i have parametric equation of an unknown surface $S(u,v)$: $x=f(u, v)$, $y=g(u,v)$ and $z=h(u,v)$, where $u$ and $v$ are two independent variables.
How can I check whether it's a sphere or not?
my attempt:
i defined a general function:
$S=x^2+y^2+z^2$
I know that the value of $x^2+y^2+z^2$ will be equal to the square of radius i.e.$R^2$ at every point $(x,y,z)$ on a sphere. Therefore to consider all the points on a sphere, if i calculate maximum and minimum values of $S=x^2+y^2+z^2$ they must always be equal for a sphere. so I can write a general condition to check where it is a sphere or not
if max$(x^2+y^2+z^2)$ $=$ min$(x^2+y^2+z^2)$, it is a spherical surface
if max$(x^2+y^2+z^2)$ $\ne$ min$(x^2+y^2+z^2)$, it is not spherical surface
Is my above condition will work to check whether it's a spherical surface or not for any arbitrary input of parametric equations, $x=f(u, v)$, $y=g(u,v)$ and $z=h(u,v)$?
i have checked that my above condition fails to distinguish between a spherical surface say $x=5\sin u\cos v$, $y=5\sin u\sin v$ and $z=5\cos u$ and a toroidal surface say $x=(15+4\cos u)\cos v$, $y=(15+4\cos u)\sin v$ and $z=4\sin u$. but I don't know why it becomes true for a toroidal surface too. initially I thought that above condition is true only for a spherical surface only but i found that it's also true for a toroidal surface.
please suggest me any generic condition to exactly check if parametric equation $S(u,v)$ is spherical surface or not for 100% certainty. thanks