Cylindrical algebraic decomposition I: the basic algorithm
SIAM Journal on Computing
Cylindrical algebraic decomposition II: an adjacency algorithm for the plane
SIAM Journal on Computing
An improved upper complexity bound for the topology computation of a real algebraic plane curve
Journal of Complexity - Special issue for the Foundations of Computational Mathematics conference, Rio de Janeiro, Brazil, Jan. 1997
An efficient method for analyzing the topology of plane real algebraic curves
Selected papers presented at the international IMACS symposium on Symbolic computation, new trends and developments
Factors of iterated resultants and discriminants
Journal of Symbolic Computation
Interval arithmetic in cylindrical algebraic decomposition
Journal of Symbolic Computation
Efficient topology determination of implicitly defined algebraic plane curves
Computer Aided Geometric Design
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
Polynomial real root isolation using Descarte's rule of signs
SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
Efficient isolation of polynomial's real roots
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
On the exact computation of the topology of real algebraic curves
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Almost tight recursion tree bounds for the Descartes method
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
High-performance implementations of the Descartes method
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Journal of Computational and Applied Mathematics
New bounds for the Descartes method
Journal of Symbolic Computation
EXACUS: efficient and exact algorithms for curves and surfaces
ESA'05 Proceedings of the 13th annual European conference on Algorithms
A descartes algorithm for polynomials with bit-stream coefficients
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
Computing the topology of a real algebraic plane curve whose equation is not directly available
Proceedings of the 2007 international workshop on Symbolic-numeric computation
On the complexity of real solving bivariate systems
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Exact and efficient 2D-arrangements of arbitrary algebraic curves
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Topology of real algebraic space curves
Journal of Symbolic Computation
Exact arrangements on tori and Dupin cyclides
Proceedings of the 2008 ACM symposium on Solid and physical modeling
Proceedings of the 2008 ACM symposium on Solid and physical modeling
Exact geometric-topological analysis of algebraic surfaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
Visualizing and exploring planar algebraic arrangements: a web application
Proceedings of the twenty-fourth annual symposium on Computational geometry
On the computation of the topology of a non-reduced implicit space curve
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Improving the topology computation of an arrangement of cubics
Computational Geometry: Theory and Applications
ACM Transactions on Graphics (TOG)
On the asymptotic and practical complexity of solving bivariate systems over the reals
Journal of Symbolic Computation
On the topology of planar algebraic curves
Proceedings of the twenty-fifth annual symposium on Computational geometry
Computer Aided Geometric Design
Root isolation for bivariate polynomial systems with local generic position method
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Isolating real roots of real polynomials
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
An efficient algorithm for the stratification and triangulation of an algebraic surface
Computational Geometry: Theory and Applications
On the Complexity of Reliable Root Approximation
CASC '09 Proceedings of the 11th International Workshop on Computer Algebra in Scientific Computing
Visualizing Arcs of Implicit Algebraic Curves, Exactly and Fast
ISVC '09 Proceedings of the 5th International Symposium on Advances in Visual Computing: Part I
On the different shapes arising in a family of plane rational curves depending on a parameter
Computer Aided Geometric Design
Computing the topology of an arrangement of quartics
Proceedings of the 12th IMA international conference on Mathematics of surfaces XII
Topology of 2D and 3D rational curves
Computer Aided Geometric Design
A deterministic algorithm for isolating real roots of a real polynomial
Journal of Symbolic Computation
Journal of Symbolic Computation
Efficient real root approximation
Proceedings of the 36th international symposium on Symbolic and algebraic computation
A generic algebraic kernel for non-linear geometric applications
Proceedings of the twenty-seventh annual symposium on Computational geometry
A worst-case bound for topology computation of algebraic curves
Journal of Symbolic Computation
On the isotopic meshing of an algebraic implicit surface
Journal of Symbolic Computation
Computing the shapes arising in a family of space rational curves depending on one parameter
Computer Aided Geometric Design
Arrangement computation for planar algebraic curves
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
On the shape of curves that are rational in polar coordinates
Computer Aided Geometric Design
Computer Aided Geometric Design
On exact rasterization of real algebraic plane curves
Proceedings of the 27th Spring Conference on Computer Graphics
From approximate factorization to root isolation
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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An algorithm is presented for the geometric analysis of an algebraic curve f(x, y) = 0 in the real affine plane. It computes a cylindrical algebraic decomposition (CAD) of the plane, augmented with adjacency information. The adjacency information describes the curve's topology by a topologically equivalent planar graph. The numerical data in the CAD gives an embedding of the graph. The algorithm is designed to provide the exact result for all inputs but to perform only few symbolic operations for the sake of efficiency. In particular, the roots of f(∝, y) at a critical x-coordinate . The algorithm is implemented as C++ library AlciX in the EXACUS project. Running time comparisons with top by Gonzalez-Vega and Necula (2002), and with cad2d by Brown demonstrate its efficiency.