I am studying computer science so please go easy on me. I am also too bad at math to extract the mathematical essence that is needed to answer this question so I'm just gonna explain the whole setup.
We are looking at functions mapping points on a surfaces living in 3d onto a 2d plane.
$f: S\subseteq \mathcal{R}^3 \rightarrow \Omega \subseteq \mathcal{R}^2$ (a parametrization).
However we are dealing with a discretized version. Meaning the surface consists of triangles. If we look at a single triangle and how it is embedded into the plane we can decompose this mapping into a rotation that eliminates one axis and an affine mapping $\varphi$. Let the singular values of $\varphi$ be $\sigma_1$ and $\sigma_2$.
The two things that I don't understand now are
- that the diagonal entries of the jacobian are equal to those singular values. Meaning $\mathbf{J} = \begin{pmatrix}\sigma_1 & 0 \\\ 0 & \sigma_2 \end{pmatrix}$ and
- why the non-diagonal entries of the Jacobian are 0
I do understand that the Jacobian is square as, even though we are mapping from $\mathcal{R}^3\$ to $\mathcal{R}^2\$, we already eliminate one dimension by the rotation.
For more context:
I found this paper which introduces the symmetric dirichlet energy on meshes. This basically explains the whole part up until the singular values. My professor defined the dirichlet energy using the Jacobian entries
$E_D(f)=\int_\Omega \|\|\nabla f\|\|^2$
$\mathbf{J} = \begin{pmatrix}f_u & 0 \\\ 0 & f_v \end{pmatrix}\Rightarrow E_D(f)=f_u^2 + f_v^2$
From this I concluded that $\sigma_1=f_u$ and $\sigma_2 = f_v$
If there is any more context needed I will happily supply it.