Algebraic Geometry provides an impressive theory targeting the understanding of geometric objects defined algebraically. Geometric Modeling uses every day, in order to solve practical and difficult problems, digital shapes based on algebraic models. In this book, we have collected articles bridging these two areas. The confrontation of the different points of view results in a better analysis of what the key challenges are and how they can be met. We focus on the following important classes of problems: implicitization, classification, and intersection. The combination of illustrative pictures, explicit computations and review articles will help the reader to handle these subjects.

Algebraic Geometry and Geometric Modeling are two distinct domains of research, with few interactions up to now, though closely linked. On the one hand, Algebraic Geometry has developed an impressive theory targeting the understanding of geometric objects defined algebraically. On the other hand, Geometric Modeling is using every day, in practical and difficult problems, virtual shapes based on algebraic models. Could these two domains benefit from each other? Recent and interesting developments in this direction are about to convince us to answer yes. In this book, we have collected articles which reinforce, in some way, the natural bridge which exists between these two areas. The confrontation of the different points of view should result in a better analysis of the key problems and related methods to solve them. This was the aim of the workshop entitled Algebraic Geometry and Geometric Modeling, held from September 27 to September 29, 2004, at the University of Nice-Sophia Antipolis. This workshop was organized, in the context of the European project GAIA II (IST-2002-35512).

The first group of articles is about Implicitization problems, namely the conversion of a parametric representation of an algebraic variety into an implicit one. This problem is fundamental in Computer Aided Geometric Design (CAGD), where geometric objects are often given parametrically and some operations like testing if a point is in a variety, intersecting two varieties, determining the singular locus of a variety, is better understandood via implicit representations.

In his contribution, Goldman claims that the main contribution of Algebraic Geometry to Geometric Modeling is insight, not computation, and that we should not confuse concrete theoretical tools given by Algebraic Geometry with efficient computational methods needed by Geometric Modeling.