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.


Introduction
Trigonometric splines and polynomials have attracted widespread interest within computer aided geometric design (CAGD), particularly within curve design.In [1], the recurrence relation for the trigonometric B-splines of arbitrary order was established.Later, in [2], it was further shown that the trigonometric B-splines of odd order form a partition of a constant in the case of equidistant knots, and thus the associated trigonometric B-spline curve possesses the convex hull property.In [3], a family of trigonometric polynomials was introduced, which contains the trigonometric Lagrange and Bernstein polynomials.The study for the C-Bézier and C-B-spline curves can be found in [4][5][6].In [7], the totally positive property of the C-B-spline was proved.In [8][9][10][11], some quadratic trigonometric polynomial splines with shape parameters were shown.In [12], a class of cubic trigonometric Bézier (T-Bézier, for short) basis with a shape parameter was proposed.In [13], the cubic T-Bézier basis was further extended to possess two shape parameters.Recently, based on the theory of envelop and topological mapping, shape analysis of the cubic T-Bézier curve with shape parameters was given in [14,15].For the problems of shape preserving interpolation, the cubic and quadratic T-Bézier bases show great potential applications; see [16][17][18][19].Blossom is a powerful tool for studying Bézier-like bases and B-spline-like bases.Quasi Extended Chebyshev (QEC) space is the largest class of spaces of sufficient regularity suited for curve design in the sense that they do possess blossom.See the recently developed theory concerning blossom and QEC-space given in [20][21][22][23][24].
Many shape preserving properties are obtained when the normalized bases are totally positive; see [25].For instance, the length, number of inflections, and angular variation of the curve defined by normalized totally positive bases are bounded above by those of the control polygon.The optimal normalized totally positive basis (i.e., the normalized B-basis) is the unique normalized basis of a space with optimal shape preserving properties.It is well known that the Bernstein basis is the optimal normalized totally positive basis of the space of polynomials of degree less than or equal to ; see [26].Corner cutting algorithm plays a vital role in CADG since it provides simple geometric constructions of curves; see [27].A relevant example is the de Casteljau algorithm for the evaluation of a polynomial curve using the Bernstein basis.The purpose of this paper is to prove that the cubic trigonometric Bézier basis with two shape parameters  and  given in [13] forms an optimal normalized totally positive basis for ,  ∈ (−2, 1] and to show a new developed corner cutting algorithm for computing the corresponding cubic trigonometric Bézier curves.
The rest of this paper is organized as follows.Section 2 gives the proof of the optimal normalized totally positive property of the cubic trigonometric Bézier basis within the general framework of QEC-space.In Section 3, a new developed corner cutting algorithm for computing the corresponding cubic trigonometric Bézier curves is shown.Conclusions are given in Section 4.

Cubic Trigonometric Bézier Basis Functions
2.1.Preliminaries.Here, we give the necessary background on Extended Completed Chebyshev (ECC) space and Quasi Extended Chebyshev (QEC) space.Few results are described briefly for a good understanding to this paper, and more details can be found in [20][21][22][23][24]28].
A basis ( 0 , . . .,   ) is said to be normalized on [, ] if ∑  =0   () = 1 for any  ∈ [, ].And a basis ( 0 , . . .,   ) is said to be totally positive on [, ] if, for any sequence of points  ≤  0 <  1 < ⋅ ⋅ ⋅ <   ≤ , the collocation matrix (  (  )) 0≤,≤ is totally positive; that is, all its minors are nonnegative.For a given function space possessing totally positive basis, the optimal normalized totally positive basis (i.e., the normalized B-basis) is a normalized totally positive basis from which all other totally positive bases can be deduced by multiplying with a (regular) totally positive matrix.Such optimal normalized totally positive basis is unique and has optimal shape preserving properties (see [25]) in the sense that the curves defined by such basis best imitate the corresponding control polygons (e.g., the monotonicity and convexity of the control polygon are inherited by the curves defined by such basis).

Definition of the Cubic Trigonometric Bézier Basis.
Here, we recall the definition of the cubic trigonometric Bézier basis with two shape parameters given in [13] as follows.
By rewriting the expressions of  1 () and  2 () as the following forms: we can easily see that the cubic T-Bézier basis (2) is normalized and is actually a basis of the following trigonometric function space: (4)

Optimal Totally Positive
Property of the Basis.Firstly, we want to show that for any ,  ∈ (−2, 1], the following space is a 3-dimensional QEC-space on [0, /2].
Proof.For any ,  ∈ (−2, 1], from the expressions of the four cubic T-Bézier basis functions given in (2), we can easily check the following end-point properties of the basis.
In addition,   () ( = 0, 1, 2, 3) is strictly positive in (0, /2).Thus, by Definition 2.10 of [22], we can see that the cubic T-Bézier basis (2) is precisely the Quasi Bernstein-like basis of the space  , .And by Theorem 2.18 of [22], we can further conclude that the cubic T-Bézier basis (2) is exactly the optimal normalized totally positive basis of the space  , .Furthermore, by setting The expressions ( 16) and ( 18) describe a corner cutting algorithm for computing the cubic T-Bézier curve (14).See Figure 1 for an illustration of this new developed algorithm.

Conclusion
The cubic trigonometric Bézier basis with two shape parameters  and  given in [13] forms an optimal normalized totally positive basis of the space  , for ,  ∈ (−2, 1].However, the basis is not suited for curve design for  = −2 or  = −2 from the blossom point of view.The new developed corner cutting algorithm is an efficient and stable process for computing the corresponding cubic trigonometric Bézier curves. ; , ) = (1 − sin 2 )  2 0 + (1 − cos 2 )  2 1 .