Computing the topology of a real algebraic plane curve whose defining equations are available only “by values”

Abstract

This paper is devoted to introducing a new approach for computing the topology of a real algebraic plane curve presented either parametrically or defined by its implicit equation when the corresponding polynomials which describe the curve are known only "by values". This approach is based on the replacement of the usual algebraic manipulation of the polynomials (and their roots) appearing in the topology determination of the given curve with the computation of numerical matrices (and their eigenvalues). Such numerical matrices arise from a typical construction in Elimination Theory known as the Bézout matrix which in our case is specified by the values of the defining polynomial equations on several sample points.

Using implicit equations of parametric curves and surfaces without computing them: Polynomial algebra by values

“…Next we describe very briefly the approach introduced in [10] to determine the topology of f (x, y) = 0, assuming also that f (x, y) = 0 is in generic position. The x-coordinates of the critical points of the curve f (x, y) = 0 (step (1)) are determined by computing the real generalized eigenvalues of the matrix pencil associated with B(x) where B(x) is the Bézout matrix of f (x, y) and f y (x, y) with respect to y (with dimension n − 1).…”

“…In [10], we give full details of the theory and of the algorithms for solving such problems. Here, we only give a brief overview of how to determine the topology of f (x, y) = 0 by values which will be used in the sections that follow.…”

“…In the case of a simple common root, the nullspace has dimension 1, and the method works according to the following results, whose proofs are presented in [10].…”

“…This leads to a linear system of k equations and k unknowns, from which the common root is obtained efficiently. For full details, see [10].…”

“…The first section introduces the basic tools for dealing with the Polynomial Algebra by Values approach: how to compute the Bézout matrix of two polynomials presented in the Lagrange Basis (i.e., by values) and how to determine the companion matrix pencil for a polynomial matrix when this matrix is also presented in the Lagrange Basis. The second section summarizes the results in [10] in order to compute the topology of the curve f (x, y) = 0 when it is presented by values. The third section shows how to solve "by values" six geometric problems for plane curves: the point position problem for parametric plane curves and their offsets, the intersection of two parametric curves, the self-intersections of a parametric curve, the intersections of a parametric curve and the offsets to another curve, and the determination of the topology of the offset to a given parametric plane curve.…”