What kind of shape is this? Ramanujan ellipse estimation.

Question

The image shows Ramanujan's second approximation for the perimeter of an ellipse in 3 dimensions, where x and y represent a and b, and z represents the perimeter. So when plotted it turns out that all possible perimeters lie on a type of cone whose base is the shape in the image. I was therefore thinking that a better approximation can be found if the equation for the 2d shape can be determined, as then extending it to 3 dimensions will let us calculate the perimeter of the ellipse quite accurately. So do you know what the shape is? What is the equation?

Answer 1

If you want the equation of a cross section of your cone, parallel to $xy$ plane, then just substitute $z=k$ into its equation. Squaring to eliminate the root leads to the following quartic equation: $$ 9(x-y)^{4}+96(x+y)^{2}(k-x-y)^{2}-60(x-y)^{2}(x+y)(k-x-y) +3(x-y)^{2}(k-x-y)^{2}=0. $$ Keep in mind, however, that squaring the root introduces spurious points: only the outer edge of this implicit curve should be considered. I also included in the figure (made for $k=5\pi$) the limiting lines $y=\big(-7\pm\sqrt{48}\big)x$.