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In the polar regions air navigation is not by the Lat/Long convention, but by a square grid oriented parallel to the 0/180 meridian, and tangent to the earth near the mid point of the intended flight path.Compases are set to O° parallel to the Greenwich meridian (180° when heading north along the 180° meridian). This enables navigation to "look" reasonably normal, and gives a fixed heading to a track between points near, but not crossing, the pole.
Suppose you are initially near the south pole and flying directly toward
it. Your true heading will be 180. As you pass over the pole, your
true heading will snap to 360. There is no ambiguity about this whatsoever.
Longitude has got nothing to do with it.
If you are at the south pole and wish to fly toward a particular point,
you cannot specify it in terms of heading. You can specify it in terms
of longitude, but that is not the same thing.
There is a singularity in the coordinate system at the pole. Wishful
thinking will not make the singularity go away.