Bicubic G1 interpolation of arbitrary quad meshes using a 4-split

We present a piecewise bi-cubic parametric G1 spline surface interpolating the vertices of a quadrangular surface mesh of arbitrary topological type. While tensor product surfaces need a chess boarder parameterization they are not well suited to model surfaces of arbitrary topology without introducing singularities. Our spline surface consists of tensor product patches, but they can be assembled with G1-continuity to model any non-tensor-product configuration. At common patch vertices an arbitrary number of patches can meet. The parametric domain is built by 4-splitting one unit square for each input quadrangular face. This is the key idea of our method since it enables to define a very low degree surface, that interpolates the input vertices and results from an explicit and local procedure: no global linear system has to be solved. It differs from all existing schemes since it computes four piecewise polynomial surfaces in correspondence to a mesh face instead of only one polynomial surface while interpolating the mesh vertices. Furthermore, the surface scheme is completely general in the sense that it doesn’t impone any restriction on both, the valence of the mesh vertices and the input data in general. It interpolates any 2-manifold quadrangular mesh.