In order to solve an optimization problem with the goal of reducing the distance between a bunch of 3D points and lines, I was looking for the correct way of finding the distance between 3D points and a Plucker line representation.
The Plucker line \(L\) passing through two lines \(A\) and \(B\) is defined as \(L = AB^T – BA^T\) (for more details refer to ). After a lot of looking, I found that there is a simple method for finding this distance in . 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.
 Hartley, Richard, and Andrew Zisserman. Multiple view geometry in computer vision. Cambridge university press, 2003.
 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.