Geometric Modeling and Algebraic Geometry

Transcription

1 Geometric Modeling and Algebraic Geometry

2 Bert Jüttler Ragni Piene Editors Geometric Modeling and Algebraic Geometry 123

3 Bert Jüttler Institute of Applied Geometry Johannes Kepler University Altenberger Str Linz, Austria Ragni Piene CMA and Department of Mathematics University of Oslo P.O.Box 1053 Blindern 0136 Oslo, Norway ISBN: e-isbn: Library of Congress Control Number: 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 springer.com

4 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 , Nice ) and on Computational Methods for Algebraic Spline Surfaces (Kefermarkt , 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 ) 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 Steiner surfaces, surfaces with many real singularities, monoid hypersurfaces, canal surfaces, and tensor-product surfaces of bidegree (1,2). The third part describes various algorithms for geometric computing. This includes chapters on parameterization, computation and analysis of ridges and umbilical points, surface-surface intersections, topology analysis and approximate implicitization. 1 R. Goldman and R. Krasauskas, Topics in Algebraic Geometry and Geometric Modeling, Contemporary Mathematics, American Mathematical Society M. Elkadi, B. Mourrain and R. Piene, Algebraic Geometry and Geometric Modeling, Springer T. Dokken and B. Jüttler, Computational Methods for Algebraic Spline Surfaces, Springer GAIA

5 VI Preface The editors are indebted to the reviewers, whose comments have helped greatly to identify the manuscripts suitable for publication, and for improving many of them substantially. Thanks to Springer for the constructive cooperation during the production of this book. Special thanks go to Ms. Bayer for compiling the L A TEX sources into a single coherent manuscript. Oslo and Linz, August 2007 Bert Jüttler Ragni Piene

6 Contents Part I Survey of the European project GAIA II 1 The GAIA Project on Intersection and Implicitization T. Dokken... 5 Part II Some special algebraic surfaces 2 Some Covariants Related to Steiner Surfaces F. Aries, E. Briand, C. Bruchou Real Line Arrangements and Surfaces with Many Real Nodes S. Breske, O. Labs, D. van Straten Monoid Hypersurfaces P. H. Johansen, M. Løberg, R. Piene Canal Surfaces Defined by Quadratic Families of Spheres R. Krasauskas, S. Zube General Classification of (1,2) Parametric Surfaces in P 3 T.-H. Lê, A. Galligo Part III Algorithms for geometric computing 7 Curve Parametrization over Optimal Field Extensions Exploiting the Newton Polygon T. Beck, J. Schicho Ridges and Umbilics of Polynomial Parametric Surfaces F. Cazals, J.-C. Faugère, M. Pouget, F. Rouillier...141

7 VIII Contents 9 Intersecting Biquadratic Bézier Surface Patches S. Chau, M. Oberneder, A. Galligo, B. Jüttler Cube Decompositions by Eigenvectors of Quadratic Multivariate Splines I. Ivrissimtzis, H.-P. Seidel Subdivision Methods for the Topology of 2d and 3d Implicit Curves C. Liang, B. Mourrain, J.-P. Pavone Approximate Implicitization of Space Curves and of Surfaces of Revolution M. Shalaby, B. Jüttler Index