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 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 expedite the resultant and gcd computation. We also compared our implementation with the current reference implementation, that is, Cgal's curve analysis and arrangement for algebraic curves. For various series of challenging instances, our experiments show that the new implementation outperforms the existing one.

A worst-case bound for topology computation of algebraic curves

“…The reason for this choice was better practical performance compared to the computation and isolation of the square-free part, so we do not expect our method to be faster than AlciX in practice. Moreover, [6] and [3] have recently presented new approaches which generally outperform AlciX. It is an interesting question whether the same complexity result as in this work can be achieved for AlciX, or for one of the two most recent methods.…”

Boundary-sampled halfspaces

“…To extract the boundary of a BSH shape, our algorithm first needs to compute the arrangement of the halfspaces. While algorithms for computing the arrangement of lines and planes are well-established [Agarwal and Sharir 2000], computing geometrically and topologically accurate arrangements of curved (parametric or algebraic) geometry is significantly more challenging, and exact algorithms are only known for 2D curves [Alberti et al 2008;Berberich et al 2012;Lien et al 2014] and 3D quadrics [Dupont et al 2007;Mourrain et al 2005;Schömer and Wolpert 2006]. Since we are computing a polygonal approximation of the shape's boundary, exact arrangements are not necessary, and we instead compute the arrangement after first polygonizing the boundary surface of each halfspace.…”

On the Complexity of Computing the Topology of Real Algebraic Space Curves

“…Then the topology information {R h , B h } of C h can be easily computed. Actually, there are many papers [1,5,14,18,19,23,26,28,33] dealing with the topology computation of a plane curve. We adopt the methods presented in [26] in this paper.…”

“…Computing the topology of an algebraic curve is also a basic step to compute the topology of algebraic surfaces [10,16]. There have been many papers studied the guaranteed topology and meshing for plane algebraic curves [1,3,5,8,14,18,19,23,28,33]. But there are only a few papers which studied the guaranteed topology of space algebraic curves [2,11,12,15,22,25].…”