Arrangement computation for planar algebraic curves

Abstract

We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition of the plane. Compared to previous approaches, we improve in two main aspects: Firstly, we significantly limit the types of involved exact operations, that is, our algorithms only use resultant and gcd computations as purely symbolic operations. Secondly, we introduce a new hybrid method in the lifting step of our algorithm which combines the use of a certified numerical complex root solver and information derived from the resultant computation. Additionally, we never consider any coordinate transformation and the output is also given with respect to the initial coordinate system.