The use of computers for the representation of 2D and 3D geometries is continuously growing. The dominant method for representation has been parametric curves and surface, with the exception of low degree algebraic curves and surfaces, e.g. planes, spheres, conics, ellipsoids and torus. For these lower degree curves and surfaces the duality between the parametric and algebraic descriptions has been an efficient and useful fool. For free form curves and sculptured surfaces the use of the duality has been very limited. This is due to a number of reasons:

• A parametric tensor product surface of degrees (m,n) has an algebraic degree of 2nm. Thus a bicubic surface, degrees (3,3), has algebraic degree 18. Degree eighteen polynomials are resource demanding when used on digital computers.
• The parametric described geometries are located in a limited part of space, while the algebraic description continues into infinity in most cases.
• The theory within algebraic geometry is based on the complex plane and projective space, while the current need for geometry on computers in dominated by real numbers and affine space.

Thus there is a big gap between development within algebraic geometry and the current needs for the representation of geometry on computers. In his doctorate thesis in 1984 Prof. Thomas W. Sederberg identified the benefit of combining parametric and algebraic descriptions of geometry. In the start of the 90s Tor Dokken from SINTEF Applied Mathematics started the development of what is now known as Approximate Algebraic Geometry. This is a new approach addressing algebraic geometry from the viewpoint of approximation theory. The first results were presented in Dokken doctorate thesis form 1997. Based on this the EU-sponsored RTD projects GAIA I and GAIA II including seven different European partners have been started .