Adaptive fitting with a cubic spline over planar hierarchical quadrilateral meshes

Automatic fitting techniques are required in many industrial applications, for example instrument calibration, data analysis, geometric modeling, and reverse engineering. In this paper, we present a surface construction algorithm for a cubic spline over planar hierarchical quadrilateral meshes. The surface was piecewisely constructed by interpolating the cubic spline surface of the 12 parameters at four vertices on each quadrilateral cell of the hierarchical quadrilateral mesh. For a given hierarchical quadrilateral mesh and geometric information (function values and two first-order partial derivatives) at the corresponding basis vertices of the hierarchical quadrilateral mesh, the surface can be constructed simply. Moreover, we give an adaptively refined surface algorithm for fitting scattered data points based on cubic spline surface construction. The numerical results show that the proposed adaptive algorithm is efficient in fitting scattered data points within a polygonal domain.