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**A mechanical interpretation of least squares fitting in 3D.**
*(English)*
Zbl 1141.65009

The paper is devoted to the development of a particular method for finding the least squares distance line in the Euclidean 3-space. A mechanical interpretation of this line, providing a description in terms of Plücker coordinates is given. The starting point are screw centers, real 6-vectors that represent infinitesimal rigid motions in 3-space. The necessary fundamentals of kinematics are provided. Then, a solution of the problem of finding the optimal (normalized) screw center for the given set of \(n\) points, in the sense that it minimizes the sum of the squared velocities in each point, is presented. The conclusion is that the optimal screw center will correspond to the “smallest” generalized eigenvalue of a pair of symmetric 6 by 6 matrices.

The key observation tells that the optimal screw center for a set of points has always pitch 0, and so it represents the Plücker coordinates of a line. Furthermore, after normalizing the screw center, the implied velocities are just the Euclidean distances between the given points and this line. It shows that the proposed algorithm can compute the least squares distance line. The generalized eigenvalue associated with this solution exactly equals the residue of the least squares distance line with respect to the given set of points.

The key observation tells that the optimal screw center for a set of points has always pitch 0, and so it represents the Plücker coordinates of a line. Furthermore, after normalizing the screw center, the implied velocities are just the Euclidean distances between the given points and this line. It shows that the proposed algorithm can compute the least squares distance line. The generalized eigenvalue associated with this solution exactly equals the residue of the least squares distance line with respect to the given set of points.

Reviewer: Tzvetan Semerdjiev (Sofia)

### MSC:

65D10 | Numerical smoothing, curve fitting |

65F20 | Numerical solutions to overdetermined systems, pseudoinverses |

65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |