Finding principal curvatures at points on a surface without parametrization

Question

I have some points in 3D space. I can fit the equation of a surface z = f(x,y) to them either globally or locally.
However:

The surface does not pass through the origin
I can't parametrize it.

How can I find principal curvatures at any point on this surface?
I am finding this to be very difficult. All the examples I've seen so far parametrize the surface.
I've tried a "shortcut" method of finding the normal at each point and looking at how much the angle of the normal changes from a point to its neighbor, and estimating arc length ~ distance between the two points. This didn't work too well because the points aren't evenly spaced and I don't have neighbors in all directions around each point.
Online resources point me to the shape operator, which requires the first and second fundamental forms -- which I have no clue how to get from an equation like the one I showed above. I don't have a curvilinear coordinate system - do I have to have one?
Thank you

Kajelad · Accepted Answer

If surface $M$ defined by the graph of a function $z=f(x,y)$, then the function $(x,y)mapsto (x,y,f(x,y))$ is a global parameterization $mathbb{R}^2to M$. In this parameterization, the shape operator $s$ (w.r.t. the positive $z$ direction) takes a particularly simple form
$$
s=left(I-frac{operatorname{grad}f(operatorname{grad}f)^T}{1+|operatorname{grad}f|^2}right)frac{operatorname{Hess}(f)}{sqrt{1+|operatorname{grad}(f)|^2}}
$$
Where $operatorname{Hess}(f)$ is the matrix of second partial derivatives of $f$ and $operatorname{grad}(f)$ is the vector of first partials of $f$. In index notation, this is
$$
s^i{}_j=C(partial_ipartial_j f)-C^3(partial_i f)(partial_k f)(partial_kpartial_j f),     C=[1+partial_i fpartial_i f]^{-1/2}
$$
The principal curvatures of $M$ are the eigenvalues of $s$.

Finding principal curvatures at points on a surface without parametrization

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