Multi-degree reduction of tensor product Bézier surfaces with general boundary constraints

Abstract We propose an efficient approach to the problem of multi-degree reduction of rectangular Bezier patches, with prescribed boundary control points. We observe that the solution can be given in terms of constrained bivariate dual Bernstein polynomials. The complexity of the method is O ( mn 1 n 2 ) with m ≔ min(m1, m2), where (n1, n2) and (m1, m2) is the degree of the input and output Bezier surface, respectively. If the approximation—with appropriate boundary constraints—is performed for each patch of several smoothly joined rectangular Bezier surfaces, the result is a composite surface of global Cr continuity with a prescribed r ⩾ 0. In the detailed discussion, we restrict ourselves to r ∈ {0, 1}, which is the most important case in practical application. Some illustrative examples are given.

[1]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[2]  Pawel Wozny,et al.  Constrained multi-degree reduction of triangular Bézier surfaces using dual Bernstein polynomials , 2010, J. Comput. Appl. Math..

[3]  Lizheng Lu.  Approximating tensor product Bézier surfaces with tangent plane continuity , 2009, J. Comput. Appl. Math..

[4]  Matthias Eck,et al.  Degree Reduction of Bézier Surfaces , 1992, IMA Conference on the Mathematics of Surfaces.

[5]  Josef Hoschek,et al.  Fundamentals of computer aided geometric design , 1996 .

[6]  W. N. Bailey.  Contiguous Hypergeometric Functions of the Type 3F2(1) , 1954 .

[7]  Pawel Wozny,et al.  Dual generalized Bernstein basis , 2006, J. Approx. Theory.

[8]  Guozhao Wang,et al.  Multi-degree reduction of triangular Bézier surfaces with boundary constraints , 2006, Comput. Aided Des..

[9]  Lian Zhou,et al.  Constrained multi-degree reduction of Bézier surfaces using Jacobi polynomials , 2009, Comput. Aided Geom. Des..

[10]  Qian-qian Hu,et al.  Optimal multi-degree reduction of triangular Bézier surfaces with corners continuity in the norm L2 , 2008 .

[11]  Hu Shimin,et al.  Approximate degree reduction of triangular bezier surfaces , 1998 .

[12]  Pawel Wozny,et al.  Multi-degree reduction of Bézier curves with constraints, using dual Bernstein basis polynomials , 2009, Comput. Aided Geom. Des..

[13]  Lizheng Lu.  A note on constrained degree reduction of polynomials in Bernstein-Bézier form over simplex domain , 2009 .

[14]  Matthias Eck,et al.  Least squares degree reduction of Bézier curves , 1995, Comput. Aided Des..

[15]  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.

[16]  Rene F. Swarttouw,et al.  The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue Report Fac , 1996, math/9602214.

[17]  J. A. Gregory.  The Mathematics of Surfaces. , 1987 .

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

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

[20]  Young Joon Ahn,et al.  Constrained degree reduction of polynomials in Bernstein-Bézier form over simplex domain , 2008 .

[21]  R. Askey.  Orthogonal Polynomials and Special Functions , 1975 .

[22]  Wang Guo-jin,et al.  A novel algorithm for explicit optimal multi-degree reduction of triangular surfaces , 2008 .