Near-Optimal Parameterization of the Intersection of Quadrics: II. A Classification of Pencils

Laurent Dupont 1 Daniel Lazard 2 Sylvain Lazard 1 Sylvain Petitjean 1
1 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
2 SALSA - Solvers for Algebraic Systems and Applications
LIP6 - Laboratoire d'Informatique de Paris 6, Inria Paris-Rocquencourt
Abstract : We present here the first classification of pencils of quadrics based on the type of their intersection in real projective space and we show how this classification can be used to compute efficiently the type of the real intersection. This classification is at the core of the design of the algorithms, presented in Part~III, for computing, in all cases of singular intersection, a near-optimal parameterization with polynomial functions, that is a parameterization in projective space whose coordinates functions are polynomial and such that the number of distinct square roots appearing in the coefficients is at most one away from the minimum.
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Laurent Dupont, Daniel Lazard, Sylvain Lazard, Sylvain Petitjean. Near-Optimal Parameterization of the Intersection of Quadrics: II. A Classification of Pencils. Journal of Symbolic Computation, Elsevier, 2008, 43 (3), pp.192--215. ⟨10.1016/j.jsc.2007.10.012⟩. ⟨inria-00186090⟩

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