Computer algebra: symbolic and algebraic computation (2nd ed.)
Geometric approaches to nonplanar quadric surface intersection curves
ACM Transactions on Graphics (TOG)
Geometric and solid modeling: an introduction
Geometric and solid modeling: an introduction
Automatic parsing of degenerate quadric-surface intersections
ACM Transactions on Graphics (TOG)
Using multivariate resultants to find the intersection of three quadric surfaces
ACM Transactions on Graphics (TOG)
An algorithm for constructing the convex hull of a set of spheres in dimension d
Computational Geometry: Theory and Applications
Handbook of discrete and 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
On Euclid's Algorithm and the Theory of Subresultants
Journal of the ACM (JACM)
LEDA: a platform for combinatorial and geometric computing
LEDA: a platform for combinatorial and geometric computing
A parametric algorithm for drawing pictures of solid objects composed of quadric surfaces
Communications of the ACM
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
A toolkit for algebra and geometry
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SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
ESOLID---A System for Exact Boundary Evaluation
Proceedings of the seventh ACM symposium on Solid modeling and applications
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
Near-optimal parameterization of the intersection of quadrics
Proceedings of the nineteenth annual symposium on Computational geometry
Complete, exact, and efficient computations with cubic curves
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
The predicates of the Apollonius diagram: algorithmic analysis and implementation
Computational Geometry: Theory and Applications - Special issue on robust geometric algorithms and their implementations
An exact and efficient approach for computing a cell in an arrangement of quadrics
Computational Geometry: Theory and Applications - Special issue on robust geometric algorithms and their implementations
Complete subdivision algorithms, I: intersection of Bezier curves
Proceedings of the twenty-second annual symposium on Computational geometry
An approximate arrangement algorithm for semi-algebraic curves
Proceedings of the twenty-second annual symposium on Computational geometry
Intersecting quadrics: an efficient and exact implementation
Computational Geometry: Theory and Applications
A robust algorithm for finding the real intersections of three quadric surfaces
Computer Aided Geometric Design
Handling degeneracies in exact boundary evaluation
SM '04 Proceedings of the ninth ACM symposium on Solid modeling and applications
Computational Geometry: Theory and Applications
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
The predicates of the Apollonius diagram: Algorithmic analysis and implementation
Computational Geometry: Theory and Applications - Special issue on robust geometric algorithms and their implementations
An exact and efficient approach for computing a cell in an arrangement of quadrics
Computational Geometry: Theory and Applications - Special issue on robust geometric algorithms and their implementations
Intersecting quadrics: an efficient and exact implementation
Computational Geometry: Theory and Applications
A robust algorithm for finding the real intersections of three quadric surfaces
Computer Aided Geometric Design
Determining the topology of real algebraic surfaces
IMA'05 Proceedings of the 11th IMA international conference on Mathematics of Surfaces
Real solving of bivariate polynomial systems
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
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We present two approaches to the problem of calculating a cell in a 3- dimensional arrangement of quadrics. The first approach solves the problem using rational arithmetic. It works with reductions to planar arrangements of algebraic curves. Degenerate situations such as tangential intersections and self-intersections of curves are intrinsic to the planar arrangements we obtain. The coordinates of the intersection points are given by the roots of univariate polynomials. We succeed in locating all intersection points either by extended local box hit counting arguments or by globally characterizing them with simple square root expressions. The latter is realized by a clever factorization of the univariate polynomials. Only the combination of these two results facilitates a practical and implementable algorithm.The second approach operates directly in 3-space by applying classical solid modeling techniques. Whereas the first approach guarantees a correct solution in every case the second one may fail in some degenerate situations. But with the help of verified floating point arithmetic it can detect these critical cases and is faster if the quadrics are in general position.