eBook ISBN:  9780821879245 
Product Code:  CONM/334.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9780821879245 
Product Code:  CONM/334.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 

Book DetailsContemporary MathematicsVolume: 334; 2003; 366 ppMSC: Primary 14; 52; 65; 68; Secondary 13; 41; 58
Algebraic geometry and geometric modeling both deal with curves and surfaces generated by polynomial equations. Algebraic geometry investigates the theoretical properties of polynomial curves and surfaces; geometric modeling uses polynomial, piecewise polynomial, and rational curves and surfaces to build computer models of mechanical components and assemblies for industrial design and manufacture.
The NSF sponsored the fourday “Vilnius Workshop on Algebraic Geometry and Geometric Modeling”, which brought together some of the top experts in the two research communities to examine a wide range of topics of interest to both fields. This volume is an outgrowth of that workshop. Included are surveys, tutorials, and research papers. In addition, the editors have included a translation of Minding's 1841 paper, “On the determination of the degree of an equation obtained by elimination”, which foreshadows the modern application of mixed volumes in algebraic geometry.
The volume is suitable for mathematicians, computer scientists, and engineers interested in applications of algebraic geometry to geometric modeling.
ReadershipGraduate students, research mathematicians, computer scientists, and engineers interested in applications of algebraic geometry to geometric modeling.

Table of Contents

Modeling Curves and Surfaces [ MR 2046835 ]

Ron Goldman — Polar forms in geometric modeling and algebraic geometry [ MR 2039963 ]

Wenping Wang and Rimvydas Krasauskas — Interference analysis of conics and quadrics [ MR 2039964 ]

Raimundas Vidūnas — Geometrically continuous octahedron [ MR 2039965 ]

Multisided Patches [ MR 2046835 ]

Jörg Peters — Smoothness, fairness and the need for better multisided patches [ MR 2039966 ]

Rimvydas Krasauskas and Ron Goldman — Toric Bézier patches with depth [ MR 2039967 ]

Joe Warren — On the uniqueness of barycentric coordinates [ MR 2039968 ]

Kȩstutis Karčiauskas — Rational $M$patches and tensorborder patches [ MR 2039969 ]

Implicitization and Parametrization [ MR 2046835 ]

David Cox — Curves, surfaces, and syzygies [ MR 2039970 ]

Jianmin Zheng, Thomas W. Sederberg, EngWee Chionh and David A. Cox — Implicitizing rational surfaces with base points using the method of moving surfaces [ MR 2039971 ]

Tor Dokken and Jan Brede Thomassen — Overview of approximate implicitization [ MR 2039972 ]

Josef Schicho — Algorithms for rational surfaces [ MR 2039973 ]

Tonic Varieties [ MR 2046835 ]

David Cox — What is a toric variety? [ MR 2039974 ]

Frank Sottile — Toric ideals, real toric varieties, and the moment map [ MR 2039975 ]

David Cox, Rimvydas Krasauskas and Mircea Mustaţǎ — Universal rational parametrizations and toric varieties [ MR 2039976 ]

Claire Delaunay — Real structures on smooth compact toric surfaces [ MR 2039977 ]

Mixed Volume and Resultants [ MR 2046835 ]

J. Maurice Rojas — Why polyhedra matter in nonlinear equation solving [ MR 2039978 ]

Laurent Busé, Mohamed Elkadi and Bernard Mourrain — Using projection operators in computer aided geometric design [ MR 2039979 ]

Ivan Soprounov — On combinatorial coefficients and the GelfondKhovanskii residue formula [ MR 2039980 ]

Ferdinand Minding — On the determination of the degree of an equation obtained by elimination [ MR 2039981 ]


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Algebraic geometry and geometric modeling both deal with curves and surfaces generated by polynomial equations. Algebraic geometry investigates the theoretical properties of polynomial curves and surfaces; geometric modeling uses polynomial, piecewise polynomial, and rational curves and surfaces to build computer models of mechanical components and assemblies for industrial design and manufacture.
The NSF sponsored the fourday “Vilnius Workshop on Algebraic Geometry and Geometric Modeling”, which brought together some of the top experts in the two research communities to examine a wide range of topics of interest to both fields. This volume is an outgrowth of that workshop. Included are surveys, tutorials, and research papers. In addition, the editors have included a translation of Minding's 1841 paper, “On the determination of the degree of an equation obtained by elimination”, which foreshadows the modern application of mixed volumes in algebraic geometry.
The volume is suitable for mathematicians, computer scientists, and engineers interested in applications of algebraic geometry to geometric modeling.
Graduate students, research mathematicians, computer scientists, and engineers interested in applications of algebraic geometry to geometric modeling.

Modeling Curves and Surfaces [ MR 2046835 ]

Ron Goldman — Polar forms in geometric modeling and algebraic geometry [ MR 2039963 ]

Wenping Wang and Rimvydas Krasauskas — Interference analysis of conics and quadrics [ MR 2039964 ]

Raimundas Vidūnas — Geometrically continuous octahedron [ MR 2039965 ]

Multisided Patches [ MR 2046835 ]

Jörg Peters — Smoothness, fairness and the need for better multisided patches [ MR 2039966 ]

Rimvydas Krasauskas and Ron Goldman — Toric Bézier patches with depth [ MR 2039967 ]

Joe Warren — On the uniqueness of barycentric coordinates [ MR 2039968 ]

Kȩstutis Karčiauskas — Rational $M$patches and tensorborder patches [ MR 2039969 ]

Implicitization and Parametrization [ MR 2046835 ]

David Cox — Curves, surfaces, and syzygies [ MR 2039970 ]

Jianmin Zheng, Thomas W. Sederberg, EngWee Chionh and David A. Cox — Implicitizing rational surfaces with base points using the method of moving surfaces [ MR 2039971 ]

Tor Dokken and Jan Brede Thomassen — Overview of approximate implicitization [ MR 2039972 ]

Josef Schicho — Algorithms for rational surfaces [ MR 2039973 ]

Tonic Varieties [ MR 2046835 ]

David Cox — What is a toric variety? [ MR 2039974 ]

Frank Sottile — Toric ideals, real toric varieties, and the moment map [ MR 2039975 ]

David Cox, Rimvydas Krasauskas and Mircea Mustaţǎ — Universal rational parametrizations and toric varieties [ MR 2039976 ]

Claire Delaunay — Real structures on smooth compact toric surfaces [ MR 2039977 ]

Mixed Volume and Resultants [ MR 2046835 ]

J. Maurice Rojas — Why polyhedra matter in nonlinear equation solving [ MR 2039978 ]

Laurent Busé, Mohamed Elkadi and Bernard Mourrain — Using projection operators in computer aided geometric design [ MR 2039979 ]

Ivan Soprounov — On combinatorial coefficients and the GelfondKhovanskii residue formula [ MR 2039980 ]

Ferdinand Minding — On the determination of the degree of an equation obtained by elimination [ MR 2039981 ]