By searching relevant materials, I found a method to calculate the principal curvature of the fitted surface, which calculates the Hessian matrix, then obtains its eigenvalues, and finally directly takes two eigenvalues as two principal curvatures. Why can it use its eigenvalue as the principal curvature?
Also, if my fitting surface is a quadric, then the Hessian matrix is a definite matrix(the second derivative of a quadric is a constant), which means that the curvature of any point on the fitting surface is the same. But it is impossible. Can someone tell me where I'm getting this wrong?
As a student engaged in geoscience research, I also want to ask something that I do not understand, which is about curvature calculation in ArcGIS. The formula in ArcGIS for calculating the curvature is
$-100×(d^2z/dx^2+d^2z/dy^2)$.
What is the type of curvature that ArcGIS calculates and why can it just substitute curvature with the second derivative and what does that do when you multiply it by -100? It's so confusing.
I assume by a quadric fit you mean a graph $$r(u,v) = u \mathbf{b}_1 + v\mathbf{b}_2 + F(u,v)\mathbf{b}_3$$ where $F(u,v)$ is a quadratic function and the $\mathbf{b}_i$ some orthonormal basis?
You can compute the first and second fundamental forms:
$$\mathrm{I} = \begin{bmatrix} 1 + F_u^2 & F_uF_v\\F_uF_v & 1 + F_v^2\end{bmatrix}, \quad \mathrm{I\!I} = \frac{1}{\sqrt{1+F_u^2+F_v^2}} HF$$
where $HF$ is the Hessian of $F$. As you point out, the principal curvatures will be the generalized eigenvalues of $\mathrm{I\!I}$ with respect to $\mathrm{I}$ (or equivalently, the eigenvalues of $\mathrm{I}^{-1}\mathrm{I\!I}$) which are not the same as (but also not totally unrelated to) the eigenvalues of $HF$.
Yes. The fit lets you estimate the curvature at one point on the surface; if you want to estimate how the curvature varies in a neighborhood, you need to fit using a higher-order approximation (or compute several different fits at different points). (Compare to estimating the second derivative of a 1D function by fitting a quadratic).