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.

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