Estimating Gaussian Curvature from 3x3 surface grid without parameterization

Question

Given a 3x3 fixed grid of surface coordinates (the center points of the cells of an elevation raster), is it possible to estimate the Gaussian curvature for the center point of this grid without parameterizing all 9 points into an equation?

Answer 1

It's unclear what your data exactly mean. If they somehow relate to a surface $z=f(x,y)$ the Gaussian curvature computes to $$\kappa={f_{xx}f_{yy}-f_{xy}^2\over(1+f_x^2+f_y^2)^2}\ .$$ Try to estimate this quantity from your data. I don't think that a mere nine points will give a reasonably precise approximation.