On a sphere, such as the Earth, the shortest path between two points is not a straight line because of the curved nature of the surface. Let's imagine you are trying to find the shortest path between two cities on Earth.

If you were to draw a straight line on a flat map, it would seem like the shortest route. However, when you consider that the Earth is a three-dimensional object, that straight line would actually pass through the Earth's interior, which is not physically possible. We need to stay on the surface of the sphere.

To understand the concept of the shortest path on a sphere, think about it like walking on the surface of a globe. If you were to walk from one point to another, you would follow a curved path along the globe's surface. This curved path is the shortest distance between those two points on the sphere.

To visualize this, imagine taking a piece of string and placing it on the surface of a globe, connecting the two cities. You will notice that the string curves along the surface, mimicking the shortest path.

This curved path is known as a great circle. A great circle is a circle formed by the intersection of the sphere and a plane passing through its center. It is the largest circle that can be drawn on a sphere.

So, to find the shortest path between two points on a sphere, we need to follow the curved path along the great circle, rather than a straight line. This is because the curvature of the sphere necessitates it, and it allows us to stay on the surface of the sphere while finding the most efficient route.
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