USING THE METHOD BASED ON NORMAL FUNCTIONS FOR CONSTRUCTION AND VISUALIZATION OF LEVEL SURFACES WITH RESPECT TO POINTS, LINES, AND PLANES

Abstract

In various fields of knowledge, scientific research and applied developments, the goal is to obtain a real product or recommendations for implementation. At the beginning of research or development, as a rule, the problem as a whole is formed, and tasks are set. As a rule, this involves both studying the physical characteristics of real objects and processes and selecting existing mathematical models for their description. Due to the complexity of problems and tasks, those that can be solved separately, having in the arsenal of known or new (specially developed) methods, are distinguished. Of course, this depends on the input data about the modeling objects. The difficulties relate to the extension of such selected or specially developed methods to solving problems of modeling objects and processes of different physical content. In Euclidean space, geometric objects can be represented as zero-dimensional, one-dimensional, two-dimensional, or three-dimensional. Solutions to geometric modeling problems of objects are such models that are visualized in the form of images of points, lines, surfaces, and solids. Graphical representation of solutions allows for visual analysis of results, detection of errors, and determination of further stages of work. The same applies to the development of models that represent phenomena and processes in geometric language and are implemented in the form of visual pictures. In this case, it is important to properly form the input conditions of the modeling problem. Methods for geometric representation of input information and visualization of results depend on the space dimension and level of complexity of the modeling object. The article identifies the features of using the method based on normal functions (a method from the theory of generalized parallel sets) in constructing and visualizing level surfaces (which are graphical representations of normal functions) relative to points, lines, and planes; examples are given. Limitations of use apply only to the classical method and the mathematical models underlying it. It is indicative that for the specified types of geometric objects of Euclidean space, the proposed practical implementation of the method based on normal functions is acceptable, where these objects can be arranged arbitrarily, and visual analysis of the existing intersections of level surfaces with coordinate planes can be carried out.

Keywords: geometric modeling, normal functions, visualization of models, level surfaces, parallel lines and surfaces.