I'm stuck at the following problem I'd like to solve and first and foremost to comprehend:
Given two points $p_1, p_2 \in \mathbb{R}^{2}$ how can the equation of a line in polar form $\delta = x \cdot \cos(\alpha) + y\cdot \sin(\alpha)$ be computed? I currently went the path of computing the slope-intercept form $y = mx+b$ and then converting it to the mentioned polar form. However: is there a way of computing directly the polar given two points without computing the slope and intercept first?
Thank you in advance for any hints and with best regards
If you have two points $(x_1,y_1)$ and $(x_2,y_2)$, you have to solve the following system of equations
$$\begin{cases}\delta = x_1\cos \alpha + y_1 \sin\alpha & ,\\ \delta = x_2\cos \alpha + y_2 \sin\alpha & .\\ \end{cases}$$
Subtracting the equations from one another and rearranging, you can determine that
$$\cot\alpha=-\frac{y_2-y_1}{x_2-x_1}$$
Once you have $\alpha$ you can substitute back into one of the previous previous equations to find $\delta$.