We consider the problem of creating a smooth parametric polynomial surface interpolating the vertices of an irregular quadrilateral mesh. Surfaces of any topological type can be designed. They are overall tangent plane continuous without any singular points and without any singular parameterization. In particular we will show that using a four-split of the parameter domain solves the problem of joining an arbitrary number of patches together at a common vertex with G1-continuity, known as the vertex consistency problem. The resulting surface consists of a small number of very low degree surface patches, bi-cubic tensor product Bezier patches. Explicit formulas of the Bezier control points are derived. Several degrees of freedom are available. They can either be used as additional design parameters or they can be used for shape optimizations.