Approximate Reconstructions of Perturbed Rational Planar Cubics

Abstract

This paper is devoted to a problem from geometric modelling and related applications when exact symbolic computations are sometimes used also on objects given inexactly, i.e., when it is not adequately respected that numerical or input errors may significantly influence fundamental properties of considered algebraic varieties, including e.g. their rationality. We formulate a simple algorithm for an approximation of a non-rational planar cubic which is assumed to be a perturbation of some unknown rational planar cubic. The input curve is given by a perturbed polynomial or by perturbed points sampled from the original curve. The algorithm consists of two main parts. First, we suggest geometric methods for the estimation of a singular point of the original curve. Then we select from the six-dimensional subspace of all rational cubics with a given singular point a suitable one that may also satisfy some further criteria. The designed method is presented on several commented examples.