FORMATION OF LOCATION AREA OF CURVE WITH MONOTONIC CHANGE OF CURVATURE

Abstract

The paper deals with the problem of modeling flat one-dimensional contours under specified conditions. A geometrical scheme and an algorithm have been developed for the formation of contours with a monotonic change in differential geometric characteristics: the positions of the tangents to the contour and the values of curvature at its points. The initial data for the formation of the contour are the coordinates of the points belonging to it, the order of smoothness and the nature of the change in characteristics along the contour. The options for controlling the shape of the path are the positions of the centers of curvature and normals to the curve, which are assigned at the origin. The curve is modeled according to a previously formed evolve, which is a convex outline of the first order of smoothness. The evolution of a monotone curve is formed taking into account the following requirements: the evolution is a convex curve; the normals to the curve are tangent to the evolution that defines it; the length of the evolute is equal to the difference between the radii of curvature at the points bounding the corresponding section of the curve. The contour is formed inside the area of possible location of the curve corresponding to the task. The limited range of the solution allows you to control the absence of oscillations and provide the necessary requirements for the characteristics and smoothness of the bypass. A feature of the method is the multiple repetition of the computational algorithms, which leads to the replacement with a given accuracy of the original geometric image by the accompanying broken line The software developed on the basis of the algorithms proposed in the work can be used to simulate linear elements of the frame of surfaces with increased dynamic qualities. Increased dynamic properties are required for surfaces that interact with the environment and restrict body products of aircraft, automobile, shipbuilding, turbine blades, channels of internal combustion engines, pipelines, working bodies of agricultural machines.

Key words: discretely represented curve, evolve, involute, monotonic change of curvature, normal, radius of curvature, center of curvature.