An efficient method for forming parabolic curves and surfaces

Abstract

A new method for the formation of parabolic curves and surfaces is proposed. It does not impose restrictions on the relative positions in space of the sequence of reference points relative to each other, meaning it compares favorably with other prototypes. The disadvantages of the Overhauser and Brever-Anderson methods are noted. The method allows one to eectively form and edit curves and surfaces when changing the coordinates of any given point. This positive eect is achieved due to the appropriate choice of local coordinate systems for the mathematical description of each parabola, which together define a composite interpolation curve or surface. The paper provides a detailed mathematical description of the method of parabolic interpolation of curves and surfaces based on the use of matrix calculations. Analytical descriptions of a composite parabolic curve and its first and second derivatives are given, and continuity analysis of these factors is carried out. For the matrix of points of the defining polyhedron, expressions are presented that describe the corresponding surfaces, as well as the unit normal at any point. The comparative table of the required number of pseudo-codes for calculating the coordinates of one point for constructing a parabolic curve for the three methods is given.