A higher dimensional generalization of cylindrical coordinate system?

Question

Does a higher dimensional generalization of a cylindrical coordinate system exist such that there is $N$ scalar dimensions and $M$ polar dimensions, and consequently does there exist a higher-dimensional generalization of a sphere characterized by $N$ lattitudinal and $M$ longitudinal dimensions?

Answer 1

Yes, I am confident that what you propose can be achieved. For example in four dimensional space $(x,y,z,t)$ one may introduce these coordinate systems:

• use 4-dimensional spherical coordinates

• keep $x$ cartesian, use spherical coordinates instead of $y,z,t$

• keep $x,y$ cartesian, use polar coordinates instead of $z,t$

• use polar coordinates for $x,y$ and also for $z,t$.

Note that the first has $1$ length and $3$ angles, the second and fourth have $2$ of each and the third has $3$ lengths and $1$ angle. As long as each point in four-dimensional space is uniquely defined, the choice of coordinate system is free.