# Computing the Distance Between a 3D Point and a Plücker Line

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 [1]). After a lot of looking, I found that there is a simple method for finding this distance in [2]. 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.

[1] Hartley, Richard, and Andrew Zisserman. Multiple view geometry in computer vision. Cambridge university press, 2003.

[2] 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.