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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:

$$l_{s, \boldsymbol{\theta}} = \left\{ s\boldsymbol{\theta} + t \boldsymbol{\theta}^\perp : t \in \mathbb{R} \right\}$$