S-λ bases and S-λ curves

In this paper, a class of discrete distributions called [email protected] distributions is presented and the corresponding [email protected] basis functions are constructed from these distributions by means of the technique of generating functions and transformation factors. These basis functions generate [email protected] curves. We show that [email protected] curves include Bezier curves, Poisson curves, rational Bezier curves and a lot of other curves. Therefore the research in this paper provides a unified scheme for dealing with these well-known curves. We study some important properties of the [email protected] basis functions and [email protected] curves. Furthermore, by means of the technique of generating functions, a new convenient and practical method for local changes of [email protected] curves is proposed.

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

[2]  L Hahn,et al.  Approximation by operators of probabilistic type , 1980 .

[3]  Ron Goldman,et al.  A subdivision scheme for Poisson curves and surfaces , 2000, Comput. Aided Geom. Des..

[4]  J. Delacal.  On Stancu-Mühlbach Operators and Some Connected Problems Concerning Probability Distributions , 1993 .

[5]  Guozhao Wang,et al.  A class of Bézier-like curves , 2003, Comput. Aided Geom. Des..

[6]  Xiao-Ming Zeng,et al.  Approximation properties of Gamma operators , 2005 .

[7]  Ron Goldman,et al.  Poisson approximation , 2000, Proceedings Geometric Modeling and Processing 2000. Theory and Applications.

[8]  Ron Goldman,et al.  Urn models and B-splines , 1988 .

[9]  Xiao-Ming Zeng,et al.  Bézier curves and surfaces with shape parameters , 2009, Int. J. Comput. Math..

[10]  Ron Goldman,et al.  Pyramid algorithms - a dynamic programming approach to curves and surfaces for geometric modeling , 2002, Morgan Kaufmann series in computer graphics and geometric modeling.

[11]  Xiao-Ming Zeng,et al.  On the Rates of Approximation of Bernstein Type Operators , 2001, J. Approx. Theory.

[12]  R. Khan,et al.  Some probabilistic methods in the theory of approximation operators , 1980 .

[13]  Wanqiang Shen,et al.  A basis of multi-degree splines , 2010, Comput. Aided Geom. Des..

[14]  A. W. Kemp,et al.  Univariate Discrete Distributions , 1993 .

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

[16]  Ron Goldman,et al.  The rational Bernstein bases and the multirational blossoms , 1999, Comput. Aided Geom. Des..