CUTTING A PLANE BY A REGULAR CURVE SYSTEM

Abstract

In the issues of computational geometry and electrical circuit automatic design, an essential role is played by the problems of dividing the Euclidean plane by a finite system of continuous plane curves (lines) of a given shape. Often this system has a regular structure, and the curves can have the same number of pairwise mutual intersections. Each curve consists of inner and at most two endpoints. The article considers four types of curves: bounded, semi-bounded, unbounded and closed (cyclic). In accordance with this, a bounded, semi-bounded, unbounded and closed (cyclic) systems of curves are defined. The concept of a k-regular system of lines is also defined: each pair of lines from this system intersects at exact k points. A theorem is proved which contains the largest number of domains (parts) of the plane, which is divided by a k-regular system of n curves for the cases when the system is bounded, semi-bounded, unbounded or closed (cyclic). The consequence of this theorem is to find the largest number of regions into which the plane is divided by n segments, rays, straight lines, bounded (semi-bounded or unbounded) angles, semicircles, circles, ellipses, parabolas, triangles, quadrangles, convex m-gons. In some practical aspects of projection on a plane, curves can take the form of broken lines. The theorem proof is carried out in a rather intelligible language of graph theory, the methods of which are widely used in many applied problems of physics, chemistry, biology, geography, economics, sociology, linguistics, and electrical engineering (in particular, the electrical circuits design). An essential fail in the proof is that we can consider the possibility of constructing the needed form plane curve system, which will be k-regular.