Application of Chebyshev II–Bernstein basis transformations to degree reduction of Bézier curves

Abstract

A polynomial curve on [0, 1] can be expressed in terms of Bernstein polynomials and Chebyshev polynomials of the second kind. We derive the transformation matrices that map the Bernstein and Chebyshev coefficients into each other, and examine the stability of this linear map. In the p = 1 and ∞ norms, the condition number of the Chebyshev-Bernstein transformation matrix grows at a significantly slower rate with n than in the power-Bernstein case, and the rate is very close (somewhat faster) to the Legendre-Bernstein case. Using the transformation matrices, we present a method for the best multi-degree reduction with respect to the t − t 2 -weighted square norm for the unconstrained case, which is further developed to provide a good approximation to the best multi-degree reduction with constraints of endpoints continuity of orders r, s (r, s ≥ 0). This method has a quadratic complexity, and may be ill-conditioned when it is applied to the curves of high degree. We estimate the posterior L 1 -error bounds for degree reduction.

An Approximation Method of Bézier Curve

“…For instance in reference (L Z. Lu, and G Z Wang 2008) propose to utilize the Chebyshev Polynomials for base function conversion. In reference (Sánchez-Reyes J 1997), Sánchez-Reyes uses the S power base to accurately represent the Bernstein base function.…”

Approximate Multidegree Reduction ofλ-Bézier Curves

“…The issue of degree reduction of Bézier curves is concerned with the solution of the following problem: for a given Bézier curve R ( ) of degree with Bézier points {r } =0 , find an approximate Bézier curveR ( ) of lower degree , where < , with the set of Bézier points {r } =0 , so that R andR satisfy boundary conditions at the end points, and the error between R andR is minimum. For degree reduction of Bézier curves, many scholars have done a lot of research that can be classified into three categories: geometry of approximate control point [5][6][7][8], algebraic means of basis function transformations [9][10][11][12][13][14], and B net and constrained optimization [15,16]. Watkins and Worsey [9] presented an algorithm for generating ( − 1)st degree approximation to th degree Bézier curve.…”

“…Zheng and Wang [12] proved that the problem of finding a best 2 -approximation over the interval [0,1] for constrained degree reduction is equivalent to that of finding a minimum perturbation vector in a certain weighted Euclidean norm. Using the transformation matrices, Lu and Wang [13] presented a method for the best multidegree reduction with respect to √ − 2 -weighted square norm for the unconstrained case. Tan and Fang [14] proposed three methods for degree reduction of interval generalized Ball curves of Wang-Said type.…”

The transformation matrix of Chebyshev III – Bernstein polynomial basis

“…Explicit forms of bases' transformations have been derived; between Legendre and Bernstein bases in [19], between Chebyshev polynomials of the first kind and the Bernstein bases in [8], between Jacobi and Bernstein polynomial bases in [6], between the Chebyshev of the second kind and the Bernstein bases in [16], and between the Chebyshev of the Third kind and the Bernstein bases in [2]. Applications to bases transformations can be found in [3], [7], [5], [4].…”