, each with the circle as its boundary.