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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.

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closed as too broad by The Great Duck, астон вілла олоф мэллбэрг, Daniel W. Farlow, JMP, zhoraster Jan 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.

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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.)

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  • $\begingroup$ I'm going to wait to accept, as it's a general courtesy, but I have to give you plus one. Seeing "in TheGreatDuck-coordinates" just gave me a good chuckle. Definitely deserving of a plus one. And I will look at barycentric coordinates. That sounds intriguing. $\endgroup$ – The Great Duck Oct 2 '16 at 23:10
  • $\begingroup$ @TheGreatDuck Thanks! What other coordinate systems were you looking at? The only ones I've ever used are barycentric, trilinear, cartesian, cylindrical, and spherical. $\endgroup$ – Carl Schildkraut Oct 2 '16 at 23:13
  • $\begingroup$ There's actually a whole page on wikipedia with a massive chart of them en.wikipedia.org/wiki/… I don't necessarily know how to graph in all of them (after all, I literally just read about them), but I got curious and noticed they all had some similarities. :) $\endgroup$ – The Great Duck Oct 2 '16 at 23:15
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    $\begingroup$ @TheGreatDuck Oh god that's a lot of coordinate systems. Looks interesting though! $\endgroup$ – Carl Schildkraut Oct 2 '16 at 23:15
  • $\begingroup$ I hear they are all used for very specific physics problems; however, they doesn't prevent me in using them for my purposes. All I care about is getting the Cartesian equivalent of their points (and I suppose whatever my curiosity leads me to read about). $\endgroup$ – The Great Duck Oct 2 '16 at 23:17

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