Approximate implicitization and CAD-type intersection algorithms

When developing CAD-type intersection algorithms for NonUniform Rational B-splines surfaces (NURBS) is important to ensure that the algorithm identifies all intersection branches. For transversal intersections this is fairly straight forward. For singular or near singular intersection, when the surfaces are close and near parallel, the traditional approach of recursive subdivision does not suffice on its own. The use of algebraic surfaces to separate near parallel not intersecting surfaces reduces the depth of recursion. Approximate implicitization denotes methods that approximate piecewise parametric manifolds p(s) with an algebraic hypersurface q(x)=0. The talk will address the approach to approximate implicitization already published by the author and further report on how approximate implicitization has helped the development of a CAD-type intersection algorithms addressing self-intersecting CAD-surfaces, and singular and near singular intersection of NURBS surfaces. 