Total positivity of the cubic trigonometric Bézier basis. (English) Zbl 1406.65011

Han, Xuli; Zhu, Yuanpeng

J. Appl. Math. 2014, Article ID 198745, 5 p. (2014).

Summary: 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 [the first author et al., Appl. Math. Lett. 22, No. 2, 226–231 (2009; Zbl 1163.65307)] forms an optimal normalized totally positive basis for \(\lambda,\mu\in(-2,1]\). Moreover, we show that for \(\lambda=-2\) or \(\mu=-2\) 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.

MSC:

65D17

Computer-aided design (modeling of curves and surfaces)

Citations:

Zbl 1163.65307