The GAIA II project is based on the idea that there is knowledge and approaches within algebraic geometry that will improve the quality of algorithms and approaches within CAGD.

However, approaches to geometry of the two communities are very different:

Within algebraic geometry problems are rather solved in complex numbers than real numbers, problems are rater solved in projective space than affine space. As exact arithmetic is assumed the choice of coordinate system and polynomial basis is not important.

Within CAGD problems exist and are solved in real affine space and floating point arithmetic is used. The choice of coordinate systems and polynomial bases is essential to the performance and quality of algorithms.

There is a big gap in knowledge and terminology between the communities, and to bridge this gap has proven a greater challenge than expected. The focus for the work on algebraic geometry within CAD within GAIA II has been on:

In addition to this work, we have successfully employed approaches from algebraic geometry on CAGD problems, and shown that knowledge from algebraic geometry increase the quality of theory related to CAGD. One example of such work is “Closest point calculations based on foot poins” described on the Applications page.