Great circle

Lore & Background

For any pair of distinct non-antipodal points on a sphere, there is exactly one great circle passing through both. However, every great circle through any point also passes through its antipodal point, so infinitely many great circles connect two antipodal points. The shorter of the two arcs between distinct points is called the minor arc, and its length—the great-circle distance—is proportional to the central angle formed by the two points and the sphere's center. A great circle is the largest circle that can be drawn on a given sphere. Any diameter of a great circle coincides with a diameter of the sphere, making every great circle concentric with the sphere and of the same radius. In contrast, any other circle on the sphere is a small circle, which is the intersection of the sphere with a plane not passing through its center. Every circle in Euclidean 3-space is a great circle of exactly one sphere. The disk bounded by a great circle is called a great disk, defined as the intersection of a ball and a plane through its center. In higher dimensions, great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in Euclidean space R^(n+1). Half of a great circle may be called a great semicircle, as seen in parts of a meridian in astronomy.

Reader's Guide

The concept of a great circle is fundamental to spherical geometry and navigation. Because the minor arc of a great circle is the shortest surface path between two points on a sphere, great-circle distance is the intrinsic distance on a sphere. This property is proven using calculus of variations: by introducing spherical coordinates with one point as the north pole, the length functional is minimized when the Euler–Lagrange equations yield a constant C that must be zero, forcing the longitudinal derivative to vanish and thus the path to lie along a meridian—a great circle. This derivation confirms that great circles are geodesics, making them the spherical analog of straight lines in Euclidean space. Their significance extends to astronomy (e.g., meridians as great semicircles) and to higher-dimensional geometry, where great circles on the n-sphere are defined by intersections with 2-planes through the origin.

Did You Know?

• A great circle is the largest circle that can be drawn on any given sphere.
• Any diameter of a great circle coincides with a diameter of the sphere, so every great circle is concentric with the sphere and shares the same radius.
• Every circle in Euclidean 3-space is a great circle of exactly one sphere.
• Half of a great circle may be called a great semicircle, as in parts of a meridian in astronomy.