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
Computer algebra: symbolic and algebraic computation (2nd ed.)
Computational geometry: an introduction
Computational geometry: an introduction
On approximations and incidence in cylindrical algebraic decompositions
SIAM Journal on Computing
Geometric approaches to nonplanar quadric surface intersection curves
ACM Transactions on Graphics (TOG)
The complexity of robot motion planning
The complexity of robot motion planning
Algebraic decomposition of regular curves
Journal of Symbolic Computation
An adjacency algorithm for cylindrical algebraic decompositions of three-dimenslonal space
Journal of Symbolic Computation
A polynomial-time algorithm for the topological type of a real algebraic curve
Journal of Symbolic Computation
Geometric and solid modeling: an introduction
Geometric and solid modeling: an introduction
Automatic parsing of degenerate quadric-surface intersections
ACM Transactions on Graphics (TOG)
Singular points of algebraic curves
Journal of Symbolic Computation
A fast planar partition algorithm, II
Journal of the ACM (JACM)
SMA '91 Proceedings of the first ACM symposium on Solid modeling foundations and CAD/CAM applications
On the planar intersection of natural quadrics
SMA '91 Proceedings of the first ACM symposium on Solid modeling foundations and CAD/CAM applications
A singly exponential stratification scheme for real semi-algebraic varieties and its applications
Theoretical Computer Science
Using multivariate resultants to find the intersection of three quadric surfaces
ACM Transactions on Graphics (TOG)
Algorithms for computer algebra
Algorithms for computer algebra
Graphical Models and Image Processing
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
Computational real algebraic geometry
Handbook of discrete and computational geometry
Modern computer algebra
A core library for robust numeric and geometric computation
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
MAPC: a library for efficient and exact manipulation of algebraic points and curves
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Efficient algorithms for line and curve segment intersection using restricted predicates
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
On Euclid's Algorithm and the Theory of Subresultants
Journal of the ACM (JACM)
Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
LEDA: a platform for combinatorial and geometric computing
LEDA: a platform for combinatorial and geometric computing
Algebraic methods and arithmetic filtering for exact predicates on circle arcs
Proceedings of the sixteenth annual symposium on Computational geometry
Robust Plane Sweep for Intersecting Segments
SIAM Journal on Computing
Reporting curve segment intersections using restricted predicates
Computational Geometry: Theory and Applications
A parametric algorithm for drawing pictures of solid objects composed of quadric surfaces
Communications of the ACM
Computing a 3-dimensional cell in an arrangement of quadrics: exactly and actually!
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
High-Level Filtering for Arrangements of Conic Arcs
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
A Computational Basis for Conic Arcs and Boolean Operations on Conic Polygons
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Computations with Algebraic Curves
ISAAC '88 Proceedings of the International Symposium ISSAC'88 on Symbolic and Algebraic Computation
A toolkit for algebra and geometry
A toolkit for algebra and geometry
Exact boundary evaluation for curved solids
Exact boundary evaluation for curved solids
Complete, exact, and efficient computations with cubic curves
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Intersecting quadrics: an efficient and exact implementation
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Towards and open curved kernel
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
On the computation of an arrangement of quadrics in 3D
Computational Geometry: Theory and Applications - Special issue on the 19th European workshop on computational geometry - EuroCG 03
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Intersecting quadrics: an efficient and exact implementation
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
An approximate arrangement algorithm for semi-algebraic curves
Proceedings of the twenty-second annual symposium on Computational geometry
Exact, efficient, and complete arrangement computation for cubic curves
Computational Geometry: Theory and Applications
Intersecting quadrics: an efficient and exact implementation
Computational Geometry: Theory and Applications
The voronoi diagram of three lines
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Near-optimal parameterization of the intersection of quadrics: I. The generic algorithm
Journal of Symbolic Computation
The visual hull of piecewise smooth objects
Computer Vision and Image Understanding
Complete subdivision algorithms, II: isotopic meshing of singular algebraic curves
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Proceedings of the twenty-fifth annual symposium on Computational geometry
Exact, efficient, and complete arrangement computation for cubic curves
Computational Geometry: Theory and Applications
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We present an approach for the exact and efficient computation of a cell in an arrangement of quadric surfaces. All calculations are based on exact rational algebraic methods and provide the correct mathematical results in all, even degenerate, cases. By projection, the spatial problem is reduced to the one of computing planar arrangements of algebraic curves. We succeed in locating all event points in these arrangements, including tangential intersections and singular points. By introducing an additional curve, which we call the Jacobi curve, we are able to find non-singular tangential intersections. We show that the coordinates of the singular points in our special projected planar arrangements are roots of quadratic polynomials. The coefficients of these polynomials are usually rational and contain at most a single square root. A prototypical implementation indicates that our approach leads to good performance in practice.