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 different ways. While algebraic geometry has developed impressive - sults for understanding the theoretical nature of these objects, geometric modeling focuses on practical applications of virtual shapes de? ned by algebraic equations. Recently, however, interaction between the two ? elds has stimulated new research. For instance, algorithms for solving intersection problems have bene? ted from c- tributions from the algebraic side. The workshop series on Algebraic Geometry and Geometric Modeling (Vilnius 1 2 2002 , Nice 2004 ) and on Computational Methods for Algebraic Spline Surfaces 3 (Kefermarkt 2003 , Oslo 2005) have provided a forum for the interaction between the two ? elds. The present volume presents revised papers which have grown out of the 2005 Oslo workshop, which was aligned with the ? nal review of the European project GAIA II, entitled Intersection algorithms for geometry based IT-applications 4 using approximate algebraic methods (IST 2001-35512) .

Survey of the European project GAIA II. - The GAIA Project on Intersection and Implicitization. - Some special algebraic surfaces. - Some Covariants Related to Steiner Surfaces. - Real Line Arrangements and Surfaces with Many Real Nodes. - Monoid Hypersurfaces. - Canal Surfaces Defined by Quadratic Families of Spheres. - General Classification of (1, 2) Parametric Surfaces in ? 3. - Algorithms for geometric computing. - Curve Parametrization over Optimal Field Extensions Exploiting the Newton Polygon. - Ridges and Umbilics of Polynomial Parametric Surfaces. - Intersecting Biquadratic Bézier Surface Patches. - Cube Decompositions by Eigenvectors of Quadratic Multivariate Splines. - Subdivision Methods for the Topology of 2d and 3d Implicit Curves. - Approximate Implicitization of Space Curves and of Surfaces of Revolution.