Determination of the intersection points of two plane curves by means of differential equations

Abstract

A new method is proposed to calculate the intersection points of two plane curves. The theory of singular points of a system of two differential equations is used in developing the method. The intersection point to be determined is identified with such a singular point and appropriate modifications are applied to the system to ensure that the singular point be stable, i.e. all integrals which start in the neighborhood of the singular point will always approach this point if the integral parameter tends to infinity. In addition a method is described for systematically searching for all intersection points in a prescribed rectangular area.