## Geometric Modeling and Algebraic Geometry (Google eBook)The two fields of Geometric Modeling and Algebraic Geometry, though closely related, are traditionally represented by two almost disjoint scientific communities. Both fields deal with objects defined by algebraic equations, but the objects are studied in different ways. This contributed book presents, in 12 chapters written by leading experts, recent results which rely on the interaction of both fields. Some of these results have been obtained in the frame of the European GAIA II project (IST 2001-35512) entitled “Intersection algorithms for geometry-based IT applications using approximate algebraic methods”. |

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### Contents

The GAIA Project on Intersection and Implicitization | 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 | |

### Common terms and phrases

afﬁne Aided Geometric Design algebraic curves algebraic geometry algebraic surfaces approximate implicitization Bernstein basis bidegree 1,2 biquadratic canal surfaces certiﬁed circulant matrices classiﬁcation coefﬁcients complex components Computer Aided Geometric conic coordinates corresponding covariant critical points curves and surfaces decomposition deﬁned deﬁnition denote divisor Dokken domain efﬁcient eigenvalues eigenvectors example ﬁber ﬁnd ﬁnite ﬁrst function GAIA GAIA II Geometric Modeling hypersurface implicit curves implicit equation implicit representation intersection algorithms intersection curve J¨uttler Lemma line arrangements linear Math Mathematics matrix monoid monoid surfaces Mourrain multiplicity Newton polygon nodes orbits parameter line parametric surfaces planar plane 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 surfaces of revolution Symbolic Comput tangent cone theorem topology toric umbilics vector zero