Polygonal Approximations for Curved Problems: An Application to Arrangements

Abstract

The most of the authors who proposed algorithms dealing with curved objects used a set of oracles allowing to perform basic geometric operations on curves (as computing the intersection of two curves). If the handled curves are algebraic, the oracles involve algebraic equations resolution, and in a geometric computing framework no method of solving algebraic equations is considered available. In this paper, we address the problem of the incidence graph of an arrangement of curves and we propose a method that completely avoids algebraic equations, all the computations to be done concerning linear objects. This will be done via suitable polygonal approximations of the given curves; we start by presenting a “polygonal” method in a case where the required polygonal approximations exist by definition, the case of composite Bézier curves, and then we show how we can construct these polygonal approximations in the general case of Jordan arcs.