Open AccessThe unit of cubic matrices in modeling of micro-relief
Author(s) -
E. A. Belkin,
V. N. Poyarkov,
O. I. Markov
Publication year - 2021
Publication title -
iop conference series. materials science and engineering
Language(s) - English
Resource type - Journals
eISSN - 1757-899X
pISSN - 1757-8981
DOI - 10.1088/1757-899x/1047/1/012150
Subject(s) - paraboloid , mathematics , cuboid , curvature , riemann curvature tensor , matrix (chemical analysis) , surface (topology) , square (algebra) , invariant (physics) , square matrix , mathematical analysis , geometry , eigenvalues and eigenvectors , physics , symmetric matrix , materials science , composite material , mathematical physics , quantum mechanics
The apparatus of cubic matrices allows us to give a classification of the contiguous cuboid. Invariant theory is used for this purpose. A cubic matrix is a generalization of the concept of a square or rectangular matrix. Increasing the dimension of a square matrix by one requires a new definition of the product of three-dimensional matrices. It is necessary to search for new invariants for the operator of parallel transfer of the coordinate system and the operator of rotation around the origin. The contiguous cuboloid allows us to give a three-dimensional geometric model of the microrelief an order of magnitude more accurately than the contiguous paraboloid does. The geometrical structure of the contiguous coboloid is investigated by the method of sections. The curvature of the surface at the point of contact with the cuboid is estimated by the curvature tensor, which is analogous to the Riemann-Christoffel tensor. The curvature of the surface in the local area of the point of contact is estimated with certain assumptions. There is no more precise geometric object that defines the curvature of the surface.