A mechanical interpretation of least squares fitting in 3D

Abstract

We address the computation of the line that minimizes the sum of the least squared distances with respect to a given set of $n$ points in 3-space. This problem has a well known satisfying solution by means of PCA. We offer an alternative interpretation for this optimal line as the center of the screw motion that minimizes the sum of squared velocities in the given points. The numerical translation of this viewpoint is a generalized eigenproblem, where the total residue of the optimal line appears as the smallest generalized eigenvalue.