A Cartesian coordinate system is formally defined on an affine Euclidian space by an ordered basis of orthogonal unit vectors and a coordinate origin, such that the coordinates of any point $P$ in the space are the components of the vector $\vec{OM}$ with respect to this basis. We can then intepret the coordinates as the signed perpendicular distances from the coordinate axes to $P$.
The Wikipedia article on Euclidian spaces says this about defining non-Cartesian coordinate systems:
Many other coordinates systems can be defined on a Euclidean space $E$ of dimension $n$, in the following way. Let $f$ be a homeomorphism (or, more often, a diffeomorphism) from a dense open subset of $E$ to an open subset of ${\displaystyle \mathbb {R} ^{n}.}$ The coordinates of a point $x$ of $E$ are the components of $f(x)$. The polar coordinate system (dimension $2$) and the spherical and cylindrical coordinate systems (dimension $3$) are defined this way.
I'm trying to determine how to define a polar coordinate system on a Euclidian space using the definition above, but can't decipher what it is getting at. I was wondering if someone could explain how we set up such a homeomorphism?
Thank you!
Consider the case of polar coordinates $\langle r, \theta\rangle$ for a point in $E=\mathbb{R}^2$. Then $f()$ here is the function that takes a point in Euclidean coordinates (i.e., $\langle x, y\rangle$) and produces polar coordinates for it; in other words $f(\langle x, y\rangle) = \langle\sqrt{x^2+y^2},\mathrm{atan2}(x,y)\rangle$. Here the $\mathrm{atan2}()$ function is the 'cs version' that's basically $\arctan(x/y)$ but extended in ways that recognize the signs of the coordinates, so that e.g. $\mathrm{atan2}(-x, -y)=\mathrm{atan2}(x,y)\pm\pi$.
Now, this function isn't 1-to-1 on all of $E$; in particular, it's not well-defined at the origin (and there's some fuzziness about coordinates on the x axis). However, $E-\{\langle x,0\rangle: x\geq 0\}$ is a dense, open subset of $E$, and $f()$ is defined on this dense open subset: it takes it to $\{\langle r, \theta\rangle: r\gt 0, 0\lt\theta\lt2\pi\}$, which is an open subset of $\mathbb{R}^2$.