No abstract

Multidimensional interpolation is an important scientific task in demand of various science fields and technology. The geometrical theory of multidimensional interpolation has been further developed in terms of extending the shaping tools of geometrical modeling at the expense of the contour arcs passing through the predefined points. The proposed method is based on modification of Bezier’s curve with preservation of tangents in its initial and/or final points. Then the Bezier curve retains its geometrical properties of the arc, but additionally has the possibility to pass through several points set in advance. Examples are given of modifying a 5th-order Bezier curve arc into a curve arc passing through four predefined points and a 3rd-order Bezier curve arc passing through three predefined points and having a tangent in the initial point, which is proposed to be used as the onion dome surface forming arc. Such a method helps to reduce the piecewise character of composite curves when building curves. The introduction of such research results into computeraided design and solid-state modeling (CAD/CAM) systems will allow to expand their toolbox in terms of shaping surfaces and solids of technical shapes of various purposes.

The paper implements a general approach to conjugate curves in the point calculus, which includes geometric algorithms for conjugating curves and their analytical description in the form of point equations. This is the mathematical basis for building high-performance computer-aided design systems and providing them with the necessary geometric modeling tools. Geometric algorithms for conjugating planar and spatial curves in general form involving, respectively, two or three free functions continuous and deferential within the parameter variation interval are considered. A point definition of tangents for planar and spatial curves is presented, which consists in differentiating the equation of the original curve by the current parameter followed by a parallel transfer of the resulting segment to the tangent point. Directly conjugation is provided by means of arcs of the first order of smoothness. Several examples of the definition of the arcs of the outline in the point calculus and recommendations on their use for the conjugation of curves are presented. The point equations given in the article and the computational algorithms based on them are valid not only for conjugation of curves, but also for straight lines, proceeding from the fact that a straight line is a special case of a zero-curvature curve that does not require a tangent to be constructed. The prospect of further research is for the conjugation of surface bays, in which two tangents to the original surface bays will form surface bays with cross-sections in the form of arcs of the outline of the chosen shape.

The article proposes an approach to systematization, modeling and optimization of multidimensional statistical data based on the use of projection algorithms for computer modeling. The proposed approach is presented on the example of computer modeling and optimization of socio-economic data, but it can also be effectively used to systematize and analyze other experimental statistical data. It consists the fact that the original multidimensional data are presented in the form of projections on the Radishchev’s complex drawing in the form of curved lines system. Then, on the indicator curve, the optimal value of the socio-economic indicator is selected (as a rule, this is one of the extrema of the function) and the value of the time at which it was reached is fixed. Here, the indicator curve is understood as the curve corresponding to the response function, and the factor curve is the curves corresponding to the factors influencing the response function. Further, a scientific hypothesis is put forward that the joint interaction of factors recorded at a given moment in time ensures the optimal value of the socio-economic indicator. Thus, we obtain the optimal values of the factors influencing the response function, which in this case is the socio-economic indicator. The interaction between the indicator curve and the factor curves is carried out through the line of interprojection connection. The proposed scientific hypothesis is fully justified, provided that all possible factors affecting the behavior of the socio-economic indicator are taken into account. The implementation of the proposed approach was carried out using the Radishchev’s complex drawing, which displays both the values of the factors and the socio-economic indicator. At the same time, on the Radishchev’s complex drawing, the most favorable conditions for the socio-economic indicator are selected by methods of mathematical analysis. Further, with the help of the line of inter-projection communication, by means of standardization, the desired weight coefficients are determined, corresponding to the most favorable conditions for the socio-economic indicator. This approach is completely independent of the subjective opinion of experts and based solely on the initial statistical information.