| Abstract: | Based on polyhedral splines, some multivariate splines of different orderswith given supports over arbitrary topological meshes are developed. Schemes forchoosing suitable families of multivariate splines based on pre-given meshes arediscussed. Those multivariate splines with inner knots and boundary knots fromthe related meshes are used to generate rational spline shapes with related controlpoints. Steps for up to $C^2$-surfaces over the meshes are designed. The relationshipamong the meshes and their knots, the splines and control points is analyzed. Toavoid any unexpected discontinuities and get higher smoothness, a heart-repairingtechnique to adjust inner knots in the multivariate splines is designed.With the theory above, bivariate $C^1$-quadratic splines over rectangular meshes aredeveloped. Those bivariate splines are used to generate rational $C^1$-quadratic surfaces over the meshes with related control points and weights. The properties ofthe surfaces are analyzed. The boundary curves and the corner points and tangentplanes, and smooth connecting conditions of different patches are presented. The $C^1$−continuous connection schemes between two patches of the surfaces are presented. |
