So I'm thinking about a space exploration game where the primary mechanic is to fly a space ship around the surface of various 4-dimensional surfaces. The way I'd like to render this is by ray casting along the parameters of a surface; my possibly faulty assumption being that the ray would follow the geodesics of the surface (This appears to be true as far as I have seen). I've been able to look many parameterizations up such as ones for the unit 3-sphere, and a Klein bottle and a couple other things in lower dimensions that I might play with for surfaces of planets (terrestrial walking). I was hoping for parameterization of more spaces so that I could explore those as well. The tours seems most interesting because it seems to me like you would only see discreetly many copies of yourself and those copies would be different sizes. For instance you would see two quite large copies of yourself in the two cardinal directions and some smaller ones in between. This feels a lot more mentally manageable than seeing an exact copy of myself in every direction all of which are equally far away as in the 3-sphere case. If possible I'd also like to explore n-fold toruses as well.
So my question is a) what are the geodesics of a 3-torus and b) what is an example of a parameterization of a 3-torus. Do the rays cast though the parameterization match up with the geodesics of the torus?
bonus question: what are the geodesics and parameterizations of 2-fold and 3-fold toruses? What about 2-fold and 3-fold toruses in higher dimensions?