Computer algebra: symbolic and algebraic computation (2nd ed.)
Journal of Symbolic Computation
A polynomial-time algorithm for the topological type of a real algebraic curve
Journal of Symbolic Computation
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
Modern computer algebra
Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
Efficient topology determination of implicitly defined algebraic plane curves
Computer Aided Geometric Design
Complete subdivision algorithms, I: intersection of Bezier curves
Proceedings of the twenty-second annual symposium on Computational geometry
Proceedings of the 4th international conference on Computer graphics and interactive techniques in Australasia and Southeast Asia
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
Fast and exact geometric analysis of real algebraic plane curves
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Topology of real algebraic space curves
Journal of Symbolic Computation
Exact geometric-topological analysis of algebraic surfaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
Complete subdivision algorithms, II: isotopic meshing of singular algebraic curves
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
ACM Transactions on Graphics (TOG)
Topology determination and isolation for implicit plane curves
Proceedings of the 2009 ACM symposium on Applied Computing
On the asymptotic and practical complexity of solving bivariate systems over the reals
Journal of Symbolic Computation
Proceedings of the twenty-fifth annual symposium on Computational geometry
On the topology of planar algebraic curves
Proceedings of the twenty-fifth annual symposium on Computational geometry
Lower bounds for zero-dimensional projections
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
Visualizing Arcs of Implicit Algebraic Curves, Exactly and Fast
ISVC '09 Proceedings of the 5th International Symposium on Advances in Visual Computing: Part I
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
Connectivity queries on curves in Rn
ACM Communications in Computer Algebra
Complete subdivision algorithms, II: Isotopic meshing of singular algebraic curves
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 worst-case bound for topology computation of algebraic curves
Journal of Symbolic Computation
Real solving of bivariate polynomial systems
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
Homeomorphic approximation of the intersection curve of two rational surfaces
Computer Aided Geometric Design
On the shape of curves that are rational in polar coordinates
Computer Aided Geometric Design
Non-local isotopic approximation of nonsingular surfaces
Computer-Aided Design
On exact rasterization of real algebraic plane curves
Proceedings of the 27th Spring Conference on Computer Graphics
Journal of Symbolic Computation
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We consider the problem of computing a representation of the plane graph induced by one (or more) algebraic curves in the real plane. We make no assumptions about the curves, in particular we allow arbitrary singularities and arbitrary intersection. This problem has been well studied for the case of a single curve. All proposed approaches to this problem so far require finding and counting real roots of polynomials over an algebraic extension of Q, i.e. the coefficients of those polynomials are algebraic numbers. Various algebraic approaches for this real root finding and counting problem have been developed, but they tend to be costly unless speedups via floating point approximations are introduced, which without additional checks in some cases can render the approach incorrect for some inputs.We propose a method that is always correct and that avoids finding and counting real roots of polynomials with non-rational coefficients. We achieve this using two simple geometric approaches: a triple projections method and a curve avoidance method. We have implemented our approach for the case of computing the topology of a single real algebraic curve. Even this prototypical implementation without optimizations appears to be competitive with other implementations.