On the exact computation of the topology of real algebraic curves

Abstract

We consider the problem of computing a representation of the plane graph induced by one (or more) algebraic curves in the real plane. We make no assumptions about the curves, in particular we allow arbitrary singularities and arbitrary intersection. This problem has been well studied for the case of a single curve. All proposed approaches to this problem so far require finding and counting real roots of polynomials over an algebraic extension of Q, i.e. the coefficients of those polynomials are algebraic numbers. Various algebraic approaches for this real root finding and counting problem have been developed, but they tend to be costly unless speedups via floating point approximations are introduced, which without additional checks in some cases can render the approach incorrect for some inputs.We propose a method that is always correct and that avoids finding and counting real roots of polynomials with non-rational coefficients. We achieve this using two simple geometric approaches: a triple projections method and a curve avoidance method. We have implemented our approach for the case of computing the topology of a single real algebraic curve. Even this prototypical implementation without optimizations appears to be competitive with other implementations.