Multi-sided surface patch generation and vertex blending

A new kind of multi-sided surface macropatch is proposed: the M-patch, which provides great flexibility for freeform surface construction and can be applied to both vertex and edge blending effectively. An n-sided M-patch combines n 4-sided subpatches at shared edges and a common vertex in the middle. The generation of the M-patch is based on the moving average concept. Each 3-sided subpatch is a moving average in the form of a weighted sum of n geometric primitives. A new conceptual tool, the control pattern, provides an intuitive facility for surface shaping. For an M-patch or an M-patch complex, design constraints are seen in terms of pattern geometry and weight matrices. The overall shape of an M-patch is altered by the location of the control patterns. Because the constructed weight functions are always positive, the resulting surface patch is in the convex hull of its control patterns. Visual interrogation of the M-patch complex via contour curves and isophotes has confirmed the generally pleasing shape obtained from the M-patch approach. A number of pathological examples and challenge cases are illustrated, along with many conventional ones. The continuity related computation during patch construction time is eliminated by using pre-designed control patterns and weight functions, and interactive construction is well facilitated by control patterns. The M-patch is of low polynomial or rational polynomial degree; biquintic G{dollar}sp1{dollar} M-patch and approximate G{dollar}sp1{dollar} (AG{dollar}sp1){dollar} M-patch complexes are obtained for generic n-sided patches, and bicubic G{dollar}sp1{dollar} M-patch and M-patch complexes are obtainable for 4-sided patches. Each subpatch of an M-patch is convertible to a tensor product Bezier patch or rational Bezier patch; therefore, it is easy to integrate into existing modeling systems.