Injectivity of B-spline biquadratic maps

The application of isogeometric analysis requires a full parametrization map of the physical domain. This map is suitable for the analysis only if it is injective. In this paper we give a necessary condition for the injectivity of a B-spline biquadratic map. Moreover, sufficient injectivity conditions, which are easy to check in the practice, are introduced. A method for parametrizing planar regions using a biquadratic map is also proposed. The method is simpler than others reported in the literature, since it is based on the solution of an optimization problem without restrictions and only requires the construction of a quadrilateral mesh with optimal geometric properties. Finally we show that the proposed method can be successfully used to obtain high quality parametrizations of simply connected planar regions with irregular boundary.