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Those are the only coordinate systems that I've heard of. Clearly we can invent other coordinate systems that have the two properties

a) All points in Euclidean space are represented by the system (surjectivity).

b) A point in Euclidean space cannot be represented in two or more different ways by the system (injectivity).

To take an example of such a coordinate system, I thought of this: Say that for any point $p$ in Euclidean space with Cartesian coordinates $(x_1,y_1)$, we associate it with a square with a center in the origin whose side length is $s=2 \cdot |\max(x_1,y_1)|$. Then starting at the point with Cartesian coordinates $(s,0)$, we travel on the square in a counterclockwise way until we find $p$, then calculate the distance $d$ that we have traveled. Using this coordinate system, we can say that the coordinate is expressed as $(s/2,d)$. Hence, the points $(3,2)$ and $(-4,9)$ in Cartesian coordinates would have the coordinates $(3,2)$ and $(9,22)$ respectively using this system. Clearly this coordinate system satisfies the above properties (it just uses squares instead of the circles in the polar coordinate system), but I can't think of a situation where it would be useful. So back to my question, do there exist any coordinate systems besides the Cartesian and polar ones that are useful for some situations (for some problems maybe or applications)?

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  • $\begingroup$ Polar doesn't satisfy those conditions. The point with Cartesian coordinates $(x,y) = (1,0)$ can be represented in lots of ways in polar corodinates: $(r,\theta) = (1,0)$ or $(1, 2\pi)$ or $(-1,\pi)$ or .... $\endgroup$ – Hurkyl Nov 24 '13 at 21:26
  • $\begingroup$ Like Hurkyl suggested, usually the best you can get is a local change of coordinates. The inverse function theorem tells you of the possibilities, and allows you to decide if and where a function $f$ satisfies the conditions. For a more global approach, you can see, e.g., the mapping class group of $\mathbb R^2$, and the MCG of general spaces. $\endgroup$ – user99680 Nov 24 '13 at 21:36
  • $\begingroup$ @Hurkyl You are right. However we can make the polar coordinates satisfy the conditions by requiring $r$ to be non-negative and the angle to be larger or equal to $0$ but less than $2 \pi$. $\endgroup$ – Sid Nov 24 '13 at 21:41
  • $\begingroup$ @Sid: The trick to choose just one coordinate for each actual point has some significant drawbacks, and also works for everything, not just polar coordinates, and often "nicely". $\endgroup$ – Hurkyl Nov 24 '13 at 21:48
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For example: Elliptic coordinates may be useful in a boundary value problem where the boundary is an ellipse.

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There are affine coordinate systems in $\mathbb{R}^2$. Basis vectors are any two linearly independent vectors. The origin need not be $(0,0)$. For example $e_1=(1,0)$, $e_2=(2,3)$. A special case of this is when $e_1$ is orthogonal to $e_2$ and both are unit vectors, that is the Descartes coordinate system. They are useful for representing affine transformations with nice matrices.

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For a more general take, of when and if there is a coordinate system, like Hurkyl suggested, usually the best you can get is a local change of coordinates. The inverse function theorem (IFT) tells you of the possibilities, and allows you to decide if and where a function $f$ satisfies the conditions; . For a more global approach, you can see, e.g., the mapping class group of $\mathbb R^2$, and the general MCG ,which is the group of self-diffeomorphisms of a space tells you of the possible general coordinate changes .

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