Content of the “Geometric Modeling” Course for the “Mathematics and Computer Science” Training Program

Abstract

In this paper has been considered the main content and distinctive features of the “Geometric Modeling” training course for the “Mathematics and Computer Science” training program 02.03.01 (“Mathematical and Computer Modeling” specialization). The goal of the “Geometric Modeling” course study is the assimilation of mathematical methods for construction of geometric objects with complex curved shapes, and techniques for their computer visualization by using polygons of curves and surfaces. Methods for construction of structures’ curved shapes using spline representations, as well as techniques for construction of surfaces and volumetric geometries using motion operations and basic logical operations on geometric objects are considered. The spline representations include linear and bilinear splines, Hermite cubic splines and Hermite surfaces, natural cubic and bicubic interpolation splines, Bezier curves and surfaces, rational Bezier splines, B-splines and B-spline surfaces, NURBS-curves and NURBS-surfaces, transfinite interpolation methods, and splines of surfaces with triangular form. Logical operations for intersection of two spline curves, and intersection of two parametric surfaces are considered. The principles of scientific visualization and computer animation are considered in this course as well. Some examples for visualization of initial data and results of curves and surfaces construction in two- and three-dimensional spaces through the software shell developed by authors and used by students while doing tests have been demonstrated. The software shell has a web interface with the WebGL library graphic support. Tasks for four practical studies in a computer classroom, as well as several variations of homework are represented. The problems occurring in preparation materials for some course sections are discussed, as well as the practical importance of acquired knowledge for the further progress of students. The paper may be interesting for teachers of “Geometric Modeling” and “Computer Graphics” courses aimed to students with a specialization in mathematics and information, as well as to those who independently develop software interfaces for algorithms of geometric modeling.