Two points on equator

If you have two points both lying on the equator of the Earth, why is following the equator (in whichever direction is closer) not always a shortest path between them? (Ignore mountains and such!)

Solution to metapuzzle: show

The most obvious mathematical representation of the Earth is as a ball. This won't allow a solution to the puzzle, as following the equator (in the better direction) will always be at least as short as any other route. So we are going to have to come up with something better.

A much more accurate, and only slightly more complicated, mathematical representation of the Earth is as an oblate spheroid: see this Wikipedia section. The distance from the earth's centre to a point on its equator is 0.33% greater than to either of its poles.

Restated puzzle: show

You have two points both lying on the equator of the Earth, regarded as an oblate spheroid. Why is following the equator (in whichever direction is closer) not always a shortest path between them?

Solution to puzzle: show

If the two points are antipodal or nearly so, the shortest route between them is via a pole.

Comments: show

I believe that this is a good metapuzzle-puzzle pair, and that it was intended as such by the setter.

The simple model of the earth as a sphere does not provide a solution. The next-simplest model does provide a solution.

Two points on the equator

Problem statement

Solution to metapuzzle: show

Restated puzzle: show

Solution to puzzle: show

Comments: show