Within the general framework of Quasi Extended Chebyshev space, we prove that the cubic trigonometric Bézier basis with two shape parametersλandμgiven in Han et al. (2009) forms an optimal normalized totally positive basis forλ,μ∈(-2,1]. Moreover, we show that forλ=-2orμ=-2the 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.