I was just thinking about how a if you've got a circle, in cartesian coordinates that is given by the equation $x^2+y^2=k^2$. You can then move over to polar coordinates and this same circle when represented under polar coordinates, looks like a straight line.
I was just wondering if it was always the case that for any arbitrary curve, there always existed some way to transform the coordinates such that you could view that curve as a straight line?
$\newcommand{\R}{\mathbb{R}}$
Locally, any curve in $\R^2$ is a graph of some function $f \colon \R \to \R$, i.e., as the set $\Gamma = \{ (x,f(x)): x \in \R \}$. To make it straight, one only needs a very simple change of coordinates: $$ \Phi(x,y) = (x,y-f(x)), \quad \Phi^{-1}(x,y) = (x,y+f(x)). $$ It's simple to check that $\Phi$ transform $\Gamma$ into the line $(x,0)$.
Of course it only works locally, and the circle is a good reason why.