A Class of Trigonometric Bernstein-Type Basis Functions with Four Shape Parameters

In this work, a family of four new trigonometric Bernstein-type basis functions with four shape parameters is constructed, which form a normalized basis with optimal total positivity. Based on the new basis functions, a kind of trigonometric Bézier-type curves with four shape parameters, analogous to the cubic Bézier curves, is constructed. With appropriate choices of control points and shape parameters, the resulting trigonometric Bézier-type curves can represent exactly any arc of an ellipse or parabola. The four shape parameters have tension control roles on adjusting the shape of resulting curves. Moreover, a new corner cutting algorithm is also proposed for calculating the trigonometric Bézier-type curves stably and efficiently.


Introduction
In computer aided geometric design and computer graphics, parametric curves and surfaces are often expressed by linearly combining control points and basis functions.Generally speaking, basis functions with good properties play a vital role in parametric curves and surfaces design.For instance, if the basis functions have partition of unity, nonnegativity, and total positivity, the resulting parametric curves will possess affine invariance property, convex hull property, and variation diminishing property, which are important in curves design.In engineering, the classical B-spline basis functions have been widely applied in modeling parametric curves; see [1,2].However, with the fixed knot vectors, the shape of B-spline curves is determined totally by their control points.One may use the weights in the nonuniform rational B-spline curves to modify the shape of the resulting parametric curves; however, rational form may be unstable and its derivatives and integrals are hard to compute.
In order to adjust the shape of the parametric curves flexibly, some basis functions with shape parameters have been proposed; see [3][4][5].These methods have a common idea that new basis functions are constructed by incorporating shape parameters into the classical Bézier or B-spline basis functions.In [6,7], quadratic and cubic trigonometric Bernstein-type basis functions with shape parameters were shown.In [8], a kind of cubic trigonometric Bernsteintype basis functions possessing two shape parameters was presented, which includes the cubic trigonometric Bernsteintype basis functions with a shape parameter given in [6] as a special case.In [9], shape analysis of the cubic trigonometric Bézier-type curve with a shape parameter given in [6] was presented by using the theory of envelop and topological mapping.Later, in [10], the totally positive property of the cubic trigonometric Bernstein-type basis functions with two shape parameters given in [8] was proved, which implies that the cubic trigonometric Bernstein-type basis with two shape parameters is suitable for conformal design.Recently, a class of cubic trigonometric nonuniform B-spline basis functions having a local shape parameter was proposed in [11], which is an extension of the cubic trigonometric nonuniform spline basis functions with a global shape parameter given in [6].In [12], a class of C-Bézier basis of the space span{1, , sin , cos } was constructed, where the length of the interval serves as shape parameter.Later, in [13], geometric interpretation of the change of the shape parameter on C-Bézier curves was given.In [14], it was proved that the critical length for the span{1, , sin , cos } is 2, which implies that in the span{1, , sin , cos }, Extended Complete Chebyshev-(ECC-) system exists only on interval of length less than 2.Later, in [15], it was shown that this restriction can be overcome by replacing ECC-system with the Canonical Complete Chebyshev-(CCC-) system.
For controlling the parametric curves efficiently, basis functions with tension shape parameters have aroused great interest among the researchers.In [16], a class of polynomial splines with variable degree was constructed in the space spanned by span{1, , (1 − )  ,   }, in which  and  serve as tension shape parameters.In [17,18], it was proved that the polynomial splines with variable degree form a Quasi Extended Chebyshev-(QEC-) system.Later, in [19], the approximation power, the existence of a normalized B-basis, and the structure of a degree-raising process for spaces of the form {1, ,  2 , . . .,  −2 , (), V()} were given.Within the general framework of QEC-system, the dimension elevation algorithm for the space {1, ,  2 , . . .,  −2 , (1 − )  ,   } was studied via blossom theory; see [20][21][22][23].Recently, in [24], the total positivity of the polynomial splines with variable degree was proved based on the theory of CCC-systems.The variable degree polynomial splines have been widely used for constructing shape preserving interpolation and approximation splines; see [25][26][27].In [28], based on some truncated polynomial functions, the explicit representations of changeable degree spline basis functions were given.In [29], a kind of five trigonometric blending functions with two exponential shape parameters  and  was proposed in the space spanned by span{1, sin (1 − sin ) −1 , cos (1 − cos ) −1 , (1 − sin )  , (1 − cos )  }.Later, in [30], a generalization of these five trigonometric blending functions was presented.Some exponential splines and rational splines with tension shape parameters have also developed for curve design, see [31][32][33] for example.Recently, four trigonometric Bernstein-type basis functions of the space spanned by span{1, sin 2 , (1−sin )  , (1−cos )  } were constructed in [34], which form a normalized basis with optimal total positivity.In [35], a family of rational trigonometric basis functions with denominator shape parameters of the space spanned by span{1, sin 2 The purpose of this paper is to present four new trigonometric Bernstein-type basis functions constructed in the space spanned by span{1, sin 2 , (1 − sin )  (1 −  sin ), (1 − cos )  (1 −  cos )}, which form a normalized optimal totally positive basis and include the bases given in [6][7][8]34] as special cases.The parametric curves constructed by this new basis have shape preserving property.The four shape parameters , , , and  have tension control property on modifying the shape of parametric curves.Compared with the four polynomial Bernstein-type basis functions with variable degree constructed in the space span{1, , (1−)  ,   } ( see [16]), the new constructed trigonometric Bernsteintype basis functions have more computation complexity for the same degree, while any arc of an ellipse or parabola can be represented exactly by using the new trigonometric Bernstein-type basis functions.And compared with the four rational trigonometric basis functions with two denominator shape parameters constructed in the space spanned by span{1, sin 2  [35]), the new constructed trigonometric Bernstein-type basis functions possess four shape parameters and thus have more flexibility in free-form curves shape design.
The rest of this paper is organized as follows.In Section 2, the construction and properties of the trigonometric Bernstein-type basis functions are given.Section 3 gives the definition and properties of the trigonometric Béziertype curves.For computing the trigonometric Bézier-type curves stably and efficiently, a new corner cutting algorithm is developed.Comparison between the trigonometric Béziertype curves and the variable degree Bézier polynomial curves given in [16] is shown.And tensor product Bézier-type patches are also shown.Conclusions are given in Section 4.

Trigonometric Bernstein-Type
Basis Functions The corresponding mother-function is given as follows: We shall prove the totally positive property of the new basis functions by using the theory of Quasi Extended Chebyshev (QEC) space.The related concepts concerning ECC-space, QEC-space, blossom, and Quasi Bernstein-type basis can be found in [17,18,[20][21][22][23]34].
In the second step, we further prove that the space  ,,, forms a QEC-space on [0, /2].To this end, we need to prove that any nonzero element of the space  ,,, has at most 2 roots on [0, /2] (keep it in mind that in a QEC-space, we count multiplicities as far as possible up to 2).Consider any nonzero function where  ∈ [0, /2].Since the space  ,,, is an ECC-space in (0, /2), () has at most two roots in (0, /2).Suppose that the function () has a root at 0; then we get  1 = 0.In this case, if  2 = 0, then () has a singular root at 0 and a singular root at /2.If  0 = 0, we can check that 0 is a double root of () (we count multiplicities as far as possible up to 2).If  0  2 > 0, () has singular one root at 0 and it does not vanish anywhere on (0, /2].If  0  2 < 0, () has singular one root at 0 and it does not vanish at /2.Moreover, for the following function by directly computing, we obtain and it follows that () is a monotone increasing function on [0, /2].From these together with (0)(/2) = 2 0  2 ( + ) < 0, we can see that () has exactly singular one root in (0, /2); thus we can immediately conclude that () = sin () (notice that  1 = 0 for the current case) has exactly one root in (0, /2).Similarly, for the case that () has a root at /2, we can also derive that the function () has at most 2 roots on [0, /2] (we count multiplicities as far as possible up to 2).Summarizing the above analysis, we can conclude that the space  , is a QEC-space on [0, /2].
Since the space  ,,, forms a QEC-space on [0, /2], from Theorem 3.1 of [21], we can conclude that blossom exists in  ,,, , which indicates that the new space  ,,, is suited for curve design.In addition, from Theorem 2.18 of [21], we can also know that the space  ,,, has a normalized basis of Quasi Bernstein-type on [0, /2].In the next Theorem 3, we will compute the associated Chebyshev-Bézier points of the mother-function Φ() defined in (2) and construct the associated trigonometric Bernstein-type basis   fl   (0,/2) of the space  ,,, .Before further discussion, we want to prove the following lemma, which will be used to discuss the positivity of the trigonometric Bernstein-type basis.
It can be easily checked that the new basis functions have the following important end-point property.

Construction of the Trigonometric
Bézier-Type Curves is called a trigonometric Bézier-type (TB-type for short) curve with four shape parameters , , , and .

Shape Control and Corner Cutting Algorithm.
For  ∈ [0, /2], we rewrite the expression of the TB-type curve (24) as the following form: Obviously,  0 (t; , ) decreases with the increase of  or  for any fixed  ∈ (0, /2).This implies that the resulting TBtype curve moves in the same direction of the edge  0 −  1 as  or  increases.On the contrary, when  or  decreases, the resulting TB-type curve will move in the opposite direction to the edge  0 −  1 .The shape parameters  and  have the similar effects on the edge  3 −  2 .As ,  or ,  increase, respectively, the TB-type curve will tend to the point  1 or  2 , respectively.And when the shape parameters satisfy  = ,  = , the TB-type curve will move in the same direction or the opposite direction to the edge  2 − 1 when  or  increases or decreases, respectively.These imply that the four shape parameters , , , and  serve as local tension parameters.Figures 2 and 3 give some examples that the shape of TB-type curves can be modified conveniently by using the four shape parameters under the fixed control points.

The Representation of Elliptic and Parabolic Arcs.
In this subsection, we shall show that with appropriated choices of control points and shape parameters, any arc of an ellipse or parabola can be represented exactly by using the new TB-type curves (; , , , ) given in (24).

Approximation.
Control polygons provide an important tool in geometric modeling.It is an advantage if the curve being modeled tends to preserve the shape of its control polygon.Now we show some relations of the TB-type curves (24) and the cubic Bézier curves corresponding to their control polygons.Theorem 6. Suppose the control points  0 ,  1 ,  2 ,  3 are not collinear.For  ∈ [0,1], the relationships between where  * = ( 1 +  2 )/2, ℎ(, , , ) = max{(, ), (, )}, and Proof.By direct computation, we have (0; , , , ) =  0 = (0), (1; , , , These imply the theorem. From Theorem 6, we can see that TB-type curve ((/2); , , , ) is closer to the control polygon than the cubic Bézier curve if ℎ(, , , ) < 1. Figure 6 shows the comparison between the TB-type curves and the classical cubic Bézier curves under the same control points.It can be seen that as  =  increases at the same time, the resulting TB-type curves will be totally more close to the control polygon than the cubic Bézier curves.These indicate that the TB-type curves can better maintain the characteristic of the control polygon than the cubic Bézier curves.
Figure 7 shows the comparison between the TB-type curves and the polynomial Bézier curves with variable degree generated by the four polynomial Bernstein-like basis functions with variable degree given in the space spanned span{1, , (1 − )  ,   } given in [16] under the same control points.For the same degree, the TB-type curves have more computation complexity than the variable degree polynomial Bézier curves; however the resulting TB-type curves (solid lines) are nearer to the control polygon than the variable degree polynomial Bézier curves (dashed lines).
Figure 8 shows the comparison between the TB-type curves and the DTB-like curves generated by the four rational trigonometric basis functions constructed in the rational trigonometric space span{1, sin 2 , (1 [35] under the same control points.From Figure 8, clearly, the TB-type curves (solid lines) are closer to the control polygon than the DBT-like curves (dashed lines).

Composite Trigonometric Bézier-Type Curves.
In practical curve design, we often composite several TB-type curves to generate curves with complex shapes.In piecing TBtype curves together, we need to handle the smoothness connection conditions of the resulting curve.Give two TBtype curve segments as follows: and It is obvious that if the point  3 =  0 , the two segments form a curve with  0 continuity.We shall derive some sufficient smoothness connection conditions for two TB-type curve segments forming a curve with  1 or  2 continuity.For knots  1 <  2 <  3 , we denote the resulting curve () constructed by ( 36) and (37) as follows: where ℎ  =  +1 −   ,  = 1, 2.
Theorem 7.For   ,   ∈ [1, +∞),  = 1, 2, the resulting curve () is  1 continuous at the knot  2 , if the following condition holds  And for   ,   ∈ (1, +∞),  = 1, 2, () is  2 continuous at the knot  2 if the condition (39) together with the following condition holds at the same time Proof.After some direct computation, we get Thus under the conditions of Theorem 7, we can easily conclude the result.
Since the univariate TB-type basis functions given in (19) can represent exactly arc of an ellipse or parabola, we expect that the associated TB-type patch can represent exactly ellipsoid patch and paraboloid patch if the control points and the shape parameters are chosen appropriately.In fact, suppose we choose control points of the TB-type patch as  00 = (0, 0, ) ,  01 = (0, 0, ) ,  02 = (0, 0, ) ,  03 = (0, 0, ) , in which , ,  > 0, and take shape parameters  Tensor product TB-type patches (, V) have similar properties to that of TB-type curves given in (24) and we omit the details. Figure 11 shows some TB-type patches with different shape parameters under the fixed control points.

Conclusion
The four new developed trigonometric Bernstein-type basis functions with four shape parameters form a normalized basis with optimal total positivity and are useful for constructing parametric curves in CAGD, which include the bases given in [6][7][8]34] as special cases.The four shape parameters have tension control property on modifying the shape of curves.By using the four shape parameters, the resulting trigonometric Bézier-type curves can be nearer to the given control polygon than the cubic Bézier curves.With appropriated choices of control points and shape parameters, any arc of an ellipse or parabola can be represented exactly by using the trigonometric Bézier-type curves.The new proposed corner cutting algorithm is useful for calculating the trigonometric Bézier-type curves efficiently and stably.There are also some work worthy of further study, such as subdivide algorithm for the new trigonometric Bézier-type curves.These will be our future work.

Figure 4 :
Figure 4: Corner cutting algorithm with different shape parameters.

Figure 5 :
Figure 5: The representation of elliptic and parabolic arcs.