Great circle

A great circle is a specific type of circle found on a sphere. In mathematics, it is also called an orthodrome. A great circle forms when a flat plane passes directly through the center point of a sphere. This intersection creates the largest possible circle that can exist on that sphere.

Because the plane passes through the center, the circle is concentric with the sphere. This means the circle and the sphere share the exact same center and radius. Every great circle also shares its diameter with the diameter of the sphere. This unique geometry makes great circles essential for understanding spherical geometry.

To understand how a great circle works, we must look at how it divides a sphere. When a plane cuts through the center, it splits the sphere into two equal parts. These two equal halves are known as hemispheres.

Any other circle formed by a plane that does not pass through the center is called a small circle. While small circles are common, they do not divide the sphere into equal halves. A great circle is much more significant because it represents the maximum width of the sphere. In higher dimensions, these shapes are called great circles on an n-sphere. They occur when a 2-plane passes through the origin of the Euclidean space.

Great circles serve a very important role as geodesics. A geodesic is the shortest path between two points on a curved surface. In standard flat geometry, we use straight lines to connect points. On a sphere, however, the natural analog to a straight line is a great circle.

If you pick two distinct points on a sphere that are not directly opposite each other, there is exactly one unique great circle that passes through both. If you pick two points that are antipodal, meaning they are directly opposite, there are infinitely many great circles that can connect them.

When traveling between two points on a great circle, the path is divided into two arcs. The shorter of these two paths is called the minor arc. This minor arc is the shortest surface-path between the two points. The length of this arc is known as the great-circle distance. This distance is the intrinsic distance on the surface of the sphere. The length of the arc is proportional to the central angle. This angle is formed by the two points and the center of the sphere.

Mathematicians use advanced tools to prove that the minor arc is indeed the shortest path. One method involves using the calculus of variations. This process examines a class of all regular paths between two points. By introducing spherical coordinates, researchers can analyze curves on the sphere. They use the Euler–Lagrange equation to find which path minimizes the distance.

The math shows that the shortest path must lie on a meridian. This means the curve must exist on a plane that passes through the origin. Since the origin is the center of the sphere, the path must follow a great circle.

We can see great circles in many practical and scientific applications. On the Earth, the equator is a famous example of a great circle. Any meridian and its opposite meridian also form a great circle. There is even a great circle that divides the Earth into land and water hemispheres.

In astronomy, great circles appear on the celestial sphere. Examples include the celestial equator, the celestial horizon, and the ecliptic. Pilots and sailors use great circles to navigate the Earth's surface. While the Earth is not a perfect sphere, these circles provide very accurate approximations for air and sea travel.

Great circles connect many different fields of study. They are a fundamental part of spherical trigonometry and Riemannian geometry. They also play a role in the Funk transform, which integrates functions along all great circles of a sphere. Understanding these circles helps us navigate our world and the stars. They turn the complex curves of a sphere into predictable, mathematical paths.