How do you calculate y coordinates on a rotated super ellipse

For my work, I am working on a model to predict the length of different bars that you can cut from a super ellipse.

Is it possible to calculate with an analytical solution the yy-coordinate if xx is given on a rotated super ellipse?
I have found already this analytical solution for ordinary ellipses with

Where:
a=xcosαa=x \cos\alpha;
b=sinαb=\sin\alpha;
c=a2c=a^2;
d=xsinαd=x \sin\alpha;
e=cosαe=\cos\alpha;
f=b2f=b^2.

So my question is: is it possible to derive a similar equation for yy, given the general equation for a super ellipse is:

(xcosα+ysinα)nan+(xsinα−ycosα)nbn=1.
{(x\cos\alpha+y\sin\alpha)^n\over a^n}+
{(x\sin\alpha-y\cos\alpha)^n\over b^n}=1.

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