TECHNIQUE FOR COMPOSING POINT EQUATIONS B-SURFACES USING GEOMETRIC BN MATRIX

Abstract

With the advent of a point BN-number, new possibilities were found for solving interpolation problems.  Further development of the geometric-mathematical apparatus of the point BN-calculation led to the need for the introduction of geometric matrixes (geomatrixes).  The formation of point equations of surfaces interpolating the initial points causes some difficulties.  The use of a geomatrixes for the purpose of error-free receiving of the point equation of a segment of a surface by predetermined points on it, greatly simplifies the search for an interpolant for input points. 

The article presents the development of the mathematical apparatus of a point BN-calculation for the purpose of its further use in the of multifactorial processes modeling.  The method of formation of point equations of B-surfaces is proposed.  For this purpose, the concept of a geometric matrix (geomatrix) is introduced and the technique of the use of geomatrix for the construction of segments of surfaces with predetermined starting points is developed.  The research is based on the method of moving simplex, by which the function-parameters of the point equation of the segment B-surface are obtained.  The term "B-surface" is used to represent the shape of any surface at the reference points taken on it.  The B-surface dot equation is defined as a superposition of elements of a geomatrix, which is the product of the geomatrix of the points and the geometric parameters.  The point equations for B-surfaces of 3×3 (nine reference points) are obtained.  Thus, an algorithm for the construction of point equations for B-surfaces with the use of operations over geomatrixes is proposed.  The main advantage of the method is that interpolation formulas are obtained in a point form without solving systems of linear equations.  This significantly simplifies the modeling process.

The main advantage of B models is that any multi-dimensional problem can be translated into an appropriate number of one-, two-, or three-dimensional problems due to the fact that the B-models mainly use parametric communication unlike the traditional projective connection between the image and its projection

Keywords: B-functions generation, B-curves forming, parameter, geometric shape, geometric matrix.