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I recently had to deal with polar coordinates and thus wondered: "Polar coordinates" is just a special name for some bijection from $\mathbb{R}^2$ to $\mathbb{R}^2$ that can be very easily visualized and that is rather important.

There are of course lots of other bijections (like $\binom{x} {y} \mapsto \binom{x} {y} +\binom{1} {1} $).

My questions are: 1) What are some other (interesting! - not like the above translation) bijections of the plane ?

2) Is there a way to classify the bijections of the plane such that this classification explains why the polar coordinates bijection is so important ? Or is there at least a mathematical argument that argues for polar coordinates ?

(I'm thinking of an argument that could go like "an equivalence class can be established on the set of all bijections of the plane and the polar coordinates bijections is a represantative of an important equivalence class" and NOT an argument like "polar coordinates are used a lot in mechanics and thats why they are important", that is based on how often in mathematics/physics polar coordinates are used, because the latter does not contain any mathematical property that gives evidence for supporting polar coordinates)

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Strictly speaking, polar coordinates are not bijective because different values of $(r, \theta)$ can map to the same point. But I'm just nitpicking here. – Scaramouche Jan 13 '12 at 10:37
They are bijective to $\{(r,\theta):x\in\mathbb{R},\theta\in[0,2\pi]\}$. Which itself is neat to me because it's a like taking the plane and squeezing it into a skinny little rectangle (I know that's nothing unique but I still like it). – crf Jun 9 '12 at 8:03
up vote 0 down vote accepted

There are lots of interesting bijections of the plane:

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