DESIGNATION OF CENTERS OF CURVATURE OF THE INTERPOLATING CURVA

Abstract

The article proposes a method for assigning initial data to form an evolution of a monotonic curve interpolating a given sequence of points.

The tasks that can be solved using the proposed method include the formation of graphs describing phenomena or processes, as well as modeling, based on a set of starting points, of linear frames that define a surface. Controlling the presence of special points is an important condition for solving these problems qualitatively.

The solution to the problem of controlling the presence of special points in the curve interpolating a given point series can be based on the rejection of the analytical representation of the curve sections. The interpolating curve is formed as a region of possible location of its monotonic parts. The main condition for solving the problem is the correct assignment of curve characteristics at the starting points. These characteristics are the position of the normals and the value of the curvature at the starting points. The problem can be solved by forming an evolution of a monotonic curve that interpolates a given point series.

The problem is solved in two stages. First, the position of the normal of the interpolating curve is assigned at all starting points. Secondly, the positions of the curvature centers on the already assigned normals are assigned.

The position of each normal is assigned within a region whose boundaries are determined based on the specified properties of the interpolating curve line. The initial area of possible location of each normal is determined by the coordinates of five consecutive starting points. This initial area is refined by simultaneously assigning the position of the normal at all starting points.

The position of each curvature center is assigned within a predefined segment that takes into account the entire area of its possible location.

Normals assigned at the starting points and segments connecting the curvature centers assigned on these normals delimit a chain of triangles. This sequence of triangles is the region of possible location of the monotonic curve evolution that interpolates the given sequence of points. The task of further forming the evolute is to form its sections in the form of arcs of convex curves that touch the normals at the corresponding centers of curvature. The length of these arcs should be equal to the difference in curvature radii at the points that bound the corresponding section of the interpolating curve.

Setting the position of the normals and curvature centers within the specified ranges makes it possible to create an evolution that defines a monotonic regular curve line along which the curvature values change uniformly and that interpolates the original sequence of points.

Keywords: interpolation, monotonic curved line, special points, normal, center of curvature, evolution, radius of curvature.