Krdytkyu

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:

$$R = sqrt{big(a^2 cos^2thetabig)+big(b^2 sin^2thetabig)}$$

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

$$R = sqrt{1 - 2Delta cos(2theta)+ Delta^2}$$

Which as given in Solving for Ellipse Parameters given a radius and angle (Challenge 1)
has the simplest solution as $$Delta = cos(2theta) pm sqrt{R^2-sin^2(2theta)}$$

I am trying to similarly solve for $Delta$ given a modified ellipse with the following relationship:

$$R = (1-Delta)sqrt{1 - 2Delta cos(2theta)+ Delta^2}$$

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.)