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This book summarizes research carried out in workshops of the SAGA project, an Initial Training Network exploring the interplay of Shapes, Algebra, Geometry and Algorithms. Written by a combination of young and experienced researchers, the book introduces new ideas in an established context. Among the central topics are approximate and sparse implicitization and surface parametrization; algebraic tools for geometric computing; algebraic geometry for computer aided design applications and problems with industrial applications. Readers will encounter new methods for the (approximate) transition between the implicit and parametric representation; new algebraic tools for geometric computing; new applications of isogeometric analysis, and will gain insight into the emerging research field situated between algebraic geometry and computer aided geometric design.

Shape theory (Topology) --- Homotopy theory --- Mappings (Mathematics) --- Topological manifolds --- Topological spaces --- Geometry. --- Algebra. --- Computer science. --- Computer aided design. --- Mathematics of Computing. --- Computer-Aided Engineering (CAD, CAE) and Design. --- Mathematical Modeling and Industrial Mathematics. --- Informatics --- Science --- CAD (Computer-aided design) --- Computer-assisted design --- Computer-aided engineering --- Design --- Mathematics --- Mathematical analysis --- Euclid's Elements --- Computer science—Mathematics. --- Computer-aided engineering. --- Mathematical models. --- Models, Mathematical --- Simulation methods --- CAE --- Engineering --- Data processing

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This book summarizes research carried out in workshops of the SAGA project, an Initial Training Network exploring the interplay of Shapes, Algebra, Geometry and Algorithms. Written by a combination of young and experienced researchers, the book introduces new ideas in an established context. Among the central topics are approximate and sparse implicitization and surface parametrization; algebraic tools for geometric computing; algebraic geometry for computer aided design applications and problems with industrial applications. Readers will encounter new methods for the (approximate) transition between the implicit and parametric representation; new algebraic tools for geometric computing; new applications of isogeometric analysis, and will gain insight into the emerging research field situated between algebraic geometry and computer aided geometric design.

Mathematics --- Algebra --- Geometry --- Planning (firm) --- Computer science --- Programming --- Computer architecture. Operating systems --- Artificial intelligence. Robotics. Simulation. Graphics --- Computer. Automation --- algebra --- computers --- informatica --- mathematische modellen --- wiskunde --- wiskunde --- computerkunde --- CAD (computer aided design) --- geometrie

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The papers included in this volume provide an overview about the state-of-the-art in approximative implicitization and various related topics, including both the theoretical basis and the existing computational techniques. The novel idea of approximate implicitization has strengthened the existing link between Computer Aided Geometric Design and classical algebraic geometry. There is a growing interest from researchers and professionals both in CAGD and Algebraic Geometry, to meet and combine knowledge and ideas, in order to better solve industrial--type challenges, as well as to initiate new directions for basic research. This volume will support this exchange of ideas between the various communities.

Differential geometry. Global analysis --- Geometry --- Numerical analysis --- Computer architecture. Operating systems --- Computer. Automation --- landmeetkunde --- differentiaal geometrie --- visualisatie --- informatica --- wiskunde --- numerieke analyse

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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) .

Mathematics. --- Computer graphics. --- Algebraic geometry. --- Visualization. --- Mathematical models. --- Computational intelligence. --- Algebraic Geometry. --- Mathematical Modeling and Industrial Mathematics. --- Computer Graphics. --- Computational Intelligence. --- Intelligence, Computational --- Artificial intelligence --- Soft computing --- Models, Mathematical --- Simulation methods --- Visualisation --- Imagery (Psychology) --- Imagination --- Visual perception --- Algebraic geometry --- Geometry --- Automatic drafting --- Graphic data processing --- Graphics, Computer --- Computer art --- Graphic arts --- Electronic data processing --- Engineering graphics --- Image processing --- Math --- Science --- Digital techniques --- Curves on surfaces --- Geometry, Algebraic. --- Geometrical models. --- Models and modelmaking --- Surfaces, Curves on --- Models