I was looking at $-z = cos(x)cos(y)$ in math3d, and I noticed that if you take a plane -z = T (T is a slider), the intersection (or level set) kind of looks like some $x^n + y^n = r^n$.
The math3d link is here.
For example:
When z=-0.90, it looks similar to some $x^2 + y^2 = 0.5^2$.
When z=-0.25, it looks similar to some $x^4 + y^4 = 1.3^4$.
When z=-0.01, it looks similar to some $x^{20} + y^{20} = \frac{\pi}{2}^{20}$
And of course, when z=0, the level set is a square.
In desmos, I tried creating a function that has this type of behavior, link here, where the slider $n$ is supposed to vary the level set depth. I asked on reddit and someone also tried variations of $ (1-x^2/a)(1 - y^2/b) = k$. But I can't figure out if it's possible to get an exact model.
If you try to solve for the level sets from $cos(x)cos(y) = c$, you get $y = cos^{-1}(c*sec(x))$. I'm not sure if you can get past this point.
Is it possible to describe the intersection of z=c and $cos(x)cos(y)$ with a non-trig function? Or is it just an illusion that $y = cos^{-1}(c*sec(x))$ looks like it can be described by some $x^n + y^n$ curve?
Thanks!