For a straighline in 2D space $y=mx+b$, the normal parametrization of the line gives an alternate representation of the line as $R=x\cos\theta +y\sin\theta$, where $R$ is the algebraic distance from the origin to the line and $\theta$ is the angle $R$ makes with the line.
In the paper Use of the Hough Transform to Detect Lines and Curves, the author states that restricting $\theta$ to $[0,\pi)$ gives a unique representation of all possible lines.
My question is how would $\theta$ be defined for lines passing through the origin such that they are also uniquely identified by $R$ and $\theta$ ?
PS: Also I am pretty sure that $R$ is allowed to be negative in this definition but would be nice if anyone could confirm this ?