2 research outputs found
Conjugacy and geometry I : foot of the perpendicular, distance and gram determinant
In this note on space geometry, the Gram determinant is used for
expressing distances, vectors whose magnitude equals those distances and
best approximation points. Three cases are considered: distances from a
point to a line and to a plane and distances between two skew lines.
(Symbolic) determinants occur in the expressions of the feet of
perpendiculars and in the representation of the vectors materializing the
distances. Because best approximation problems often require the use of
subspaces, in order to solve the general cases of the proposed problems,
we make extensive use of the conjugacy principle much present in
Mathematics. The main purpose of this paper, focused on the resolution of
distance problems in tridimensional geometry, is to provide the
acquisition of spatial abilities through the proposed constructive approach.
The obtained results, which could be a starting point and give clues for
solving more advanced geometry problems, are applicable in several fields
of practical sciences, such as the Coordinate Metrology, for instance.
Moreover, this paper may be a window for coming across with a diversity
of scalar products.info:eu-repo/semantics/publishedVersio
Conjugacy and geometry II : moore-penrose inverse and feet of the perpendiculars
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.info:eu-repo/semantics/publishedVersio