Algorithms for Rational Real Algebraic Curves

Abstract

In this paper, we study fundamental properties of real curves, especially of rational real curves, and we derive several algorithms to decide the reality and rationality of curves in the complex plane. Furthermore, if the curve is real and rational, we determine a real parametrization. More precisely, we present a reality test algorithm for plane curves, and three different types of real parametrization algorithms that we call: direct parametrization algorithms (they compute a rational real parametrization, if it exists), algebraically optimal parametrization algorithms (they compute a rational real parametrization over the smallest possible real field extension, if the curve is rational and real), and hybrid parametrization algorithms (they combine parametrization and reparametrization techniques to derive algebraically optimal rational real parametrizations).