Great circle
Largest circle on a sphere, shortest path between two points.
A great circle, also called an orthodrome, is the circular intersection of a sphere and a plane that passes through the sphere's center. In spherical geometry, great circles serve as the natural analog of straight lines in Euclidean space, and the minor arc between two points on a sphere is the shortest surface path connecting them.
- field
- Mathematics
- known_for
- Shortest path on a sphere (geodesic), largest circle on a sphere, analog of straight lines in spherical geometry
Lore & Background
A great circle is defined as the intersection of a sphere with a diametral plane—a plane passing through the sphere's center. Any arc of a great circle is a geodesic of the sphere, making great circles the spherical-geometry equivalent of straight lines in Euclidean space. For any two distinct non-antipodal points on a sphere, there is exactly one great circle passing through both; for antipodal points, infinitely many great circles exist. The shorter of the two arcs between two points is called the minor arc, and its length is the great-circle distance, proportional to the central angle formed by the points and the sphere's center.
Reader's Guide
Great circles are fundamental in spherical geometry and navigation because they represent the shortest path between two points on a sphere's surface. The derivation of this property uses calculus of variations: by introducing spherical coordinates and applying the Euler–Lagrange equation to the arc length functional, one shows that the minimizing curve satisfies conditions leading to a constant longitude, meaning the path lies along a great circle. Every great circle is concentric with the sphere and shares its radius; any other circle on the sphere is a small circle, the intersection with a plane not through the center. In higher dimensions, great circles on the n-sphere are intersections with 2-planes through the origin in Euclidean space R^(n+1). The disk bounded by a great circle is called a great disk, and half of a great circle is a great semicircle, as seen in parts of a meridian in astronomy.
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.
- Every circle in Euclidean 3-space is a great circle of exactly one sphere.
- The minor arc of a great circle is the shortest surface-path between two distinct points on a sphere.
Frequently Asked Questions
What exactly is a great circle?
A great circle is the largest circle you can trace on a sphere, created where a plane cuts through the sphere's center. It plays the same role on a curved surface that a straight line plays on a flat plane.
How is a great circle different from a small circle like a latitude line?
A great circle always passes through the sphere's center, making it the widest circle the surface can contain, while a small circle (such as most latitude lines) does not. Only great circles split a sphere into two perfectly equal hemispheres.
Why do long-haul pilots and ocean navigators rely on great circles?
The minor arc of a great circle between two points is the shortest surface route, so it defines the most efficient flight or sailing path. This is why the term 'orthodrome' is used interchangeably with great circle in navigation and cartography.
What makes a great circle the geodesic on a sphere?
In spherical geometry, no other curve on the surface connects two points with a shorter length than the minor arc of the great circle passing through them. That property is what earns it the title of geodesic, the curved-surface analog of a straight line.
Where can I spot real-world great circles on Earth?
The equator and every meridian (line of longitude) are natural great circles on our planet. The International Date Line also roughly traces a great-circle path, though it zigzags to avoid cutting through countries.
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