Classification and resolution of critical cases in Grandine and Klein's topology determination using a perturbation method

We classify and resolve all critical cases in the topology determination method proposed in [Grandine, T.A., Klein IV, F.W., 1997. A new approach to the surface intersection problem. Computer Aided Geometric Design 14 (2), 111–134]. Their algorithm for finding the intersection of two parametric surfaces has two steps: determining the topology of the intersection curves and using that information to find the curves themselves. The essence of the first step is to decide whether the boundary points and the turning points are at the start or the end of a contour. However, there are several cases in which the decision criteria proposed by Grandine and Klein are not applicable. We classify all these cases, which include the tangential intersection of two surfaces, the tangential intersection of the contour with the boundary of the domain of a surface, and the vanishing of (u″(τ),v″(τ)) in the interior of the (u,v) domain. Then we resolve all these cases using a perturbation method, which is based on the fact that transversality is a stable and generic property of an intersection. We also present a classical method for resolution, which uses higher-order derivatives.