I have a blue curve, a simple circle $x^2+y^2=1$, onto which I am projecting, like a shadow, an exact function $F$. Let's say it's a simple Gaussian $F(x)=e^{-x^2}$ visualized in red below.
As the function is projected onto the shape, it does not wrap around it, but so-to-speak casts a shadow as if it were a cylinder as i put the sun behind a Gaussian function, resulting in a bent green function following the shape of the blue circle.
I now proceed to unwrap the shadow. This gives me a new straight green function that differs from the original red one and would range from $[-\pi,\pi]$.
What is the formula for getting the shape of the unwrapped green function?
Desmos link involving my guess: https://www.desmos.com/3d/1b8ed83fc3
Additional more general question: what if my blue curve is a different equation, for example $x^4+y^4=1$ instead of a perfect circle?