On the computation of an arrangement of quadrics in 3D

In this paper, we study a sweeping algorithm for computing the arrangement of a set of quadrics in R^3. We define a ''trapezoidal'' decomposition in the sweeping plane, and we study the evolution of this subdivision during the sweep. A key point of this algorithm is the manipulation of algebraic numbers. In this perspective, we put a large emphasis on the use of algebraic tools, needed to compute the arrangement, including Sturm sequences and Rational Univariate Representation of the roots of a multivariate polynomial system.

[1]  Arjeh M. Cohen,et al.  Some tapas of computer algebra , 1999, Algorithms and computation in mathematics.

[2]  Kurt Mehlhorn,et al.  Infimaximal Frames: A Technique for Making Lines Look Like Segments , 2003, Int. J. Comput. Geom. Appl..

[3]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[4]  Daniel Lazard,et al.  Resolution des Systemes d'Equations Algebriques , 1981, Theor. Comput. Sci..

[5]  Bernard Mourrain,et al.  Using projection operators in computer aided geometric design , 2003 .

[6]  李幼升,et al.  Ph , 1989 .

[7]  Sylvain Lazard,et al.  Intersecting quadrics: an efficient and exact implementation , 2004, SCG '04.

[8]  Leonidas J. Guibas,et al.  A Singly Exponential Stratification Scheme for Real Semi-Algebraic Varieties and its Applications , 1991, Theor. Comput. Sci..

[9]  Micha Sharir,et al.  Davenport-Schinzel sequences and their geometric applications , 1995, Handbook of Computational Geometry.

[10]  Robert M. Corless,et al.  A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots , 1997, ISSAC.

[11]  P. Zimmermann,et al.  Efficient isolation of polynomial's real roots , 2004 .

[12]  Dan Halperin,et al.  Improved construction of vertical decompositions of three-dimensional arrangements , 2002, SCG '02.

[13]  David Eugene Smith,et al.  A source book in mathematics , 1930 .

[14]  Bernard Mourrain,et al.  Computing the Isolated Roots by Matrix Methods , 1998, J. Symb. Comput..

[15]  Robert H. Anderson,et al.  A Source Book , 1995 .

[16]  Leonidas J. Guibas,et al.  A Singly-Expenential Stratification Scheme for Real Semi-Algebraic Varieties and Its Applications , 1989, ICALP.

[17]  Joseph O'Rourke,et al.  Handbook of Discrete and Computational Geometry, Second Edition , 1997 .

[18]  Fabrice Rouillier,et al.  Bernstein's basis and real root isolation , 2004 .

[19]  D. S. Arnon,et al.  Algorithms in real algebraic geometry , 1988 .

[20]  Michael T. Goodrich,et al.  Dynamic trees and dynamic point location , 1991, STOC '91.

[21]  Philippe Trébuchet.  Vers une résolution stable et rapide des équations algébriques , 2002 .

[22]  Sylvain Lazard,et al.  Near-Optimal Parameterization of the Intersection of Quadrics : III . Parameterizing Singular Intersections , 2005 .

[23]  Fabrice Rouillier,et al.  Symbolic Recipes for Polynomial System Solving , 1999 .

[24]  Fabrice Rouillier,et al.  Solving Zero-Dimensional Systems Through the Rational Univariate Representation , 1999, Applicable Algebra in Engineering, Communication and Computing.

[25]  Jeremy R. Johnson,et al.  Polynomial real root isolation using approximate arithmetic , 1997, ISSAC.

[26]  Michael N. Vrahatis,et al.  On the Complexity of Isolating Real Roots and Computing with Certainty the Topological Degree , 2002, J. Complex..

[27]  S. Demri,et al.  Notes de cours , 2003 .

[28]  Elmar Schömer,et al.  Complete, exact, and efficient computations with cubic curves , 2004, SCG '04.

[29]  Elmar Schömer,et al.  Computing a 3-dimensional cell in an arrangement of quadrics: exactly and actually! , 2001, SCG '01.

[30]  George E. Collins,et al.  Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975, Automata Theory and Formal Languages.

[31]  F. S. Macaulay.  On the resolution of a given modular system into primary systems including some properties of Hilbert numbers , 1913 .

[32]  H. Stetter,et al.  An Elimination Algorithm for the Computation of All Zeros of a System of Multivariate Polynomial Equations , 1988 .