## Geometric Modeling and Algebraic GeometryThe two ?elds of Geometric Modeling and Algebraic Geometry, though closely - lated, are traditionally represented by two almost disjoint scienti?c communities. Both ?elds deal with objects de?ned by algebraic equations, but the objects are studied in different ways. While algebraic geometry has developed impressive - sults for understanding the theoretical nature of these objects, geometric modeling focuses on practical applications of virtual shapes de?ned by algebraic equations. Recently, however, interaction between the two ?elds has stimulated new research. For instance, algorithms for solving intersection problems have bene?ted from c- tributions from the algebraic side. The workshop series on Algebraic Geometry and Geometric Modeling (Vilnius 1 2 2002 , Nice 2004 ) and on Computational Methods for Algebraic Spline Surfaces 3 (Kefermarkt 2003 , Oslo 2005) have provided a forum for the interaction between the two ?elds. The present volume presents revised papers which have grown out of the 2005 Oslo workshop, which was aligned with the ?nal review of the European project GAIA II, entitled Intersection algorithms for geometry based IT-applications 4 using approximate algebraic methods (IST 2001-35512) . |

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### Contents

5 | |

Some Covariants Related to Steiner Surfaces | 30 |

Real Line Arrangements and Surfaces with Many Real Nodes | 47 |

Monoid Hypersurfaces | 55 |

Canal Surfaces Deﬁned by Quadratic Families of Spheres | 79 |

General Classification of 12 Parametric Surfaces in ℙ³ | 93 |

Curve Parametrization over Optimal Field Extensions Exploiting the Newton Polygon | 118 |

Ridges and Umbilics of Polynomial Parametric Surfaces | 141 |

Intersecting Biquadratic Bézier Surface Patches | 161 |

Cube Decompositions by Eigenvectors of Quadratic Multivariate Splines | 181 |

Subdivision Methods for the Topology of 2d and 3d Implicit Curves | 198 |

Approximate Implicitization of Space Curves and of Surfaces of Revolution | 215 |

228 | |

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Aided Geometric Design algebraic curves algebraic geometry algebraic surfaces approximate implicitization Bernstein basis bidegree 1,2 biquadratic canal surfaces circulant matrices classification coefficients complex components Computer Aided Geometric conic coordinates corresponding covariant critical points curves and surfaces decomposition defined denote divisor Dokken domain eigenvalues eigenvectors example fiber function G-circulant GAIA GAIA project Geometric Modeling hypersurface implicit curves implicit equation implicit representation intersection algorithms intersection curve J¨uttler Lemma line arrangements linear locus Math Mathematics matrix monoid monoid surfaces Mourrain multiplicity Newton polygon nodes obtained orbits parameter line parametric surfaces planar preimage problem profile curve Quadric quartic monoid quartic surfaces rational parametric ridge curve roots ruled surface self-intersection Sendra singular point space curve Steiner surface study points subdivision surface of degree surface patches surfaces of revolution Symbolic Comput tangent cone theorem topology toric umbilics vector vertices zero

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Page 11 - Combining the parametric and implicit representation the intersection is described by q(p(t)) = 0, a linear equation in the variable t. Using exact arithmetic it is easy to classify the solution as: • An empty set, if the lines are parallel. • The whole line, if the lines coincide. • One point, if lines are non-parallel. Next we look at the intersection of two rational parametric curves of degree n and d, respectively. From algebraic geometry...

Page 8 - CAD-vendors are conservative, and new technology has to be backward compliant. Improved intersection algorithms have thus to be compliant with STEP representation of geometry and the traditional approach to CAD coming from the late 1980s. For research within CAD-type intersection algorithms to be of interest to producing industries and CAD-vendors backward compatibility and the legacy of existing CAD-models have not to be forgotten.