How to generate random points from a custom curve? my question is how to generate random points ($\theta_1,\phi_1$) on a curve determined by :
$
\arccos\left(\cos(\theta_1)cos(\dfrac{\pi}{6})+\sin(\theta_1)\sin(\dfrac{\pi}{6})\cos(\phi_1)\right) = \dfrac{\pi}{6}
$
Thank you!

Given that the curve equation
$$\arccos\left(\cos(\theta_1)\cos(\dfrac{\pi}{6})+\sin(\theta_1)\sin(\dfrac{\pi}{6})\cos(\phi_1)\right) = \dfrac{\pi}{6}$$
can be inverted into
$$\cos(\phi_1)=\dfrac{1-\cos(\theta_1)}{\sin(\theta_1)\tan(\pi/6)}=\tan(\theta_1/2)/\tan(\pi/6)\tag{1}$$
with an existing solution in $\phi_1$ when $\theta_1\in(-\pi/3,\pi/3)$, a solution to draw points "at random" (i.e., non deterministically) over that curve is to draw uniformly $\theta_1$ in $(-\pi/3,\pi/3)$ and to derive $\phi_1$ from (1), with two solutions in $(-\pi,\pi)$. As in the realisation below with $10^5$ generations:

Be aware, though, that this distribution does not return points uniformly distributed over the curve.
