Geometric Modeling and Algebraic Geometry
The two ?elds of Geometric Modeling and Algebraic Geometry, though closely - lated, are traditionally represented by two almost disjoint scienti?c communities. Both ?elds deal with objects de?ned by algebraic equations, but the objects are studied in diff
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Geometric Modeling and Algebraic Geometry
Geometric Modeling and Algebraic Geometry
Bert J¨uttler • Ragni Piene Editors
Geometric Modeling and Algebraic Geometry
123
Bert J¨uttler Institute of Applied Geometry Johannes Kepler University Altenberger Str. 69 4040 Linz, Austria [email protected]
ISBN: 978-3-540-72184-0
Ragni Piene CMA and Department of Mathematics University of Oslo P.O.Box 1053 Blindern 0136 Oslo, Norway [email protected]
e-ISBN: 978-3-540-72185-7
Library of Congress Control Number: 2007935446 Mathematics Subject Classification Numbers (2000): 65D17, 68U06, 53A05, 14P05, 14J26 c Springer-Verlag Berlin Heidelberg 2008 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMX Design GmbH, Heidelberg Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
Preface
The two fields of Geometric Modeling and Algebraic Geometry, though closely related, are traditionally represented by two almost disjoint scientific communities. Both fields deal with objects defined by algebraic equations, but the objects are studied in different ways. While algebraic geometry has developed impressive results for understanding the theoretical nature of these objects, geometric modeling focuses on practical applications of virtual shapes defined by algebraic equations. Recently, however, interaction between the two fields has stimulated new research. For instance, algorithms for solving intersection problems have benefited from contributions from the algebraic side. The workshop series on Algebraic Geometry and Geometric Modeling (Vilnius 20021 , Nice 20042 ) and on Computational Methods for Algebraic Spline Surfaces (Kefermarkt 20033 , Oslo 2005) have provided a forum for the interaction between the two fields. The present volume presents revised papers which have grown out of the 2005 Oslo workshop, which was aligned with the final review of the European project GAIA II, entitled Intersection algorithms for geometry based IT-applications using approximate algebraic methods (IST 2001-35512)4 . It consists of 12 chapters, which are organized in 3 parts. The first part describes the aims and the results of the GAIA II project. Part 2 consists of 5 chapters covering results about special algebraic surfaces, such as Stein
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