At the core of recursive intersection algorithms are strategies to determine if an intersection is simple (all intersection tracks are touching the boundary of one of the surface patches intersected), or if there is a closed intersection curve in the interior of both surfaces intersected.

• In the case of transversal intersections Sinha’s theorem form 1985 states that the normal fields of the two surfaces have to overlap for a closed intersection loop to exist. Provide the intersection curves have a proper parameterization, and are transversal repeated subdivision will, provided the subdivision points are chosen carefully, identify all intersection curves..
• In the case when the intersection curves are singular or near singular finding the intersections is much more complex. The ambition of GAIA II has been to provide a solution for these cases, and also handling cases where the paremtrization of the surfaces is not proper.

A protype surface surface and surface self.intersection algorithm is part of the results of GAIA II. These protypes will be made available as Open Source under the GNU GPL license in the second half of 2005.