Abstract
The S-λ Curves have become an important research subject in computer aided geometric design (CAGD), which owes to its good geometric properties (such as affine invariance, symmetry, and locality). This paper presents a new method to approximate an S-λ curve of degree n by using an S-λ curve of degree n-1. We transform this degree reduction problem into the function optimization problem first, and then using a new genetic simulated annealing algorithm to determine the global optimal solution of the optimization problem. The method can be used to approximate S-λ curves with fixed or unconstrained endpoints. Examples are given to verify the effectiveness of the presented algorithm; and these numeric examples show that the algorithm is not only easy to implement, but also offers high precision, which makes it valuable in practical applications.
Highlights
Many methods used in Computer Aided Geometric Design (CAGD) are closely associated with probability distributions, discrete distributions (Johnson et al, 2005 [1])
The Bernstein basis functions used in Bézier curves are related to the binomial distribution, the B-spline basis functions used in B-spline curves are connected with some stochastic processes (Dahmen et al, 1986 [2], Goldman et al, 2000 [3], Hu et al, 2018 [4,5,6,7]), the Poisson basis functions introduced by Goldman and Morin (2000) were developed from the Poisson distribution, and the Bernstein basis functions of negative degree presented by Goldman (1999) are taken from the negative binomial distribution [8]
These basis functions and other basis functions generated from discrete distributions, are important in the approximation process based on approximation theory
Summary
Many methods used in Computer Aided Geometric Design (CAGD) are closely associated with probability distributions, discrete distributions (Johnson et al, 2005 [1]). The Bernstein basis functions used in Bézier curves are related to the binomial distribution, the B-spline basis functions used in B-spline curves are connected with some stochastic processes (Dahmen et al., 1986 [2], Goldman et al, 2000 [3], Hu et al, 2018 [4,5,6,7]), the Poisson basis functions introduced by Goldman and Morin (2000) were developed from the Poisson distribution, and the Bernstein basis functions of negative degree presented by Goldman (1999) are taken from the negative binomial distribution [8] These basis functions and other basis functions generated from discrete distributions, are important in the approximation process based on approximation theory The first class is geometric method based on control points, which is an inverse process of degree elevation and has been used to determine the control points of degree-reduced curves (Ren et al, 2007 [18]).
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