Does every coordinate system have a well-defined conversion to cartesian coordinates? [closed]

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.

closed as too broad by The Great Duck, астон вілла олоф мэллбэрг, Daniel W. Farlow, JMP, zhorasterJan 17 '17 at 5:24

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

They're not always defined by their transformation to Cartesian coordinates, but if we have a point $(a_1,a_2,...,a_n)$ in TheGreatDuck-coordinates, it must represent a point $(b_1,b_2,...,b_m)$ in Cartesian coordinates. (Note that $m$ does not necessarily equal $n$ - look at barycentric coordinates.)