I am struggling to understand the derivation of an equation in a paper (A Bayesian Method for Probable Surface Reconstruction and Decimation, specifically Eqn. 16).
Basically they define three vertices of a facet: $x_k, x_{k'},x_{k''}$ The normalized facet normal is defined as: $n_i = \dfrac{(x_{k'}-x_k) \times (x_{k''}-x_k)}{|(x_{k'}-x_k) \times (x_{k''}-x_k)|}$
So far so good. The problem is then that they need to compute $\frac{\partial n_i}{\partial x_k}$. Firstly ${n_i}$ and ${x_k}$ are both vectors, hence I'd expect that this partial derivative notation means in effect the Jacobian of ${n_i}$ wrt. ${x_k}$? In that case that is a 3x3 matrix. However the formula below (see Eqn 16) implies that the result is a 3x1 vector?? (confused!)
$\frac{\partial n_i}{\partial x_k} = \frac{I - n_in_i^T}{|(x_{k'} - x_k) \times (x_{k''} - x_k)|} (x_{k''} - x_{k'}) \times x_k$
I was hoping someone could shed some light on the dimensionallity confusion and also how that formula was derived, or if incorrect what is the correct forumation for $\frac{\partial n_i}{\partial x_k}$? Thanks for the help!