Great circle
The largest circle on a sphere, analogous to a straight line.
A great circle, also known as an orthodrome, is defined as the intersection of a sphere with 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 is concentric with its sphere, sharing the same center and radius, and any diameter of a great circle is also a diameter of the sphere itself. In contrast, the intersection of a sphere with a plane that does not pass through its center is called a small circle, which serves as 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 of the sphere, meaning it represents the shortest path along the surface between two points. For any two distinct points on a sphere that are not antipodal (directly opposite each other), there is exactly one great circle passing through both. However, because every great circle that passes through a point also passes through its antipodal point, there are infinitely many great circles through any pair of antipodal points. The shorter of the two arcs connecting two points along a great circle is called the minor arc; its length is the great-circle distance between the points, proportional to the central angle formed by the two points and the sphere's center. The disk bounded by a great circle is known as a great disk, and half of a great circle is sometimes referred to as a great semicircle.
The fact that the minor arc of a great circle is the shortest surface path between two points can be proven using calculus of variations. By placing one point at the north pole in spherical coordinates and considering all regular paths between the two points, the Euler–Lagrange equation shows that the curve minimizing length must lie on a meridian, which corresponds to a plane through the sphere's center. Great circles have numerous applications: on the celestial sphere, examples include the celestial horizon, celestial equator, and ecliptic. For navigation on Earth, great circles provide accurate approximations of geodesics for air and sea routes, even though Earth is not a perfect sphere. The equator is a great circle, as i
- field
- Mathematics
- known_for
- Shortest surface path between two points on a sphere; largest circle on a sphere; analog of straight lines in spherical geometry
- definition
- Intersection of a sphere and a plane through the sphere's center
- related_concept
- Small circle (intersection with a plane not through the center)
Lore & Background
For any pair of distinct non-antipodal points on a sphere, there is exactly one great circle passing through both. 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 is the great-circle distance, 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. Any other circle on the sphere is a small circle, the spherical-geometry analog of circles in Euclidean space. 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, the intersection of a ball and a plane through its center. In higher dimensions, great circles on the n-sphere are intersections of the n-sphere with 2-planes through the origin in Euclidean space R^(n+1). Half of a great circle may be called a great semicircle, as in parts of a meridian in astronomy.
Reader's Guide
The concept of the 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 routes are used for air and sea travel to minimize distance. The derivation of this shortest-path property uses calculus of variations: by introducing spherical coordinates with one point as the north pole, the arc length functional is minimized via the Euler–Lagrange equation, leading to the condition that the longitude coordinate is constant, meaning the path lies along a meridian—a great circle. This mathematical proof confirms that great circles are geodesics on a sphere. The distinction between great circles and small circles parallels the Euclidean distinction between straight lines and circles, making great circles essential for understanding spherical geometry as a non-Euclidean geometry. Their properties also extend to higher-dimensional spheres, where they remain the intersection with 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 shares the same radius as the sphere.
- 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.
Frequently Asked Questions
What exactly is a great circle on a sphere?
A great circle is the curve you get where a flat plane cuts through a sphere's center, tracing the largest possible ring on its surface. In other words, it is the sphere's own radius laid out as a full loop.
Why is it called 'great' rather than just a circle?
The word 'great' sets it apart from a small circle, which results when the cutting plane misses the sphere's center. A great circle shares the sphere's full radius, so no other circle on that surface can be larger.
How do great circles appear on maps and in navigation?
They trace the shortest surface path between two points on Earth, which is why flight and sailing routes follow them. On a flat projection, however, that 'straight' spherical arc often renders as a curve, making map routes look deceptively bent.
What is the great circle's role in spherical geometry?
It serves as the direct spherical equivalent of a Euclidean straight line, acting as the natural 'straightest' path available on a curved surface. Every arc of a great circle is a geodesic, meaning it is the locally shortest curve connecting its endpoints.
Why do cartographers and mathematicians rely on great circles?
They underpin spherical trigonometry and give meaning to concepts like the equator, meridians, and efficient global routing. Without the great-circle framework, measuring true distances and angles on a curved planet would lack a consistent geometric basis.
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