CONJUGATION OF CURVED LINES IN THE POINT CALCULUS

Abstract

The paper implements a general approach to conjugate curves in the point calculus, which includes geometric algorithms for conjugating curves and their analytical description in the form of point equations. This is the mathematical basis for building high-performance computer-aided design systems and providing them with the necessary geometric modeling tools. Geometric algorithms for conjugating planar and spatial curves in general form involving, respectively, two or three free functions continuous and deferential within the parameter variation interval are considered. A point definition of tangents for planar and spatial curves is presented, which consists in differentiating the equation of the original curve by the current parameter followed by a parallel transfer of the resulting segment to the tangent point. Directly conjugation is provided by means of arcs of the first order of smoothness. Several examples of the definition of the arcs of the outline in the point calculus and recommendations on their use for the conjugation of curves are presented. The point equations given in the article and the computational algorithms based on them are valid not only for conjugation of curves, but also for straight lines, proceeding from the fact that a straight line is a special case of a zero-curvature curve that does not require a tangent to be constructed. The prospect of further research is for the conjugation of surface bays, in which two tangents to the original surface bays will form surface bays with cross-sections in the form of arcs of the outline of the chosen shape.