Automatic parameterization of rational curves and surfaces 1: conics and conicoids
Computer-Aided Design
Geometric approaches to nonplanar quadric surface intersection curves
ACM Transactions on Graphics (TOG)
Natural quadrics: projections and intersections
IBM Journal of Research and Development
A parametric algorithm for drawing pictures of solid objects composed of quadric surfaces
Communications of the ACM
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
Using multivariate resultants to find the intersection of three quadric surfaces
ACM Transactions on Graphics (TOG)
On the lower degree intersections of two natural quadrics
ACM Transactions on Graphics (TOG)
Computing a 3-dimensional cell in an arrangement of quadrics: exactly and actually!
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
AfriGraph '01 1st International Conference on Virtual Reality, Computer Graphics and Visualization in Southern Africa ( formerly known as SAGA 2001 )
Near-optimal parameterization of the intersection of quadrics
Proceedings of the nineteenth annual symposium on Computational geometry
Enhancing Levin's method for computing quadric-surface intersections
Computer Aided Geometric Design
Intersecting quadrics: an efficient and exact implementation
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Journal of Computer Science and Technology - Special issue on computer graphics and computer-aided design
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
Handling degeneracies in exact boundary evaluation
SM '04 Proceedings of the ninth ACM symposium on Solid modeling and applications
Near-optimal parameterization of the intersection of quadrics: I. The generic algorithm
Journal of Symbolic Computation
Using signature sequences to classify intersection curves of two quadrics
Computer Aided Geometric Design
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
Multisensory learning cues using analytical collision detection between a needle and a tube
HAPTICS'04 Proceedings of the 12th international conference on Haptic interfaces for virtual environment and teleoperator systems
Circles in torus-torus intersections
Journal of Computational and Applied Mathematics
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In general, two quadric surfaces intersect in a nonsingular quartic space curve. Under special circumstances, however, this intersection may “degenerate” into a quartic with a double point, or a composite of lines, conics, and twisted cubics whose degrees, counted over the complex projective domain, sum to four. Such degenerate forms are important since they occur with surprising frequency in practice and, unlike the generic case, they admit rational parameterizations. Invoking concepts from classical algebraic geometry, we formulate the condition for a degenerate intersection in terms of the vanishing of a polynomial expression in the quadric coefficients. When this is satisfied, we apply a multivariate polynomial factorization algorithm to the projecting cone of the intersection curve. Factors of this cone which correspond to intersection components “at infinity” may be removed a priori. A careful examination of the remaining cone factors then facilitates the identification and parameterization of the various real, affine intersection elements that may arise: isolated points, lines, conics, cubics, and singular quartics. The procedure is essentially automatic (avoiding the tedium of case-by-case analyses), encompasses the full range of quadric forms, and is amenable to implementation in exact (symbolic) arithmetic.