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The use of approximate implicitization in recursive surface-intersection algorithm


Although the origin of algebraic geometry was to solve geometric problems of daily life, current research is addressing problems described in homogeneous coordinates and complex spaces. The introduction of computers more than five decades ago triggered research in Computer Aided Geometric Design (CAGD). This research has always been focused on what is feasible to compute on available commodity computers. The continuous increase in computational performance has also triggered demands for better quality and higher accuracy in Computer Aided Design systems, especially with respect to surface intersection algorithms.To try to satisfy these demands researchers in classical Algebraic Geometry and researches in CAGD have in recent years joined forces to address challenges related to geometry representation and interrogation in producing industries. The EU IST-2001-35512 project Intersection algorithms for geometry based IT-applications using approximate algebraic methods addressed the problems of problematic intersections in CAD-systems. The focus was surface self-intersections and situations where surfaces are near parallel in intervals along the intersection curves. The project addressed the challenges from the viewpoint of exact computations, semi-numeric computations and a numeric recursive subdivision approach. After the project end the project partners have continued their research. The mini-symposium will focus on experiences of such a focused research project and results from and after the project. It will address:approximate implicitization, how to approximate parametric surfaces by real algebraic surfaces; exact, numeric and recursive intersection algorithms; resolving topological questions in practice about real curves and surfaces; use of multi-core CPUs and programmable graphics cards for improving quality and performance of intersection algorithms.


Academic lecture





  • SINTEF Digital / Mathematics and Cybernetics

Presented at

6th International Congress on Industrial and Applied Mathematics (ICIAM 07)


Zurich, Switzerland


16.07.2007 - 20.07.2007


Swiss Mathematical Society (SMG)



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