Given the monkey saddle $z=x^3-3xy^2$ over the unit circle $x^2+y^2 \leq 1$, find an ellipse whose length is the same as the length of the outer edge of the monkey saddle.
I've already found a parameterization for the monkey saddle in cylindrical coordinates:
$x=r\cos \theta$
$y = r\sin\theta$
$z = r^3\cos3\theta$
And I've found the area of the monkey saddle in the region: $\frac{\pi}{6}[3\sqrt{10}+log(3+\sqrt{10})]$
And I know that the arc length of the ellipse $x^2/a^2+y^2/b^2=1$ is $\int_0^{2\pi}{\sqrt{a^2\cos^2\theta+b^2\sin^2\theta} d\theta}$