Geometric Modeling and Algebraic Geometry (Google eBook)

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Bert Jüttler, Ragni Piene
Springer Science & Business Media, Dec 24, 2007 - Mathematics - 231 pages
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The 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

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
Index
228
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