Great circle
Largest circle on a sphere, shortest path between two points.
A great circle, also known as an orthodrome, is defined as the intersection of a sphere with a diametral plane—a plane that passes through the sphere's center. This intersection is always a circle, and it is the largest possible circle that can be drawn on any given sphere. Every great circle shares its center and radius with the sphere itself, and any diameter of a great circle is also a diameter of the sphere. In contrast, the intersection of a sphere with a plane that does not pass through its center produces a small circle, which is the spherical-geometry analog of a circle in Euclidean space. Notably, any circle in Euclidean three-dimensional space is a great circle of exactly one sphere.
In spherical geometry, great circles are the natural counterpart to straight lines in Euclidean geometry. Any arc of a great circle is a geodesic on the sphere. For any two distinct points on a sphere that are not antipodal (directly opposite each other), there exists exactly one great circle passing through both. However, because every great circle that passes through a point also passes through its antipodal point, infinitely many great circles connect two antipodal points. The shorter of the two arcs between two points on a great circle is called the minor arc, and it represents the shortest surface path between them. The length of this minor arc is the great-circle distance, which is proportional to the central angle formed by the two points and the sphere's center.
A great circle divides the sphere into two equal hemispheres. The disk bounded by a great circle is termed a great disk, formed by the intersection of a ball and a plane through its center. Half of a great circle is sometimes called a great semicircle, as seen in parts of a meridian in astronomy. In higher dimensions, great circles on an n-sphere are defined as the intersection of that n-sphere with two-dimensional planes passing through the origin. The calculus of variations can be used to prove that the minor arc of a great circle is the shortest path between two points on a sphere. Examples of great circles on the celestial sphere include the celestial horizon, the celestial equator, and the ecliptic. On Earth, the equator is a great circle, and any meridian paired with its opposite meridian also forms one. Great circles are used to approximate geodesics for air and sea navigation, and the Funk tra
- 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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