I wish to take a set of points described using Cartesian coordinates ($x, y$) and map them to a set of polar coordinates ($r, \theta$), such that $r = y$ and $\theta = x$. Thus horizontal lines become circles, and vertical lines become radial lines.
1) Is there a name for this transformation? I'm hoping for something catchy like Polarisation...
2) What will happen to an angled line such as $y = mx + c$ (where $c \ne 0$) under this transformation? (I'm guessing a hyperbola...)
On
I don't know about part (1), but I can provide an answer to part (2).
You have the "polarized" equation $r=a\theta+b=a\left(\theta+\frac{b}{a}\right)$. Notice that $r=a\theta$ is just an Archimedes' spiral, and the $\frac{b}{a}$ does a counterclockwise rotation of $\frac{b}{a}$ radians, so any non-horizontal and non-vertical line is just a rotated Archimedes' spiral.
Recall that the standard change of variable is just a function $f: (x,y)\mapsto (\sqrt{x^2+y^2} ,\arctan (\frac yx ))$ with conditions on inverse tangent because it should give different values depending on signs of $x,y$. Then the inverse function is $f^{-1} :(r, \theta ) \mapsto (r\cos \theta , r\sin \theta)$
Anyway, what you described is just pluggin in $(y,x)$ (as radius and angle) into the inverse function,
$$(\tilde x, \tilde y) = (y \cos x, y \sin x)$$
Where $(\tilde{x},\tilde{y})$ is the new coordinates.
So the first equation shows that for the same $y$ values, any horizontal lines would be mapped to a circle. Whereas the same $x$ value (vertical line) would be mapped to the straight line $y=mx$.
Generally given $L=\{(t,mt+c): \forall t\in \Bbb{R} , m,c \in \Bbb{R} \}$
This gives $ (\tilde x, \tilde y) = ((mt+c)\cos t, (mt+c) \sin t)$, which is a curve parametrized by $t$. Observe if $t$ increases, the radius increases, while the angle rotates continuously. So for example if $m,c >0$, the curve starts as $(c,0)$ and spirals outwards.