Geometric Modeling and Algebraic GeometryBert Jüttler, Ragni Piene 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) . |
Contents
| 5 | |
Some Covariants Related to Steiner Surfaces | 30 |
F Aries E Briand C Bruchou 31 | 47 |
Canal Surfaces Defined by Quadratic Families of Spheres | 79 |
General Classification of 12 Parametric Surfaces in | 93 |
Curve Parametrization over Optimal Field Extensions | 118 |
T Beck J Schicho 119 | 141 |
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Aided algebraic algebraic geometry algorithm applications approach approximate implicitization associated assume basis bound branches called classification coefficients complex components compute condition conic connected consider construction contains coordinates corresponding covariant critical points curve curves and surfaces decomposition defined definition degree denote described Design determined direction divisor domain double edges eigenvectors equation equivalent exact example exist faces field four function GAIA geometric give given Hence hypersurface interest intersection intersection curve intervals invariant linear Mathematics matrix method Modeling monoid multiplicity normal Note obtained parametric patches plane polygon polynomial positive possible present problem properties quadratic quartic rational regular representation represented respect ridges roots self-intersection shape simple singular point space step subdivision surfaces tangent techniques tion topology umbilics values vector vertices zero
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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.