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?