I've been trying to figure out an explicit form for a quadratic funnel surface, but I'm not succeeding. Essentially, I'm wanting an explicit form for the surface that would be created by revolving the following curve around the y-axis:
This curve has the formula (for $x<1$): $$y=\sqrt{-\left(x-1\right)^{2}+1}$$
The closest I got was this:
$$y = \max\left(\sqrt{-(|x|-1)^2+1}, \sqrt{-(|z|-1)^2+1} \right)$$
Which gives me:
But this isn't quite right since it has a hard crease at each corner. What is the correct formula?