A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: David Iwancio
Date: 2021 Jul 20, 12:09 -0700
Consider: if you follow a rhumb line at 090° or 270°, you are effectively moving along a circle centered around the poles. If you follow such a course at latitude 89°59'59.99", you are going to get dizzy. If you're "moving" east or west at latitude 90° you are effectively spinning in place, and if your location is treated as an infinitessimal point, then we enter a philosophical debate over whether an infinitessimal point can "spin." Regardless, any direction, any change of position away from latitude 90° is necessarily going to be along a meridian (which all meet at the poles).
Another way of looking at this: you can't determine the hour angle of a body from latitude 90° because you don't have an hour angle. You are simultaneously standing on all meridians, all hour angles, so the body's "local" hour angle is meaningless.
Your longitude is indefinite, so any formula you use will produce the indefinite result of dividing by zero.