# Calculate distance between a point and a line in polar coordinates

In a 2d space, given a point in polar coordinates and a line in polar coordinates, how do you calculate the distance?

The single somewhat related question was this, which assumes a point in cartesian coordinates. The solution there is to transform the line to cartesian coordinates. I hope there's a solution without transforming everything to cartesian coordinates.

I solved the problem by just assuming a rectangular triangle.

Given a point $$P (\rho, \phi)$$ and a line $$L (r, \theta)$$, when searching for distance $$d$$:

Let's say $$r$$ is the side $$c$$ in our triangle, $$\gamma = 90°$$, $$A$$ is our origin, so $$\alpha = |\phi - \theta|$$, $$\beta = 90° - \alpha$$ and lastly the wanted distance $$d$$ as $$a$$ between $$A$$ and $$B$$:

$$a = c · cos(\beta)$$

or using our initial variables:

$$d = r · cos(|\phi - \theta|)$$

EDIT: I just noticed this only works for lines coming from the origin (which is my use case, but isn't necessarily a complete answer to the question)