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:
$a = 1 - \Delta$
$b = 1 + \Delta$
The solution for $\Delta$ as a function of $R$ and $\theta$ is determined from
$$R = \sqrt{1 - 2\Delta cos(2\theta)+ \Delta^2}$$
Which as given in Solving for Ellipse Parameters given a radius and angle (Challenge 1) has the simplest solution as $$\Delta = cos(2\theta) \pm \sqrt{R^2-sin^2(2\theta)}$$
I am trying to similarly solve for $\Delta$ given a modified ellipse with the following relationship:
$$R = (1-\Delta)\sqrt{1 - 2\Delta cos(2\theta)+ \Delta^2}$$
What is the simplest form of $\Delta$ given $R$ and $\theta$ from the above equation?
If a simple relationship does not exist, then an approximation will be acceptable given that I can limit $\Delta$ to be $0.9<\Delta<1$
(Note this is related to my answer at this link on the signal processing site where I had to find the root of the equation to solve but am hoping for a simple closed form equation: https://dsp.stackexchange.com/questions/54006/computation-of-parameter-filter-to-match-a-given-frequency-response/54008#54008)
Squaring both sides gives $$R^2 = (1 - \Delta)^2 (\Delta^2 - 2 \Delta \cos 2 \theta + 1) .$$ This is a quartic equation in $\Delta$, so there is a closed form in terms of $R, \theta$, but it's too large to reproduce here: For general values of $\theta, R$ there won't be a simple form, and given the content of the linked post, the numerical approximations you've already been using are probably the most useful. If you're interested in limiting cases, say, $R \ll 1$ or $R \gg 1$, one can analyze the above equation to produce good approximate solutions for $\Delta$.
Edit If we have the additional condition that $R \ll 1$, then substituting gives $\Delta \approx 1$. To get a next-order approximation, we can set $\Delta = 1 + \epsilon$ with $\epsilon \approx 0$ (this is close to the condition specified in the edit to the original question), giving $$\epsilon^4 + 4 \sin^2 \theta \, \epsilon^3 + 4 \sin^2 \theta \, \epsilon^2 = R^2 .$$ Provided that $\sin \theta$ is not too close to $0$, the $\epsilon^2$ term on the left dominates the others, giving $$\epsilon \approx \pm \frac{R}{2} \csc \theta,$$ so $$\Delta \approx 1 \pm \frac{R}{2} \csc \theta .$$
On the domain $\frac{9}{10} < \Delta < 1$ of interest, this is a robust approximation:
From innermost to outermost, the red curves are the graphs of the solution functions $\Delta(\theta)$ for $R = \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64}, \frac{1}{128}$, and the blue curves are the graphs of the above approximations.
On the other hand, for very small $R$, say, $R < \frac{1}{64}$, it's possible for to have $\frac{9}{10} < \Delta < 1$ but $\sin \theta$ small enough that the above approximation becomes poor: In the limit $\sin \theta \to 0$, the $\epsilon^4$ term dominates, giving a zeroth-order approximation $\Delta \approx 1 - \sqrt{R}$. Computing to second order in $\sin \theta$ gives the approximation $$\Delta \approx 1 - \sqrt{R} + \left(1 - \frac{1}{\sqrt{R}}\right) \sin^2 \theta .$$
This plot for $R = \frac{1}{256}$ is typical for $R < \frac{1}{100}$; the green curve is the small-angle approximation. (NB the expanded scale of the horizontal axis.)