SCCI-HYBRID METHODS FOR 2D CURVE TRACING

Abstract

A hybrid method for plotting 2-dimensional curves, defined implicitly by equations of the form f(x,y) = 0 is presented. The method is extremely robust and reliable and consists of Space Covering techniques, Continuation principles and Interval analysis (i.e. SCCI). The space covering, based on iterated subdivision, guarantees that no curve branches or isolated curve parts or even points are lost (which can happen if grid methods are used). The continuation method is initiated in a subarea as soon as it is proven that the subarea contains only one smooth curve. Such a subarea does not need to be subdivided further so that the computation is accelerated as far as possible with respect to the subdivision process. The novelty of the SCCI-hybrid method is the intense use of the implicit function theorem for controlling the steps of the method. Although the implicit function theorem has a rather local nature, it is empowered with global properties by evaluating it in an interval environment. This means that the theorem can provide global information about the curve in a subarea such as existence, non-existence, uniqueness of the curve or even the presence of singular points. The information gained allows the above-mentioned control of the subarea and the decision of its further processing, i.e. deleting it, subdividing it, switching to the continuation method or preparing the plotting of the curve in this subarea. The curves can be processed mathematically in such a manner, that the derivation of the plotted curve from the exact curve is as small as desired (modulo the screen resolution).