Deciding on the Regularity of a Planar Coons Map

We consider the construction of a regular map from the unit square to a general four-sided domain, a problem that arises in several applications, e.g., in the numerical solution of integral equations or when dealing with trimmed surfaces in CAD. In our approach, we consider the problem in as general a form as possible, which means that initially no assumptions are made about the type of mathematical representation of the boundary curves. This approach becomes possible by using planar Coons maps to describe the parameterization. We show that the regularity of a Coons map depends both on the waviness and the similarity of the boundary curves. Constraining these properties allows us to formulate sufficient conditions for regularity and to specify special cases in which the regularity of the Coons map is obvious. For the case of polynomial boundary curves, we present a regularity criterion that is both necessary and sufficient and can thus be used to characterize regular polynomial mappings. Our decision algorithm implements this criterion and provides a powerful tool not only for deciding the regularity of a given Coons map, but also for determining the transition between a regular and a non-regular Coons map depending on the curvature of the boundary curves.