Given an ellipse centered on the origin in an x-y plane expressed as
$$\bigg(\frac{x}{a} \bigg)^2+\bigg(\frac{y}{b} \bigg)^2 = 1$$
In polar coordinates with radius $R$ and angle = $\theta$, this can be expressed as:
$$R = \sqrt{\big(a^2 \cos ^2\theta\big)+\big(b^2 \sin ^2\theta\big)}$$
If we set a and b to be related as follows:
$a = 1 - \Delta$
$b = 1 + \Delta$
What is the simplest solution for $\Delta$ as a function of $R$ and $\theta$ as determined from the following relationship?
$$R = \sqrt{1 - 2\Delta \cos (2\theta)+ \Delta^2}$$
$\Delta^2- 2\Delta \cos(2\theta)+1-R^2=0\\\implies\displaystyle \Delta=\cos2\theta\pm\sqrt{\cos^22\theta+R^2-1}=\cos2\theta\pm\sqrt{R^2-\sin^22\theta}$