Great Circle - Derivation of Shortest Paths

Derivation of Shortest Paths

To prove that the minor arc of a great circle is the shortest path connecting two points on the surface of a sphere, one has to apply calculus of variations to it.

Consider the class of all regular paths from a point p to another point q. Introduce spherical coordinates so that p coincides with the north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by

provided we allow φ to take on arbitrary real values. The infinitesimal arc length in these coordinates is

ds=r\sqrt{\theta'^2+\phi'^{2}\sin^{2}\theta}\, dt.

So the length of a curve γ from p to q is a functional of the curve given by

S=r\int_a^b\sqrt{\theta'^2+\phi'^{2}\sin^{2}\theta}\, dt.

Note that S is at least the length of the meridian from p to q:

Since the starting point and ending point are fixed, S is minimized if and only if φ' = 0, so the curve must lie on a meridian of the sphere φ = φ0 = constant. In Cartesian coordinates, this is

which is a plane through the origin, i.e., the center of the sphere.

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