= Smooth endpoint criterion for a surface of revolution
Suppose a smooth profile $(f(v),g(v))$ has $f(a)=0$, $f>0$ just to the right of $a$, and $f'(a)\ne0$. Use $r=f(v)$ as a local parameter and write $g(v)=h(r)$. The rotated surface is smooth at the pole if and only if
$$
h(r)=h(0)+H(r^2)
$$
near zero for a smooth function $H$. It is then locally the smooth graph $z=h(0)+H(x^2+y^2)$. Equivalently, the even reflection $h(|r|)$ is smooth; in particular all odd derivatives of $h$ at zero vanish and the profile meets the rotation axis orthogonally. The analogous condition holds at the other endpoint.
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