I recently looked up the 14 different orthogonal coordinate systems, and I noticed every single one of them quite notably had three equations which basically equated to the following vector-valued function:
(x,y,z) = g(u,v,w)
Basically every coordinate system I saw was defined by a formula taking in a point in it's coordinate system and returning its cartesian equivalent.
Is this true for every coordinate system in existence?
Now obviously 2 dimensional coordinates are going to have 2 components, 4D are going to have 4 components, etc. I get that. I'm more asking if all coordinates are defined by their conversion to cartesian, or if some coordinates are defined differently (perhaps by their... conversion from cartesian)?
The reason why I asked was because I was thinking of using this information to allow one to define their own coordinates in a thing I'm working on, and I want to be a broad-scoping as possible.