The paper presents the results of multidimensional parabolic interpolation studies, (one of the special cases of the multidimensional interpolation method), applied to solve problems of modelling multifactor processes and phenomena using geometric objects of multidimensional affine space. The authors describe the technique of geometric model tree forming of the process under study and its analytical description based on computational point algorithms with subsequent implementation on a computer. Such an approach makes it possible to effectively use multidimensional interpolation instead of multidimensional approximation (based on the least squares method) for solving problems of mathematical and computer modelling of multifactor 3-level processes and phenomena of animate and inanimate nature, technology, economy, construction, and architecture The study gives an example of multidimensional parabolic interpolation application to simulate the dependence of the fine-grained tar-polymer concrete compressive strength on 4 factors: tar viscosity, polyvinyl chloride dropout concentration in coal tar, activator concentration on the mineral powder surface and temperature, followed by optimization of the composition and operating conditions road pavement.

The article describes the principles of solid modelling in point calculus, including the definition of geometric bodies in the form of an organized set of points in space. At the same time, the choice of point calculus as a mathematical apparatus for effective modelling of geometric bodies in 3-dimensional space is substantiated, expanding the instrumental capabilities of computer graphics. By means of generalization, it has been established that the dimension of the space in which the geometric body is defined is equal to the number of current parameters. On the basis of this, a new definition of a geometric body is proposed as a geometric set of points, in which the number of its determining parameters is equal to the dimension of space. Examples of the definition of a tetrahedron body and a triangular prism body in point calculus, obtained considering the proposed definition of the term “geometric body”, are given. The obtained point equations are completely invariant with respect to the choice of the coordinate system and depend only on the coordinates of the points that define the vertices of the modelled bodies. Thus, the obtained point equations determine the entire set of bodies of tetrahedrons, bodies of triangular prisms, bodies of elliptical cylinders and cones in 3-dimensional space. The prospect of further research is the definition in point calculus of geometric bodies of curvilinear and irregular shapes, considering their relative position in space, as well as more complex composite geometric bodies in 3-dimensional space.

The article presents a new vision of the process of approximating the solution of differential equations based on the construction of geometric objects of multidimensional space incident to nodal points, called geometric interpolants, which have pre-defined differential characteristics corresponding to the original differential equation. The incidence condition for a geometric interpolant to nodal points is provided by a special way of constructing a tree of a geometric model obtained on the basis of the moving simplex method and using special arcs of algebraic curves obtained on the basis of Bernstein polynomials. A fundamental computational algorithm for solving differential equations based on geometric interpolants of multidimensional space is developed. It includes the choice and analytical description of the geometric interpolant, its coordinate-wise calculation and differentiation, the substitution of the values of the parameters of the nodal points and the solution of the system of linear algebraic equations. The proposed method is used as an example of solving the inhomogeneous heat equation with a linear Laplacian, for approximation of which a 16-point 2-parameter interpolant is used. The accuracy of the approximation was estimated using scientific visualization by superimposing the obtained surface on the surface of the reference solution obtained on the basis of the variable separation method. As a result, an almost complete coincidence of the approximation solution with the reference one was established.

The paper proposes an approach to modeling geometric multifactorial processes and phenomena using mixed geometrical interpolants, for example of the steel fiber concrete strength characteristics’ simulation A key role is played by the formation of the geometric model, as developed in detail in the article. Analytical description of the geometric model obtained by means of algebraic curves (that pass through the predetermined point in advance) is realized by means of the BN-calculus mathematical formalism.

The paper describes an approach to solid modeling of geometric objects in the form of an organized three-parameter set of points in three-dimensional space. The relevance of the research topic is due to the widespread use of solid-state models in various branches of science and technology, mechanical engineering, construction and medicine. Solid-state computer models are currently one of the basic computer graphics tools and an integral part of computer- aided design and calculation systems. It is widely used as one of the control elements of CNC machines and 3D printing, the development of information systems in the design and construction of buildings and structures, finite element calculations of deformed states in aircraft and mechanical engineering, their manufacture in medicine, etc. The choice of point calculus as a mathematical apparatus for the analytical description of solid models of geometric objects is substantiated. Examples of modeling sets of elliptical bodies and toroidal bodies in a simplex of three-dimensional space are given.