Computing the topology of an arrangement of parametric or implicit algebraic curves in the Lagrange basis

Abstract

The study of arrangements of geometric objects is a problem of great interest in combinatorial and computational geometry, and in CAGD, with applications in computer-aided manufacturing, geometric modeling, robotics, NC-machining, molecular modeling, etc. [2, 3] We present a new algebraic method for computing the topology of the arrangement of a finite family of planar algebraic curves. The curves are given either by their implicit equations or by polynomial parametrizations. As far as we know, none of the previous works on this problem include the treatment of parametrized curves. The input of the method is assumed to be given as Lagrange data, i.e., by the values of the polynomials defining the curves, at a suitable set of nodes. The output is a graph representing critical points of the curves, all the intersection points between curves, and segments joining some of these points isotopic to the arcs of the curves. The method follows a sweep-line algorithm. To find critical points, intersections and self-intersections we use Bezout matrices. We take advantage of the existence of a basis of the nullspace of a Bezout matrix that has a nice and useful structure (see [4]). We use algebra by values techniques, which means that all computations (Bezout matrices, polynomial derivatives, etc.) are done in the Lagrange basis [5, 1]. This approach reduces the computational cost of manipulating high degree polynomials, and avoids conversion between different basis, which is known to be a numerically unstable procedure. To compute the roots of the determinant of a Bezout matrix we apply linearization, i.e., we use a straightforward procedure to build (in the Lagrange basis) a companion matrix pencil whose generalized eigenvalues are the roots of the aforementioned determinant. The algorithm has been tested using Maple 17, showing that it is accurate and quite efficient.