Rational Algebraic Curves

A Computer Algebra Approach

• J. Rafael Sendra
• Franz Winkler
• Sonia Pérez-Díaz
Textbook

Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 22)

1. Front Matter
Pages I-IX
2. Pages 1-13
3. Pages 15-65
4. Pages 67-86
5. Pages 87-147
6. Pages 149-186
7. Pages 187-208
8. Pages 209-237
9. Back Matter
Pages 239-269

Introduction

Algebraic curves and surfaces are an old topic of geometric and algebraic investigation. They have found applications for instance in ancient and m- ern architectural designs, in number theoretic problems, in models of b- logical shapes, in error-correcting codes, and in cryptographic algorithms. Recently they have gained additional practical importance as central objects in computer-aided geometric design. Modern airplanes, cars, and household appliances would be unthinkable without the computational manipulation of algebraic curves and surfaces. Algebraic curves and surfaces combine fas- nating mathematical beauty with challenging computational complexity and wide spread practical applicability. In this book we treat only algebraic curves, although many of the results and methods can be and in fact have been generalized to surfaces. Being the solution loci of algebraic, i. e. , polynomial, equations in two variables, plane algebraiccurvesarewellsuited forbeing investigatedwith symboliccomputer algebra methods. This is exactly the approach we take in our book. We apply algorithms from computer algebra to the analysis, and manipulation of al- braic curves. To a large extent this amounts to being able to represent these algebraic curves in di?erent ways, such as implicitly by de?ning polyno- als, parametrically by rational functions, or locally parametrically by power series expansions around a point.

Keywords

algebra algebraic curves computer computer algebra geometric computation geometry rational parametrization

Authors and affiliations

• J. Rafael Sendra
• 1
• Franz Winkler
• 2
• Sonia Pérez-Díaz
• 1