Say you want to move from A to B on a circle. A simple algorithm would be to just go the shortest way from A to B which is well defined if A and B are not antip

Say you want to move from A to B on a circle. A simple algorithm would be to just go the shortest way from A to B which is well defined if A and B are not antipodal. If they are, you could define to always go against the clock. Now this algorithm is fine, but it is not continuous (smooth), because a small variation in an antipodal point gives two vastly different paths.