I am working with a surface that has an unknown parametrization; however, we can define a function on the boundary. Thus, I can define a curve in a plane. Can I define principal curvatures along the function $y = f(x)$? Additionally, is anybody familiar with formulas which allow me to express the principal curvatures in terms of $f(x)$. I believe I should be able to use the following curvature function:
$k = \frac{f''(x)}{(1+f'(x)^2)^{3/2}}$
for the curvature in the x-direction. Anybody have suggestions/corrections? I appreciate it.