[Paper Review] Arrangement Computation for Planar Algebraic Curves

Eric Berberich, Pavel Emeliyanenko+2 more|arXiv (Cornell University)|Mar 24, 2011

Advanced Numerical Analysis Techniques26 references4 citations

TL;DR

This paper presents a certified and complete algorithm for computing arrangements of planar algebraic curves using only resultant and GCD as symbolic operations, combined with a hybrid symbolic-numeric lifting step via a certified complex root solver. The method achieves up to 120× speedup over the prior state-of-the-art (CGAL's Ak_2) on hard instances, especially for high-degree or singular curves, by leveraging GPU-accelerated arithmetic and avoiding coordinate shearing.

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.

Motivation & Objective

• To develop a certified and complete algorithm for computing arrangements of real planar algebraic curves without coordinate transformations.
• To reduce reliance on expensive symbolic operations such as subresultants and Gröbner bases in curve topology computation.
• To improve efficiency for high-degree and singular curves where previous methods suffer from computational bottlenecks.
• To enable practical performance gains through GPU acceleration of resultant and GCD computations.
• To maintain geometric accuracy and completeness while minimizing symbolic computation and avoiding shearing transformations.

Proposed method

• The algorithm computes the arrangement of algebraic curves using a cylindrical algebraic decomposition of the plane, analyzing single and pairwise curves via geometric-topological decomposition.
• It uses only resultant and GCD as symbolic operations, offloading these to graphics hardware for acceleration.
• A hybrid lifting step combines a certified numerical complex root solver with theoretical bounds from Teissier on the number of complex roots to isolate real roots efficiently.
• A symbolic-numeric filter called FastLift is used as a primary root isolation method, failing only in rare cases.
• For cases where FastLift fails, a fallback method based on [4] is used to ensure completeness.
• The entire computation is performed in the original coordinate system, preserving input sparsity and avoiding coefficient blowup from shearing.

Experimental results

Research questions

• RQ1 Can the computational cost of symbolic operations in algebraic curve arrangement computation be significantly reduced while maintaining correctness and completeness?
• RQ2 How effective is a hybrid symbolic-numeric approach for root isolation in the lifting step of curve topology computation?
• RQ3 To what extent can GPU acceleration improve performance in symbolic computation tasks like resultants and GCDs for algebraic curves?
• RQ4 How does the absence of coordinate shearing affect performance and numerical stability in arrangements of singular or non-generic curves?
• RQ5 Can a certified and complete algorithm outperform the current reference implementation (CGAL’s Ak_2) on hard benchmark instances, especially for high-degree or projected intersection curves?

Key findings

• The new implementation, FastKernel, is on average twice as fast as GPU-accelerated Ak_2 for easy instances such as non-singular curves in generic position.
• For projected intersection curves of surfaces and special curves with singularities, the speedup ranges from 5× to 120×, with improvements increasing with curve degree.
• On the most challenging test case (11 curves of type X), FastKernel completed in 2696.15 seconds, while Ak_2 timed out after 4000 seconds.
• The symbolic-numeric filter FastLift succeeded for all fibers in most cases, and only failed in rare instances where the fallback Lift method was invoked.
• The performance gain is most pronounced for high-degree curves, where Ak_2’s subresultant computation becomes the dominant bottleneck.
• The method avoids shearing, preserving coefficient sparsity and preventing the coefficient bit-size blowup that degrades performance in previous approaches.