If n given control, points b_0,…b_(n-1)∈R^d are repeated periodically by b_(i+kn)=b_i, for all k∈Z. the uform limit of the Bernstein-Bezier polynomial curves of degree r with control points b_0,….b_ for r→∞ is a...If n given control, points b_0,…b_(n-1)∈R^d are repeated periodically by b_(i+kn)=b_i, for all k∈Z. the uform limit of the Bernstein-Bezier polynomial curves of degree r with control points b_0,….b_ for r→∞ is a Poisson curve(after a suitable reparametrization). This fact reveals some interesting self-simi- lar structures in case of regular n-gons in the plane.展开更多
In this note, we establish a new formulation of smoothness conditions for piecewise polynomial (: =pp) functions in terms of the B-net representation in the general n-dimensional setting. It plays an important role fo...In this note, we establish a new formulation of smoothness conditions for piecewise polynomial (: =pp) functions in terms of the B-net representation in the general n-dimensional setting. It plays an important role for 2-dimensional setting in the constructive proof of the fact that the spaces of polynomial splines with smoothness rand total degree k≥3r+2 over arbitrary triangulations achieve the optimal approximation order with the approximation constant depending only on k and the smallest angle of the partition in [5].展开更多
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-Bal...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.展开更多
文摘If n given control, points b_0,…b_(n-1)∈R^d are repeated periodically by b_(i+kn)=b_i, for all k∈Z. the uform limit of the Bernstein-Bezier polynomial curves of degree r with control points b_0,….b_ for r→∞ is a Poisson curve(after a suitable reparametrization). This fact reveals some interesting self-simi- lar structures in case of regular n-gons in the plane.
文摘In this note, we establish a new formulation of smoothness conditions for piecewise polynomial (: =pp) functions in terms of the B-net representation in the general n-dimensional setting. It plays an important role for 2-dimensional setting in the constructive proof of the fact that the spaces of polynomial splines with smoothness rand total degree k≥3r+2 over arbitrary triangulations achieve the optimal approximation order with the approximation constant depending only on k and the smallest angle of the partition in [5].
基金Supported by the National Natural Science Foundation of China(60970097 and 11271376)Postdoctoral Science Foundation of China(2015M571931)Graduate Students Scientific Research Innovation Project of Hunan Province(CX2012B111)
文摘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.
