## Shortest path on a sphere

Show that the shortest path between two points on a sphere can be acheved by walking on a great circle of the sphere passing through those two points.

**Solution:** Pick coordinates such that the sphere is centered in , has coordinates and has coordinates . We can pick , because if it’s the other way around, then by a symmetry, the problem can be transformed to the case where . A curve which joins and and is contained in the surface of our sphere can be parametrized like this:

where and . Denote by $L$ the length from to on and get

.

A few calculations yeld

Therefore

. (1)

Considering the great circle which passes through and , we see that $u_0$ is exactly the length of the small arc with ends and , because .

If are not antipodal, then this distance can only be achieved on a unique path, namely, the one described above. We should have equality in (1), which is equivalent to

and . This would imply and increasing, which is equivalent to walking on a small arc of a great circle of the sphere.

See the following pdf for another approach. Geodesics on sphere (source: http://education.uncc.edu/droyster/courses/fall98/math4080/classnotes/geodesiceqn.pdf)

This shortest path is usually called geodesic. As a complete manifold, the sphere itself satisfies the Hopf–Rinow theorem, that proves the existence of geodesic.