Distance between a 3D point and a Plucker line (or alternative representation)

Question

Given that a 3D line passing through two points $A$ and $B$ can be written as Plucker representation $L = AB^T - BA^T$, I'm wondering if there's a way to compute the distance of a Plucker line to a 3D point in a series of pure matrix-matrix operations.

If not possible, is there an alternative way to represent a 3D line that would permit the computation of point-line distances in a pure linear algebra format?

Why I care:

I am trying to formulate an optimization problem that reduces the distance between a 3D point and a line (all are in projective geometry).

My goal is to write the formulation of the optimization in a purely matrix (linear algebra) format. What I mean is to have a purely matrix representation of the function $d$ in the following formulation:

$min~\Sigma_{i=1}^{n}~d(X.P_i, L_i')^2$

where $X$ is a transformation matrix that I'm optimizing for, each $P_i$ is a 3D point and each $L_i$ is a 3D ray (a line).

The reason I would like to write all this in matrix format is that it would look more compact and would make finding the gradients of the objective function cleaner and more expressive.

Answer 1

I figured this out and found a relation for finding the distance of a 3D point to a Plucker line in [1]. Here is a direct quote from the paper:

A Plucker line $L = (n, m)$ is described by a unit vector $n$ and a moment $m$. This line representation allows to conveniently determine the distance of a 3D point $X$ to the line

$d(X, L) = ||X \times n - m||_2$

where $\times$ denotes a cross product.

Judging by the fact that they are solving an optimization very similar to the one I am dealing with, I think my original question is solved!

[1] Brox, Thomas, et al. "Combined region and motion-based 3D tracking of rigid and articulated objects." IEEE Transactions on Pattern Analysis and Machine Intelligence 32.3 (2010): 402-415.