Total Positivity of the Cubic Trigonometric Bézier Basis

Abstract

Within the general framework of Quasi Extended Chebyshev space, we prove that the cubic trigonometric Bézier basis with two shape parameters $\lambda$ and $\mu$ given in Han et al. (2009) forms an optimal normalized totally positive basis for $\mathrm{\lambda ,}\mu \in (-\mathrm{2,1}]$. Moreover, we show that for $\lambda =-\mathrm{2}$ or $\mu =-\mathrm{2}$ the basis is not suited for curve design from the blossom point of view. In order to compute the corresponding cubic trigonometric Bézier curves stably and efficiently, we also develop a new corner cutting algorithm.