What's the easiest way to find the equation of a straight line in polar form given two points?

Question

Given the points $(r_1, \theta_1)$ and $(r_2, \theta_2)$, what is the easiest way to write a straight line equation between the two? I am also wondering if this parametrisation works: $$r(t)=(1-t)r_1 + tr_2,$$ $$\theta(t) = (1-t)\theta_1+t\theta_2$$ (Desmos seems to be struggling with that and taking them as cartesian coordinates)

Answer 1

$\text{If using } r(t)=(1-t)r_1 + tr_2 \text{ parametrization},$ the standard polar form for parametrization from triangle $ OTP$

$$ r \cos (\theta - \alpha)= p $$

$$\theta(t) = \sec^{-1}\frac{r(t)}{p} + \alpha =\cos^{-1}\frac{p}{r(t)} + \alpha $$ $$\theta(t) = \sec^{-1}\frac{(1-t)r_1 + tr_2}{p} + \alpha=\cos^{-1}\frac{p}{(1-t)r_1 + tr_2} + \alpha.$$