SOLUTION OF POSITIONAL PROBLEMS AT MODELING OF MONOTONOUS CURVES

Abstract

The paper considers the development of the method of variable discrete geometric modeling of curved lines with a monotonous change in curvature. The discretely represented curve (DRC) being modeled is represented by an ordered set of points that belong to them and the differential geometric characteristics of the curve. These characteristics must be ensured during the modeling process. The curve is formed by thickening, which involves determining the initial point series of intermediate points. At the same time, we believe that the starting points are set without error and do not change their position during the modeling process. The problem of the mutual location of the DRC and an arbitrary straight line has been solved: determining the point of intersection or contact of a curve with a straight line. The solution of the problems considered in the work involves determining the area of a possible solution based on the conditions imposed on the curve: the absence of oscillation, the given order of smoothness, the dynamics of changes in the tangent positions and the values ​​of the radii of curvature along the curves, etc. In the process of modeling, the area of a possible solution is refined.

The proposed algorithms can be used to form surfaces according to the given differential-geometric conditions on the basis of the framework, the linear elements of which are flat DRCs. Algorithms allow you to match the characteristics of the curves that define the mesh discrete frame of the surface. This makes it possible to ensure the intersection of curved lines belonging to different families of lines in the process of successive thickenings, and to control the dynamics of changes in positions tangent to the surface. Modeling the surface involves the thickening of the DRC, which are the elements of the surface frame, and the increase of these elements - the thickening of the linear frame. The elements of the model are the lines formed on the basis of the original nodes (output DRCs) and the lines formed on the basis of the original point series consisting of points of condensation of the original DRCs (condensation DRCs). The surface will be considered defined if all the DRCs that make up the frame of the surface are formed with an error that does not exceed the specified value.

Key words: discretely represented curve, region of curve location, second order of smoothness, monotonicity of curvature change.