## QI: Quadric Intersection |

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## QI's GForge |

(Model courtesy of SGDL Systems)

- Near-Optimal Parameterization of the Intersection of
Quadrics. Proc.
*19th ACM Symp. on Comput. Geom.*, pp. 246--255, 2003. - Near-Optimal
Parameterization of the Intersection of Quadrics: I. The Generic Algorithm,
*Journal of Symbolic Computation*, 43(3):168--191 2008. (Research report.) -
Near-Optimal Parameterization of
the Intersection of Quadrics: II. A Classification of Pencils,
*Journal of Symbolic Computation*43(3):192--215, 2008. (Research report.) - Near-Optimal
Parameterization of
the Intersection of Quadrics: III. Parameterizing Singular Intersections,
*Journal of Symbolic Computation*43(3):216--232, 2008. (Research report.) - Intersecting
Quadrics: An Efficient and Exact Implementation,
*Computational Geometry: Theory and Applications*, 35(1-2):74--99, 2006 (special issue of SoCG'04). (Research report.)

QI has the following features:

- it computes an exact parameterization of the intersection of two quadrics with integer coefficients of arbitrary size;
- it correctly identifies, separates and parameterizes all the connected components of the intersection and gives all the relevant topological information;
- the parameterization is rational when one exists; otherwise the intersection is a smooth quartic and the parameterization involves the square root of a polynomial;
- the parameterization is either optimal in the degree of the extension of the integers on which its coefficients are defined or, in a small number of well-identified cases, involves one extra possibly unnecessary square root;
- it is fast and efficient and can routinely compute parameterizations of the intersection of quadrics with input coefficients having ten digits in less than 50 milliseconds on a mainstream PC.

- Interactions between potential energy surfaces in photochemistry (Department of Chemistry, Imperial College, London).
- Intersections in a thermal tool for spacecraft thermal radiation analysis.
- Image of conics seen by a catadioptric camera with a paraboloidal mirror (Institut de Recherche en Informatique de Toulouse).

Additionally, the implementation used by QI of Uspensky's algorithm for isolating the real roots of a univariate polynomial (written by members of the Spaces and Vegas teams) is released under the terms of the GNU Lesser General Public License. You can get it here (with the accompanying template Makefile).