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Cover: 9783642092916 | Rational Algebraic Curves | A Computer Algebra Approach | Taschenbuch
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Rational Algebraic Curves
A Computer Algebra Approach
Taschenbuch von J. Rafael Sendra (u. a.)
Sprache: Englisch

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Beschreibung
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.
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.
Zusammenfassung
The central problem considered in this book is the determination of rational parametrizability of an algebraic curve, and, in the positive case, the computation of a good rational parametrization. This amounts to determining the genus of a curve, i.e. its complete singularity structure, computing regular points of the curve in small coordinate fields, and constructing linear systems of curves with prescribed intersection multiplicities. Various optimality criteria for rational parametrizations of algebraic curves are discussed. This book is mainly intended for graduate students and researchers in constructive algebraic curve geometry. It provides a nice and easy introduction to the topic for students.
Inhaltsverzeichnis
and Motivation.- Plane Algebraic Curves.- The Genus of a Curve.- Rational Parametrization.- Algebraically Optimal Parametrization.- Rational Reparametrization.- Real Curves.
Details
Erscheinungsjahr: 2010
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 280
Reihe: Algorithms and Computation in Mathematics
Inhalt: x
270 S.
24 s/w Illustr.
270 p. 24 illus.
ISBN-13: 9783642092916
ISBN-10: 3642092918
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Sendra, J. Rafael
Pérez-Diaz, Sonia
Winkler, Franz
Auflage: Softcover reprint of hardcover 1st ed. 2008
Hersteller: Springer-Verlag GmbH
Springer Berlin Heidelberg
Algorithms and Computation in Mathematics
Maße: 235 x 155 x 16 mm
Von/Mit: J. Rafael Sendra (u. a.)
Erscheinungsdatum: 20.11.2010
Gewicht: 0,429 kg
preigu-id: 107211838
Zusammenfassung
The central problem considered in this book is the determination of rational parametrizability of an algebraic curve, and, in the positive case, the computation of a good rational parametrization. This amounts to determining the genus of a curve, i.e. its complete singularity structure, computing regular points of the curve in small coordinate fields, and constructing linear systems of curves with prescribed intersection multiplicities. Various optimality criteria for rational parametrizations of algebraic curves are discussed. This book is mainly intended for graduate students and researchers in constructive algebraic curve geometry. It provides a nice and easy introduction to the topic for students.
Inhaltsverzeichnis
and Motivation.- Plane Algebraic Curves.- The Genus of a Curve.- Rational Parametrization.- Algebraically Optimal Parametrization.- Rational Reparametrization.- Real Curves.
Details
Erscheinungsjahr: 2010
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 280
Reihe: Algorithms and Computation in Mathematics
Inhalt: x
270 S.
24 s/w Illustr.
270 p. 24 illus.
ISBN-13: 9783642092916
ISBN-10: 3642092918
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Sendra, J. Rafael
Pérez-Diaz, Sonia
Winkler, Franz
Auflage: Softcover reprint of hardcover 1st ed. 2008
Hersteller: Springer-Verlag GmbH
Springer Berlin Heidelberg
Algorithms and Computation in Mathematics
Maße: 235 x 155 x 16 mm
Von/Mit: J. Rafael Sendra (u. a.)
Erscheinungsdatum: 20.11.2010
Gewicht: 0,429 kg
preigu-id: 107211838
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