Let's parameterize a 2D (unoriented) line by the slope $m$ and intercept $b$, and let the slope be positive. Thus, we are looking at all lines in the first quadrant. I want to know if there's a natural way to measure distances between lines or the "size" of a line in the $(m,b)$ space. I know that there's no natural metric on the space of all possible unoriented lines, but I wonder if there exists one for the restricted case I have described above. Here's how I am reasoning about it for now.
Two lines with the same slope are different only by the intercept $b$, so that's trivial. Next, two lines with the same intercept differ by the angle $\theta=\arctan(m)$. So, am I correct in thinking that the 2D polar coordinate system is the natural coordinate system here? If so, then can I use $b^2+b^2\theta^2$ as a metric?