Myltsev O. Mathematical modeling of forms of multidimensional geometric objects using cognitive computer graphics

The dissertation is devoted to solving the actual scientific and technical problem of increasing the effectiveness of mathematical modeling of the forms of complex geometric objects, given analytically by multidimensional functions, using the means of illustrative and cognitive graphics. To solve this problem, a mathematical apparatus is proposed for the synthesis, analysis and visualization of graphic images-models (M-images) of differential geometric characteristics of forms of multidimensional objects based on discrete voxel structures using cognitive computer graphics. In the dissertation, an analytical review of the current state of the problem of mathematical modeling of the shapes of geometric objects is carried out. The functional representation of geometric objects based on the theory of R-functions, discrete models for representing the shapes of three-dimensional geometric objects are considered. Methods for studying the shape of geometric objects, given analytically by functions of two variables, using graphic images-models and ways of using M-images in solving optimization problems in two-dimensional space are analyzed. Modeling of the shapes of three-dimensional geometric objects based on voxel structures is proposed. A discrete voxel model has been developed for representing the forms of multidimensional geometric objects, specified analytically using R-functions of n-variables. A recursive method for the formation of a discrete voxel model for representing the shapes of three-dimensional geometric objects, specified analytically using R-functions of three variables, has been developed, which allows obtaining for each voxel a four-component normal vector and partial derivatives of a function of order n. An iterative method for interpolation refinement of a voxel model of graphic data has been developed, which increases the detail of a geometric object. A projection method for visualizing a voxel model of graphic data has been developed, which can be used to construct images of arbitrary three-dimensional objects that can be split into parts and display a separate part independently of others. Modeling of forms of multidimensional geometric objects based on voxel structures of graphic images-models is proposed. Methods for the synthesis of graphic images-models based on voxel structures, reflecting the local differential geometric characteristics of a three-dimensional object, specified analytically using R-functions, with generalization in the n-dimensional case, have been developed. The complexity of the recursive approach in the formation of graphical M-images of multidimensional geometric objects is estimated. Methods for analyzing the shapes of geometric objects using voxel structures of graphic images-models are proposed. Methods of spatial motion along a gradient in three-dimensional space based on the voxel structures of graphic images-models have been developed. A mathematical apparatus has been developed for the automated solution of optimization problems of mathematical programming based on the voxel structures of graphic images-models.