Optimal multi-degree reduction of triangular Bézier surfaces with corners continuity in the norm L2

This paper derives an approximation algorithm for multi-degree reduction of a degree n triangular Bezier surface with corners continuity in the norm L"2. The new idea is to use orthonormality of triangular Jacobi polynomials and the transformation relationship between bivariate Jacobi and Bernstein polynomials. This algorithm has a very simple and explicit expression in matrix form, i.e., the reduced matrix depends only on the degrees of the surfaces before and after degree reduction. And the approximation error of this degree-reduced surface is minimum and can get a precise expression before processing of degree reduction. Combined with surface subdivision, the piecewise degree-reduced patches possess global C^0 continuity. Finally several numerical examples are presented to validate the effectiveness of this algorithm.

[1]  Gengzhe Chang,et al.  Converse theorems of convexity for Bernstein polynomials over triangles , 1990 .

[2]  Karsten Opitz,et al.  Hybrid Cubic Bézier Triangle Patches , 1992 .

[3]  Leon M. Hall,et al.  Special Functions , 1998 .

[4]  Abedallah Rababah,et al.  Distance for degree raising and reduction of triangular Bézier surfaces , 2003 .

[5]  Jasper V. Stokman,et al.  Orthogonal Polynomials of Several Variables , 2001, J. Approx. Theory.

[6]  Abedallah Rababah,et al.  L-2 Degree Reduction of Triangular Bézier Surfaces with Common Tangent Planes at Vertices , 2005, Int. J. Comput. Geom. Appl..

[7]  Yuesheng Xu,et al.  Degree reduction of Bézier curves by uniform approximation with endpoint interpolation , 1995, Comput. Aided Des..

[8]  Gerald Farin,et al.  Curves and surfaces for cagd , 1992 .

[9]  Philip J. Davis,et al.  The convexity of Bernstein polynomials over triangles , 1984 .

[10]  G. Farin.  Curves and Surfaces for Cagd: A Practical Guide , 2001 .

[11]  Wolfgang Böhm,et al.  A survey of curve and surface methods in CAGD , 1984, Comput. Aided Geom. Des..

[12]  Wen Lea Pearn,et al.  (Journal of Computational and Applied Mathematics,228(1):274-278)Optimization of the T Policy M/G/1 Queue with Server Breakdowns and General Startup Times , 2009 .

[13]  Guodong Chen,et al.  Multi-degree reduction of tensor product Bézier surfaces with conditions of corners interpolations , 2008, Science in China Series F: Information Sciences.

[14]  Guozhao Wang,et al.  Perturbing B ezier coefficients for best constrained degree reduction in the L 2-norm , 2003 .

[15]  Paweł Woźny,et al.  Connections between two-variable Bernstein and Jacobi polynomials on the triangle , 2006 .

[16]  孙家广,et al.  Approximate Degree Reduction of Triangular Bezier Surfaces , 1998 .

[17]  Stephen Mann,et al.  Continuity Adjustments to Triangular Bézier Patches That Retain Polynomial Precision , 2000 .

[18]  Guozhao Wang,et al.  Perturbing Bézier coefficients for best constrained degree reduction in the L2-norm , 2003, Graph. Model..

[19]  Guodong Chen,et al.  Optimal multi-degree reduction of Bézier curves with constraints of endpoints continuity , 2002, Comput. Aided Geom. Des..

[20]  Young Joon Ahn,et al.  Constrained polynomial degree reduction in the L2-norm equals best weighted Euclidean approximation of Bézier coefficients , 2004, Comput. Aided Geom. Des..

[21]  Horst Nowacki,et al.  Approximate conversion of surface representations with polynomial bases , 1985, Comput. Aided Geom. Des..

[22]  Stephen Mann.  Implementation of Some Triangular Data Fitting Schemes Using Averaging To Get Continuity , 2000 .