**Contemporary Mathematics**

Volume: 334;
2003;
366 pp;
Softcover

MSC: Primary 14; 52; 65; 68;
Secondary 13; 41; 58

Print ISBN: 978-0-8218-3420-6

Product Code: CONM/334

List Price: $103.00

AMS Member Price: $82.40

MAA Member Price: $92.70

**Electronic ISBN: 978-0-8218-7924-5
Product Code: CONM/334.E**

List Price: $103.00

AMS Member Price: $82.40

MAA Member Price: $92.70

# Topics in Algebraic Geometry and Geometric Modeling

Share this page *Edited by *
*Ron Goldman; Rimvydas Krasauskas*

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

## Topics in Algebraic Geometry and Geometric Modeling

- Contents iii4 free
- Preface v6 free
- Modeling Curves and Surfaces 112 free
- Multisided Patches 5364
- lmplicitization and Parametrization 129140
- Toric Varieties 201212
- What is a toric variety? 203214
- 1. Varieties 203214
- 2. Toric Varieties 204215
- 3. Examples of Toric Varieties 205216
- 4. Cones 206217
- 5. Affine Toric Varieties 207218
- 6. Coordinate Rings 209220
- 7. Normality 209220
- 8. Fans and Toric Varieties 210221
- 9. Properties of Toric Varieties 212223
- 10. Homogeneous Coordinates 212223
- 11. Examples of Homogeneous Coordinates 215226
- 12. The Toric Variety of a Polytope 217228
- 13. Polytopes and Homogeneous Coordinates 219230

- Toric ideals, real toric varieties, and the moment map 225236
- Universal rational parametrizations and toric varieties 241252
- Real structures on smooth compact toric surfaces 267278

- Mixed Volume and Resultants 291302
- Index 363374 free