Algebraic Geometry and Geometric Modeling

This book spans the distance between algebraic descriptions of geometric objects and the rendering of digital geometric shapes based on algebraic models. These contrasting points of view inspire a thorough analysis of the key challenges and how they are met. The articles focus on important classes of problems: implicitization, classification, and intersection. Combining illustrative graphics, computations and review articles this book helps the reader gain a firm practical grasp of these subjects.

Résumé

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.

Contenu
Algebraic geometry and geometric modeling: insight and computation.- Implicitization using approximation complexes.- Piecewise approximate implicitization: experiments using industrial data.- Computing with parameterized varieties.- Implicitization and Distance Bounds.- Singularities and their deformations: how they change the shape and view of objects.- Overview of topological properties of real algebraic surfaces.- Illustrating the classification of real cubic surfaces.- Bézier patches on almost toric surfaces.- On parametric surfaces of low degree in P3(C).- On the intersection with revolution and canal surfaces.- A sampling algorithm computing self-intersections of parametric surfaces.- Elimination in generically rigid 3D geometric constraint systems.- Minkowski decomposition of convex lattice polygons.- Reducing the number of variables of a polynomial.