Polar form relative to center
In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate $\theta$ measured from the major axis, the ellipse's equation is
$r(\theta)=\dfrac{b}{\sqrt{1-(e\cos\theta)^2}}$
Polar form relative to focus
If instead we use polar coordinates with the origin at one focus, with the angular coordinate $\theta = 0$ still measured from the major axis, the ellipse's equation is
$r(\theta)=\dfrac{a(1-e^2)}{1\pm e\cos\theta}$
Related: https://en.wikipedia.org/wiki/Ellipse
But I couldn't find polar form relative to vertex. What is the formula that provides the relationship between $\theta$ angle and $r$ length? $0 \leq \theta < \pi/2$
Begin with the cartesian equation of an ellipse with the vertex on the origin:
$$\frac{(x-a)^2}{a^2}+\frac{y^2}{b^2}=1$$
Substitute $x=r \cos \theta$ and $y = r \sin \theta$ and solve for $r$ to get
$$r = \frac{2ab^2\cos\theta}{b^2\cos^2\theta+a^2\sin^2\theta}$$