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Paramétrage quasi-optimal de l'intersection de deux quadriques : théorie, algorithmes et implantation

Laurent Dupont 1
1 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : This thesis presents a robust and efficient algorithm for the
computation of an exact parameterized form of the intersection curve
of two quadric surfaces defined by implicit equations with rational
coefficients.
For the first time, the parameterization that we obtain contains all
the topological informations of the curve and is simple enough to be
exploited in non-trivial geometric applications.

Many new improvements, in different domains, were necessary to
reach this result. We did a complete study of all possible cases of
intersection, first in $\Pp^3(\C)$ based on the work of Segre, then in
$\Pp^3(\R)$ by exploiting the results of Uhlig on the simultaneous
reduction of two real quadratic forms. This systematic study allowed
us to have a fine understanding the geometry of the intersection of two quadric
surfaces. We are now able to determine all characteristics of the
intersection curve, that is its genus, its singular points, its
algebraic components and connected components, and all the links
between these components. When one exists, we find a rational
parameterization of the components of the curve. In this sense, our
algorithm is optimal. We also made some significant improvements on
the complexity of the radical expression of the coefficients of the
obtained parameterization. Our algorithm is near-optimal in the sense
that the coefficients of the parameterization contain at most one
unnecessary square root in their expression. Our algorithm is optimal in the
worst case, in the sense that for every type of intersection curve
(for example a regular quartic, or a cubic and line, or two conics),
there exist pairs of quadrics for which the number of square
roots in the expression of the coefficients is minimal.

Finally, we made a complete implementation of our algorithm in MuPAD
which allowed us to get previously unheard of performances, in
terms of running time and in terms of the simplicity of the
result.
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Laurent Dupont. Paramétrage quasi-optimal de l'intersection de deux quadriques : théorie, algorithmes et implantation. Génie logiciel [cs.SE]. Université Nancy II, 2004. Français. ⟨tel-00103446⟩

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