Contemporary Mathematics 2003; 366 pp; softcover Volume: 334 ISBN10: 0821834207 ISBN13: 9780821834206 List Price: US$98 Member Price: US$78.40 Order Code: CONM/334
 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. Readership Graduate students, research mathematicians, computer scientists, and engineers interested in applications of algebraic geometry to geometric modeling. Table of Contents Modeling Curves and Surfaces  R. Goldman  Polar forms in geometric modeling and algebraic geometry
 W. Wang and R. Krasauskas  Interference analysis of conics and quadrics
 R. Vidūnas  Geometrically continuous octahedron
Multisided Patches  J. Peters  Smoothness, fairness and the need for better multisided patches
 R. Krasauskas and R. Goldman  Toric Bézier patches with depth
 J. Warren  On the uniqueness of barycentric coordinates
 K. Karčiauskas  Rational \(M\)patches and tensorborder patches
Implicitization and Parametrization  D. Cox  Curves, surfaces, and syzygies
 J. Zheng, T. W. Sederberg, E.W. Chionh, and D. A. Cox  Implicitizing rational surfaces with base points using the method of moving surfaces
 T. Dokken and J. B. Thomassen  Overview of approximate implicitization
 J. Schicho  Algorithms for rational surfaces
Toric Varieties  D. Cox  What is a toric variety?
 F. Sottile  Toric ideals, real toric varieties, and the moment map
 D. Cox, R. Krasauskas, and M. Mustaţǎ  Universal rational parametrizations and toric varieties
 C. Delaunay  Real structures on smooth compact toric surfaces
Mixed Volume and Resultants  J. M. Rojas  Why polyhedra matter in nonlinear equation solving
 L. Busé, M. Elkadi, and B. Mourrain  Using projection operators in computer aided geometric design
 I. Soprounov  On combinatorial coefficients and the GelfondKhovanskii residue formula
 F. Minding  On the determination of the degree of an equation obtained by elimination
 Index
