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

\theta = \theta(t),\quad \phi = \phi(t),\quad a\le t\le b

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

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

x\sin\phi_0 - y\cos\phi_0 = 0

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