Convergent inversion approximations for polynomials in Bernstein form Original Research Article

Given a monotone polynomial function λ=f(t) in Bernstein form on t∈[ 0,1 ], we formulate an algorithm that computes a sequence of rapidly convergent Bernstein-form polynomial approximations Pn(λ) to the inverse t=f−1(λ) of this function. The method is based upon a least-squares minimization of the error en(λ)=f−1(λ)−Pn(λ), using the orthogonal Legendre polynomials to yield an uncoupled sequential computation of the least-squares coefficients, and it requires only the standard arithmetic, degree elevation, and composition algorithms for polynomials in Bernstein form. An extension of the method to error minimization under the constraint of fixed end values is also presented. As an example application, we show that the algorithm can be used to derive representations for polynomial PH curves that come arbitrarily close to the ideal of arc-length parameterization.