Total Positivity of the Cubic Trigonometric Bézier Basis

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

Geometric properties and constrained modification of trigonometric spline curves of Han

New types of quadratic and cubic trigonometrial polynomial curves have been introduced in [2] and [3]. These trigonometric curves have a global shape parameter λ. In this paper the geometric effect of this shape parameter on the curves is discussed. We prove that this effect is linear. Moreover we show that the quadratic curve can interpolate the control points at λ = √2. Constrained modification of these curves is also studied. A curve passing through a given point is computed by an algorithm which includes numerical computations. These issues are generalized for surfaces with two shape parameters. We show that a point of the surface can move along a hyperbolic paraboloid

The Trigonometric Polynomial Like Bernstein Polynomial

A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are constructed. Some interesting properties of the trigonometric polynomials are given