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Compare each of the following lengths to the length of a great circle of a s

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3. compare each of the following lengths to the length of a great circle of a sphere

A. distance between any pair of pole points

B. Distance between any pole point and its equator 

Jan 8th, 2015

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

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

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

S[\gamma] \ge r\int_a^b|\theta'(t)|\,dt \ge r|\theta(b)-\theta(a)|.

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.

Jan 8th, 2015

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