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

enter image description here

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:

enter image description here OR maybe enter image description here

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. enter image description here

  • 1
    $\begingroup$ What do you think of the following: $$y^2=4(x^2-4)-4(x^2-4)^2$$ $\endgroup$ – Andrew Chin Nov 8 at 17:02
  • $\begingroup$ 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. $\endgroup$ – Sam Nov 8 at 17:17

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