Exact Symbolic-Numeric Computation of Planar Algebraic Curves
Eric Berberich, Pavel Emeliyanenko, Alexander Kobel, Michael Sagraloff

TL;DR
This paper introduces a certified, complete algorithm for computing arrangements of real planar algebraic curves, improving efficiency and robustness by avoiding coordinate changes and limiting exact operations, with implementation in CGAL.
Contribution
The paper presents a novel algorithm combining symbolic and numerical techniques for algebraic curve arrangements, avoiding coordinate changes and enhancing performance over existing methods.
Findings
Algorithm outperforms existing implementations on challenging instances.
No coordinate change needed, increasing robustness.
Utilizes graphics hardware to accelerate symbolic computations.
Abstract
We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It 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. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional bivariate system, and an algorithm denoted GeoTop to analyze a single algebraic curve. Compared to existing approaches based on elimination techniques, we considerably improve the corresponding lifting steps in both subroutines. As a result, generic position of the input system is never assumed, and thus our algorithm never demands for any change of coordinates. In addition, we significantly limit the types of involved exact…
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Taxonomy
TopicsPolynomial and algebraic computation · Computational Geometry and Mesh Generation · Advanced Numerical Analysis Techniques
