Someone asked a question on worldbuilding about navigating by the stars on a 4D planet. In thinking about it I came up with a question that seems appropriate to ask here, as it's purely a maths question.
Suppose you're at a point on the surface of a sphere in 4 dimensions. There is a stationary point in the sky, infinitely far away - call it the sun. The sphere you're standing on rotates at a constant rate and you move with it, so from your point of view the stationary point moves across the sky.
Now suppose we take your view of the sky and project it via a stereographic projection into a 3D Euclidean space. The question is, what does the path traced out by the sun look like in this three dimensional space?
We can assume without loss of generality that the rotation of the sphere preserves the $xy$ plane and the $uv$ plane. The answer to this question will depend on the location of the stationary point and of the observer, and also on the ratio of the two rotation rates that are needed to specify such a rotation in 4D. I'm interested in knowing the general answer, i.e. what kinds of paths can be traced out depending on the values of these parameters?
