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)