# Extrapolate $y^2=4x^2-4x^4$ from plot

I am trying to extrapolate a function based on the distribution of eigenvalues of certain matrices I am working with.

In a simple case I successfully described the data with the function $$y^2=4x^2-4x^4$$:

However when I consider more extreme cases my equations become very tedious to solve. I \underline{know} the shape is very similar to $$y^2=4x^2-4x^4$$, but I need the two 'wings' to be further apart without altering their height:

OR maybe

Note that they should $$\textit{not}$$ be perfect ellipses, otherwise the following equation would make the trick: $$\frac{1}{\frac{(x-a)^2}{\tau}+\frac{y^2}{\tau}}+\frac{1}{\frac{(x-b)^2}{\tau}+\frac{y^2}{\tau}}=1$$

What should I change in the following equation in order to increase the distance between the two wings without affecting their height:

$$y^2=4x^2-4x^4$$

I tried to add different coefficients here and there but I do not manage to develop the intuition on what to change. I am almost there!

Any observation or help is always appreciated. Thank you!

EDIT: based on the comments, it works indeed! now I simply need to know how to properly scale.

• What do you think of the following: $$y^2=4(x^2-4)-4(x^2-4)^2$$ – Andrew Chin Nov 8 at 17:02
• Whoah, not bad.. It works for the points $-2$ and $2$. Nice intuition! I simply need to scale the shape and it works. Well done! Thanks a lot. – Sam Nov 8 at 17:17