Conjugacy and geometry II : moore-penrose inverse and feet of the perpendiculars

Abstract

In this paper on space geometry, generalized inverses are used in the study of distances. Three cases are considered: distance from a point to a plane, distance from a point to a line and distance between two skew lines. Moore-Penrose inverses occur in the expressions of the feet of the perpendiculars and in the representation of the vectors materializing the distances. The results of this kind of problems fit in the cadre of approximation theory and, because best approximation problems often require the projection of the origin onto linear varieties, in order to solve the proposed problems, we make extensive use of the conjugacy principle, much present in Mathematics. The obtained results are not only useful for undergraduate Science and Engineering students but are also applicable in very practical sciences and techniques, notably on Coordinate Metrology, Photogrammetry, etc. Moreover, this paper could pave the way for more generalized problems demanding more sophisticated approaches.