Based on polyhedral splines, some multivariate splines of different orders with given supports over arbitrary topological meshes are developed. Schemes for choosing suitable families of multivariate splines based on pre-given meshes are discussed. Those multivariate splines with inner knots and boundary knots from the related meshes are used to generate rational spline shapes with related control points. Steps for up to C2-surfaces over the meshes are designed. The relationship among the meshes and their knots, the splines and control points is analyzed. To avoid any unexpected discontinuities and get higher smoothness, a heart-repairing technique to adjust inner knots in the multivariate splines is designed.
With the theory above, bivariate C1-quadratic splines over rectangular meshes are developed. Those bivariate splines are used to generate rational C1-quadratic surfaces over the meshes with related control points and weights. The properties of the surfaces are analyzed. The boundary curves and the corner points and tangent planes, and smooth connecting conditions of different patches are presented. The C1–continuous connection schemes between two patches of the surfaces are presented.