Great circle

A great circle is a special kind of circle that you can find on a round object, like a ball or a planet. It is the biggest circle you can draw on the surface of a sphere. This happens when a flat surface, called a plane, cuts through the center of the sphere.

In mathematics, a great circle is the circular intersection of a sphere and a plane passing through the sphere's center point. This means the circle goes all the way around the sphere and its center is the same as the sphere's center.

Great circles are very important for navigation, especially when traveling long distances over the Earth. They show the shortest path between two points on a round surface. Pilots and ship captains often use these paths to save time and fuel on their journeys.

Discussion

Any part of a great circle is the shortest path on a sphere, much like straight lines are the shortest paths on a flat surface. For any two different points on a sphere that are not directly opposite each other, there is exactly one great circle that connects them. The shorter path along this circle between the two points is called the minor arc, and it is the quickest route on the sphere's surface.

A great circle is the biggest circle that can be drawn on a sphere. Its center lines up with the center of the sphere, and it shares the same size as the sphere. Other circles on the sphere, called small circles, are smaller because they are made by a flat surface that does not go through the sphere's center.

Derivation of shortest paths

See also: Great-circle distance

To show that the smaller part of a great circle is the shortest path between two points on a sphere, we can use a special math method.

Imagine all the different paths you could draw between two points on a ball. By using special math rules, we find that the shortest path is always along a great circle. This great circle lies on a flat surface that cuts right through the center of the sphere, making it the smoothest and shortest route between any two points on the ball.

Applications

Great circles are special paths on round objects like Earth. For example, on the sky, great circles include the horizon, the equator, and the path the Sun follows. People use great circles to help guide airplanes and ships because they are the shortest paths between two places on Earth.

On an imaginary perfect Earth, the equator is a great circle. Any line going from the North Pole to the South Pole and back also makes a great circle. These circles split Earth into two halves. If a great circle passes through one place, it will also pass through the exact opposite point on Earth.