Lines Determined by Polar Coordinates
Description
Lines Determined by Polar Coordinates
A line, as shown in Figure (1), is determined by the slope $a$ and the $y$ intercept $b$. It seems that all lines on a plane can be represented only by their slope and intercept, but this is not the case. Precisely, only lines can be depicted as functions. Therefore, a line perpendicular to the $x$ axis, as shown in Figure (2), cannot be represented by its slope and intercept.
Now, observe Figure (3). This line is perpendicular to the unit vector $\boldsymbol{\theta} = (\cos \theta, \sin \theta)$ and is at a distance $s$ from the origin. Hence, we can understand that a line on a plane is determined by the polar coordinates $(s, \theta)$. Additionally, unlike the method using slope and intercept, a line perpendicular to the $x$ axis, as shown in Figure (4), can also be expressed in polar coordinates.
This notation for lines is useful in expressing line integrals in cases like Radon Transform.
Points on a Line
As illustrated in the figure above, by adding the parameter $t$, we can represent a point on a line. If we designate $\boldsymbol{\theta}=(\cos \theta, \sin \theta)$, $\boldsymbol{\theta}^\perp=(-\sin \theta, \cos \theta)$, we get:
$$ \begin{align*} P =&\ s\boldsymbol{\theta} + t \boldsymbol{\theta}^\perp \\ =&\ (s\cos\theta, s\sin\theta) + (-t \sin\theta, t \cos\theta) \\ =&\ (s\cos\theta-t \sin\theta, s\sin\theta+ t \cos\theta) \end{align*} $$
Then, a line $l_{s, \theta}$ determined by $s$ and $\theta$ is the same as the following set:
$$ l_{s, \theta} = \left\{ s\boldsymbol{\theta} + t \boldsymbol{\theta}^\perp : t \in \mathbb{R} \right\} $$
Moreover, since $-s \big(\cos(\theta + \pi), \sin (\theta + \pi)\big) = s(\cos \theta, \sin \theta)$, for a negative $s$, it is defined as follows:
$$ l_{-s, \theta+\pi} := l_{s, \theta} $$
Generalization
For $\mathbb{R}^{n}$, a line that is perpendicular to the unit vector $\boldsymbol{\theta} \in \mathbb{S}^{n-1}$ and is at a distance $s$ from the origin can be described as follows:
$$ l_{s, \boldsymbol{\theta}} = \left\{ s\boldsymbol{\theta} + t \boldsymbol{\theta}^\perp : t \in \mathbb{R} \right\} $$