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 four-day "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

• 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

• J. Peters -- Smoothness, fairness and the need for better multi-sided 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 tensor-border patches

• 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

• 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

• J. M. Rojas -- Why polyhedra matter in non-linear equation solving
• L. Busé, M. Elkadi, and B. Mourrain -- Using projection operators in computer aided geometric design
• I. Soprounov -- On combinatorial coefficients and the Gelfond-Khovanskii residue formula
• F. Minding -- On the determination of the degree of an equation obtained by elimination
• Index