I have a line in 3D space that passes through the origin and is defined by two angles, $\theta$, and $\phi$.
How do I find a parametric equation for this line?
For Example:
if $\theta = \pi / 4$ and $\phi = \pi / 4$
then I think the parametric equations would be
$x = t$, $y = t$, and $z = t$ (Where $t$ is the parameter)
Of course, I could be wrong, which is why I'm asking.
$$ (x,y,z) = r (\cos \phi \cos \theta, \cos \phi \sin \theta ,\sin \phi) $$
is a Cartesian parametric equation of a radial line through origin of variable radius length $r.$
Please note that spherical coordinates or parameters $\theta, \phi $ are given or fixed but $r$ varies.