Bicubic Surface on a Fixed Frame: Calculation and Visualization

Abstract

The paper proposes an algorithm for calculating a composite bicubic surface with a fixed frame formed by longitudinal (along the x axis) and transverse (along the y axis) cubic splines. Frame line equations are taken as the main boundary conditions. According to the proposed algorithm, the problem is divided into two stages: first, the frame line equations are found and then the coefficients included in the equations of the bicubic portions forming the bicubic surface are calculated. This approach reduces the size of the characteristic matrix of the linear equation system by reducing the number of coefficients in the surface equation. The matrix size is reduced from 16mn to 9mn, where m and n are the number of bicubic portions along the x and y axes. Surface visualization is reduced to building a grid of longitudinal and transverse generators, the equations of which are formed from the surface equation by substituting y=const (longitudinal generators) or x=const (transverse generators). In this paper we calculate and visualize bicubic surfaces with a frame formed by a mixed set of cubic splines and straight lines. The clarity of the examples is ensured by indicating the numerical values of all calculated magnitudes with an accuracy of up to nine significant figures.