Intuition for straight lines in polar coordinates

Question

In cartesian coordinates, it's very intuitive to get an idea of how a graph of a straight line comes an equation. With polar coordinates, it's not as easy. How can I start to understand how the graph of a line comes from a polar equation without converting it to cartesian form? Thanks.

Answer 1

Let $\ell$ be a line and $A(a,\alpha)$ be the point on $\ell$ that is nearest to the pole. For a point $P(r,\theta)$ on $\ell$, $OP=r$ and $\angle POA=|\theta-\alpha|$. Note that $OA=OP\cos\angle POA$. The equation of $\ell$ is $a=r\cos(\theta-\alpha)$.