Caravantes et al. / J Zhejiang Univ Sci A 2008 9(12):1685-1693 1685 Computing the topology of an arrangement of implicitly defined real algebraic plane curves §* Jorge CARAVANTES, Laureano GONZALEZ-VEGA (Department of Mathematics, Statistics and Computation, University of Cantabria, Santander, 39005, Cantabria, Spain) E-mail: jorge.caravantes@unican.es; laureano.gonzalez@unican.es Received July 16, 2008; revision accepted Aug. 20, 2008; CrossCheck deposited Oct. 27, 2008 Abstract: We introduce a new algebraic approach dealing with the problem of computing the topology of an arrangement of a finite set of real algebraic plane curves presented implicitly. The main achievement of the presented method is a complete avoidance of irrational numbers that appear when using the sweeping method in the classical way for solving the problem at hand. Therefore, it is worth mentioning that the efficiency of the proposed method is only assured for low-degree curves. Key words: Topology computation, Real plane curves, Sweeping method doi:10.1631/jzus.A08GMP01 Document code: A CLC number: TP391.72 INTRODUCTION In the study of the topology of arrangements of plane curves presented implicitly, efficient algorithms are known, while to a high degree they require sym- bolic methods. The case of a single planar curve has been studied in (Sakkalis and Farouki, 1990; Sakkalis, 1991; Hong, 1996; Gonzalez-Vega and El Kahoui, 1996; Gonzalez-Vega and Necula, 2002; Seidel and Wolpert, 2005; Eigenwillig et al., 2007; Liang et al., 2007; Cheng et al., 2008). Efficient algorithms for arrangements of straight segments can be found in (Mehlhorn and Noher, 1999; Flato et al., 2000; Seel, 2001) and for conics in (Berberich et al., 2002; Wein, 2002). This paper is inspired by the ideas in (Gon- zalez-Vega and Necula, 2002; Eigenwillig et al., 2006), and will introduce several tools allowing one to perform an exact topological analysis for ar- rangements of curves of a general degree presented implicitly. The method in (Eigenwillig et al., 2006) for n cubics f 1 , f 2 , ..., f n finds the topology of each curve f i , then computes the topology of each pair f i , f j and, finally, puts all the information together provid- ing the topology of the considered arrangement. Since the last step before mentioned does not depend on the degree of the considered curves, for completeness it will just be sketched at the end of Section 3. Here we give a general method which is valid for every degree. Unfortunately, it needs an efficient implementation of Descartes’ rule of signs to be effi- cient for a degree higher than 5. To analyze a single curve, we consider the real roots x 1 <x 2 <...<x n (that we do not determine) of the discriminant of f i with respect to y. It is well known that f i =0 in the region (x j , x j+1 )×ú consists of (topo- logically) a finite number of disjoint segments. The number of segments is exactly the number of real roots of f i (r j+1 , y) (at most 4 due to degree limitations) for any r j+1 ∈(x j , x j+1 ). Now we have to find out what happens over the x j . Since x j is a root of the dis- criminant of f, we know (or in other cases we will be warned beforehand) that f i (x i , y) has one multiple real root (representing what we call an event point of the curve) and up to two single real roots (representing what we will call uninvolved arcs). We need to sort Journal of Zhejiang University SCIENCE A ISSN 1673-565X (Print); ISSN 1862-1775 (Online) www.zju.edu.cn/jzus; www.springerlink.com E-mail: jzus@zju.edu.cn § Presented as a Keynote at Geometric Modeling and Processing 2008 (GMP 2008), April 23-25, 2008, Hangzhou, China * Project (No. MTM2005-08690-C02-02) partially supported by the Spanish Ministry of Science and Innovation Grant