ShApes, Geometry and Algebra (2008-2012)

Background: Computer Aided Design and Manufacturing (CAD/CAM) originated about 45 years ago and is now a key part of product development and production process. Current commercial technology is built on results from Classical and Computational Geometry, and Approximation Theory, Numerical Analysis, and Computer Graphics.

In the last decade technologies for information processing and data acquisition has made rapid progress. The availability of cheap computing power on desktop computers, including high-performance graphics capabilities, and the advent of laser scanners for 3D objects, which are able to digitize even complex geometric models within seconds, poses new challenges for CAD/CAM. The geometric models to be processed have become more complex and the use of this technology has become far more widespread, especially among small and medium-size enterprises. 

The technology represented by the STEP standard is no longer adequate to address the mathematical problems that occur. In the long term, this development may prove to be problematic, since it decouples the CAD/CAM industry from new applications such as mobile telecommunication and the computer game industry. CAD-systems are due for major rework to be able to exploit the computational performance of multi-core CPUs and data stream accelerators such as programmable graphics cards. Experiments show that data stream accelerators outperform CPUs by an order of magnitude for tasks that can be effectively parallelized. Current hardware has 3 to 4 orders of magnitude more computational performance than the hardware standard CAD-technology originally targeted. This opens up the opportunity to use more advanced approaches in CAD-systems.

Within the numerical analysis community, the use of higher order polynomial representations (hpFEM and the isogeometric approach) has been conceived as a new way to break the complexity barrier caused by piecewise linear representations, and to deal efficiently with free-form geometry. In order to exploit the potential of these developments, this progress has to be matched by corresponding research and development in the geometric and CAD/CAM community.

Project objectives: The consortium consists of universities, research institutes and industrial partners that all share the belief that CAD/CAM will be greatly enhanced by exploiting mathematical results and techniques they are currently working on, covering the full spectrum from Algebraic Geometry and Computer Algebra to Computer Aided Geometric Design (CAGD), Numerical Analysis and Approximation Theory. The challenges to be addressed in SAGA are organized into four scientific work packages:

• WP1: Change of Representation
• WP2: Geometric Computing – Algebraic Tools
• WP3: Algebraic Geometry for CAD Applications
• WP4: Practical Industrial Problems

Two relatively small and exploratory European projects GAIA (2001, assessment project) and GAIA II FET Open (2002—2005), co-ordinated by SINTEF established a meeting place in Europe for researchers interested in the combination of CAGD and Real Algebraic Geometry, the field of Approximate Algebraic Geometry. The SAGA-project aims at promoting this new field, and in general at strengthening interdisciplinary and inter-sectorial research and development concerning CAD/CAM, by training a new generation of researchers familiar with both academic and industry viewpoints, while supporting the cooperation among the partners and with other interested collaborators in Europe. As one of the reviewers said in the final review of the GAIA II project, he was “very impressed by the results” of that project, but still felt that we “had just scratched the surface”.