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.
