it's called the nintendo 64 because when you see it you turn 64 degrees and continue to walk towards it but at kind of an obtuse angle
i think you'd start at an acute angle but then as u pass it since you're on a diverging path now it'd end up obtuse. assuming you're travelling across the surface of a sphere you'd never be at a reflex angle from the nintendo 64 if we measure the angle from the same direction of rotation each time , BUT this might not be the case on an oblate spheroid (i'm not sure if your path would precess or not)