Arrangement Computation for Planar Algebraic Curves

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 reduce the amount of exact operations, that is, our algorithms only uses resultant and gcd as purely symbolic operations. Secondly, we introduce a new hybrid method in the lifting step of our algorithm which combines the usage 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. We implemented our algorithm as a prototypical package of the C++-library Cgal. Our implementation exploits graphics hardware to e...

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Eric Berberich, Pavel Emeliyanenko, Alexander Kobel, Michael Sagraloff
• Keywords: Algebraic Curves, Algorithm, CORR 2011, Education, Planar Algebraic Curves