This paper presents a new algorithm for solving a system of polynomials, in a domain of n. It can be seen as an improvement of the Interval Projected Polyhedron algorithm proposed by Sherbrooke and Patrikalakis [SP93]. It uses a powerful reduction strategy based on univariate root finder using Bernstein basis representation and Descarte's rule. We analyse the behavior of the method, from a theoretical point of view, shows that for simple roots, it has a local quadratic convergence speed and gives new bounds for the complexity of approximating real roots in a box of n. The improvement of our approach, compared with classical subdivision methods, is illustrated on geometric modeling applications such as computing intersection points of implicit curves, self-intersection points of rational curves, and on the classical parallel robot benchmark problem.

Abstract. In this paper, we describe a subdivision method for handling algebraic implicit curves in 2d and 3d. We use the representation of polynomials in the Bernstein basis associated with a given box, to check if the topology of the curve is determined inside this box, from its points on the border of the box. Subdivision solvers are used for computing these points on the faces of the box, and segments joining these points are deduced to get a graph isotopic to the curve. Using envelop of polynomials, we show how this method allow to handle efficiently and accurately implicit curves with large coefficients. We report on implementation aspects and experimentations on 2d curves such as ridge curves or self intersection curves of parameterized surfaces, and on silhouette curves of implicit surfaces, showing the interesting practical behavior of this approach.

We present an overview of the open source library synaps. We describe some of the representative algorithms of the library and illustrate them on some explicit computations, such as solving polynomials and computing geometric information on implicit curves and surfaces. Moreover, we describe the design and the techniques we have developed in order to handle a hierarchy of generic and specialized data-structures and routines, based on a view mechanism. This allows us to construct dedicated plugins, which can be loaded easily in an external tool. Finally, we show how this design allows us to embed the algebraic operations, as a dedicated plugin, into the external geometric modeler axel.