By using the blossom approach, we construct four new cubic rational Bernsteinlike basis functions with two shape parameters, which form a normalized B-basis and include the cubic Bernstein basis and the cubic Said-Ball basis as special cases. Based on the new basis, we propose a class of C2 continuous cubic rational B-spline-like basis functions with two local shape parameters, which includes the cubic non-uniform B-spline basis as a special case.Their totally positive property is proved. In addition, we extend the cubic rational Bernsteinlike basis to a triangular domain which has three shape parameters and includes the cubic triangular Bernstein-B′ezier basis and the cubic triangular Said-Ball basis as special cases. The G1 continuous conditions are deduced for the joining of two patches. The shape parameters in the bases serve as tension parameters and play a foreseeable adjusting role on generating curves and patches.
In CAD/CAM, mesh rather than smooth surface is only needed sometimes. A mesh-generating method from permanence principle of Coons patch is developed. A new mesh point is defined through local small subpatch and all mesh points are computed by a linear system with special symmetric block tridiagonal coefficient matrix. By simplification, the determinant of coefficient matrix is determined by determinants of submatrices. Condition of existence of solution is given. Whether coefficient matrix is singular can be judged by a simple polynomial function with the eigenvalue of submatrix as variable. Numerical examples demonstrate the effects of shape parameters.
