Ebook: Geometric Modeling and Algebraic Geometry
- Tags: Mathematical Modeling and Industrial Mathematics, Algebraic Geometry, Computer Graphics, Visualization, Computational Intelligence
- Year: 2008
- Publisher: Springer-Verlag Berlin Heidelberg
- Edition: 1
- Language: English
- pdf
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) .
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".
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".
Content:
Front Matter....Pages I-VIII
Front Matter....Pages 2-3
The GAIA Project on Intersection and Implicitization....Pages 5-25
Front Matter....Pages 28-29
Some Covariants Related to Steiner Surfaces....Pages 31-46
Real Line Arrangements and Surfaces with Many Real Nodes....Pages 47-54
Monoid Hypersurfaces....Pages 55-77
Canal Surfaces Defined by Quadratic Families of Spheres....Pages 79-92
General Classification of (1,2) Parametric Surfaces in ?3 ....Pages 93-113
Front Matter....Pages 116-117
Curve Parametrization over Optimal Field Extensions Exploiting the Newton Polygon....Pages 119-140
Ridges and Umbilics of Polynomial Parametric Surfaces....Pages 141-159
Intersecting Biquadratic B?zier Surface Patches....Pages 161-180
Cube Decompositions by Eigenvectors of Quadratic Multivariate Splines....Pages 181-197
Subdivision Methods for the Topology of 2d and 3d Implicit Curves....Pages 199-214
Approximate Implicitization of Space Curves and of Surfaces of Revolution....Pages 215-227
Back Matter....Pages 229-231
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".
Content:
Front Matter....Pages I-VIII
Front Matter....Pages 2-3
The GAIA Project on Intersection and Implicitization....Pages 5-25
Front Matter....Pages 28-29
Some Covariants Related to Steiner Surfaces....Pages 31-46
Real Line Arrangements and Surfaces with Many Real Nodes....Pages 47-54
Monoid Hypersurfaces....Pages 55-77
Canal Surfaces Defined by Quadratic Families of Spheres....Pages 79-92
General Classification of (1,2) Parametric Surfaces in ?3 ....Pages 93-113
Front Matter....Pages 116-117
Curve Parametrization over Optimal Field Extensions Exploiting the Newton Polygon....Pages 119-140
Ridges and Umbilics of Polynomial Parametric Surfaces....Pages 141-159
Intersecting Biquadratic B?zier Surface Patches....Pages 161-180
Cube Decompositions by Eigenvectors of Quadratic Multivariate Splines....Pages 181-197
Subdivision Methods for the Topology of 2d and 3d Implicit Curves....Pages 199-214
Approximate Implicitization of Space Curves and of Surfaces of Revolution....Pages 215-227
Back Matter....Pages 229-231
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