**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

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

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 = φ_{0} = constant. In Cartesian coordinates, this is

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

Read more about this topic: Great Circle

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