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.