DETERMING OF CHARACTERISTICS AT POINTS OF CONTOUR WITH MONOTONE CHANGE OF CURVATURE

Abstract

A surface model of complex shape, as a rule, is formed on the basis of a skeleton, the elements of which are curved lines. The task of providing the specified functional properties of such surfaces requires the development of methods for forming linear model elements in the form of one-dimensional contours with specified differential geometric characteristics. Such characteristics are the fixation order and the smoothness order of the contour, the dynamics of changes in the values of curvature along the curve.

In this work, we study the constraints that are imposed on the position of the tangents passing through the points of the original curve by the law of curvature changes along the contour.

The initial data for modeling the contour is an ordered points set. This point set represents a curved line, which we will call the discretely presented curve (DPC). As a result of a preliminary analysis of the initial points set, sections are determined on the basis of which a monotonic curve can be formed. Each monotonous curve is modeled separately for areas that are bounded by neighboring points. This ensures the monotone change of curvature in each section and the joining of the sections with the second order of smoothness. DPC can be formed on the basis of any point series. At the same time, it is possible to control and correct the resulting solution, and the absence of oscillations is guaranteed.

Further development of the method is aimed at increasing its versatility and adaptability to the requirements of specific applied problems. This problem can be solved by increasing the conditions imposed on the contour by increasing the number of shaping parameters.

Key words: discretely presented curve (DPC), basic triangle, tangent, curvature, barycentric coordinates.