USING PARAMETRIC SPLINE FUNCTIONS IN MODELING

Abstract

The modern approach to 2D and 3D modelling in computer graphics, web design and 3D modelling is constantly evolving due to new technologies and needs of game and audiovisual content developers, visualisation and architectural design, industry and medicine. These conditions often lead to a growing complexity of 2D and 3D modelling tasks, with increased input data and requirements for accuracy and speed. 2D and 3D modelling requires software environments to use efficient mathematical solutions that will achieve scalability of the obtained solutions, improve interpolation accuracy and reduce visualisation and rendering time. The use of parametric spline functions to generate curves and surfaces is proposed. Such curves and surfaces allow to obtain smooth, flexible shapes with a certain number of interpolation nodes, whereas spline curves are used in 3D modelling to create the shape of the modelled object, and spline surfaces can approximate its three-dimensional shape. Interpolation parametric splines will reduce computational complexity, provide scalability of solutions and significantly simplify the modelling process. The paper shows the principles of determining interpolation curves using local linear spline functions and provides an example of constructing an interpolation curve of a parametric linear spline. To determine the interpolation error by parametric linear spline functions, two theorems are proposed to determine the interpolation errors in the construction of curves and surfaces by parametric splines. It is proposed to use parametric splines of higher orders, such as parametric quadratic or parametric cubic splines, or parametric cubic B-splines, to reduce the interpolation error of curves and surfaces in modelling 2D and 3D objects. The direction of further research is to consider parametric cubic and cubic B-spline splines and perform a comparative analysis of the results obtained in order to improve the accuracy of building curves and surfaces in 2D and 3D modelling.