I am trying to find points of a surface whose normal is parallel with the 3 dimensional axes (i.e. local maxima, minima and saddle points in x, y and z dimensions).
The function of the surface is defined as $f(u,v)$ which is continuous and gives a 3D vector indicating the position of the surface for a given set of parameters u and v.
To find the points I require I understand that I need to find the derivatives of the surface with respect to u and v. The relevant points will occur when:
$\frac{\delta f_i(u,v)}{\delta u}=0$ & $\frac{\delta f_i(u,v)}{\delta v}=0$
where $i$ is the respective dimension (x,y or z) and $0\leq u\leq1$ & $0\leq v\leq1$.
What kind of strategy should I take to solving this problem? I was thinking it may be possible to use a numerical method with the partial derivatives to iteratively find solutions. But then how would I remove the solution as you would in univariate root finding?
Any insights would be really useful.