Page 5 - Geometric Modeling and Algebraic Geometry

Preface

The two fields of Geometric Modeling and Algebraic Geometry, though closely re-
lated, 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 re-
sults 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 con-
tributions from the algebraic side.
The workshop series on Algebraic Geometry and Geometric Modeling (Vilnius
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2002 , Nice 2004 ) and on Computational Methods for Algebraic Spline Surfaces
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(Kefermarkt 2003 , 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
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using approximate algebraic methods (IST 2001-35512) .
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 comput-
ing. This includes chapters on parameterization, computation and analysis of ridges
and umbilical points, surface-surface intersections, topology analysis and approxi-
mate implicitization.
R. Goldman and R. Krasauskas, Topics in Algebraic Geometry and Geometric Modeling,
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Contemporary Mathematics, American Mathematical Society 2003.
M. Elkadi, B. Mourrain and R. Piene, Algebraic Geometry and Geometric Modeling,
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Springer 2006.
T. Dokken and B. J¨uttler, Computational Methods for Algebraic Spline Surfaces, Springer
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2005.
http://www.sintef.no/IST GAIA
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