New Trigonometric Basis Possessing Denominator Shape Parameters

Four new trigonometric Bernstein-like bases with two denominator shape parameters (DTB-like basis) are constructed, based on which a kind of trigonometric Bézier-like curve with two denominator shape parameters (DTB-like curves) that are analogous to the cubic Bézier curves is proposed. The corner cutting algorithm for computing the DTB-like curves is given. Any arc of an ellipse or a parabola can be exactly represented by using the DTB-like curves. A new class of trigonometric B-spline-like basis function with two local denominator shape parameters (DT B-spline-like basis) is constructed according to the proposed DTB-like basis. The totally positive property of the DT B-spline-like basis is supported. For different shape parameter values, the associated trigonometric B-spline-like curves with two denominator shape parameters (DT B-spline-like curves) can be C2 continuous for a non-uniform knot vector. For a special value, the generated curves can be C(2n−1) (n = 1, 2, 3, . . .) continuous for a uniform knot vector. A kind of trigonometric B-spline-like surfaces with four denominator shape parameters (DT B-spline-like surface) is shown by using the tensor product method, and the associated DT B-spline-like surfaces can be C2 continuous for a nonuniform knot vector.When given a special value, the related surfaces can be C(2n−1) (n = 1, 2, 3, . . .) continuous for a uniform knot vector. A new class of trigonometric Bernstein–Bézier-like basis function with three denominator shape parameters (DT BB-like basis) over a triangular domain is also constructed. A de Casteljau-type algorithm is developed for computing the associated trigonometric Bernstein–Bézier-like patchwith three denominator shape parameters (DTBB-like patch).The condition forG1 continuous jointing two DT BB-like patches over the triangular domain is deduced.


Introduction
The construction of basis functions has always been a difficulty of computer-aided geometric design (CAGD).A class of practical basis functions often plays a decisive role in the geometric industry.Conventional cubic B-spline curves and surfaces are widely applied for CAGD due to their remarkable local adjustment properties.However, given control points keep a generated conventional cubic B-spline curve over a single location.Although cubic rational B-spline curves and surfaces can adjust positions and shapes by changing the weighting factor [1][2][3], their adjustment effect is difficult to predict due to its own defects.In recent years, trigonometric polynomials and splines with one or more shape parameters have been widely used with CAGD, especially in the design of curves and surfaces.Details can be found in [4][5][6][7] and the corresponding references therein.For example, researchers have used shape parameters to propose quadratic and cubic trigonometric polynomial splines [8,9].In [10], the extended cubic trigonometric spline curve of [8] was given.In [11], a class of C-Bézier curves was constructed in the space span {1, , sin , cos }, where the length of the interval serves as shape parameter.The sine and ellipse curves can be represented by the C-Bézier curves.
Although many improved methods are available, they are rarely applied in solving practical problems.In the final analysis, these techniques increase the flexibility of the curve by adding shape parameters compared with the traditional Bézier and B-spline methods.However, the technique itself cannot replace the traditional method, and several aspects still need improvement.For example, the majority of these methods discuss only basic properties, such as nonnegativity, partition of unity, symmetry, and linear independence.Shape preservation, total positivity, and variation diminishing, which are important properties for curve design, are often overlooked.However, the basis function, which has total positivity, ensures that the related curve contains variation diminishing and shape preservation.Therefore, possessing total positivity is highly important for basis functions.In addition, constructing cubic curves and surfaces remains the main method among the improved techniques.In general, these improved methods have  2 continuity, thereby meeting engineering requirements.However, in many practical applications, if the requirement for continuity is high, then these methods are slightly insufficient and often need to increase the number of times the curve is constructed.The B-spline curve and surface are regarded as examples.Notably, the continuity and locality of the curved surface are directly related to the number of times.The more times the curve is constructed, the higher the continuous order, but the locality is poor, and the computational complexity is high.Therefore, sacrificing the local property of its dominant position is necessary to achieve the special requirements of high-order continuity.Therefore, in the construction of curves and surfaces, the importance of meeting high-order continuity without increasing the computational complexity and without affecting its local properties is highlighted.
The traditional surface over rectangular domains, which possesses research and application value, has been widely used in CAGD.Obtaining a surface over a rectangular domain is easy because the traditional surface over such domain is a direct extension of the traditional Bézier curve by the tensor product method.However, by using the tensor product method, we fail to extend the patch over a triangular domain because it is not a tensor product surface.In many practical applications, surface modeling based on patch construction over triangular domains is important.Thus, the study of patches over triangular domains is of considerable interest.Therefore, the construction of a practical method that generates patches is important.For this reason, researchers have conducted numerous works.In [34], Cao showed a class of basis functions over a triangular domain.The related patch can be rendered flexible by an adjustment of the values.In [35], Han proposed a patch over a triangular domain, which can construct boundaries that can exactly represent elliptic arcs.A kind of quasi-Bernstein-Bézier polynomials over a triangular domain was proposed in [36].Recently, Zhu constructed the -Bernstein-Bézier-like basis, which possesses 10 functions; the related exponential parameter has tension effect.
This study proposes a class of DTB-like bases with tension effects that is based on previous studies.The proposed basis has two denominator shape parameters constructed in the space {1, sin 2 , (1 cos ]} and can form an optimal normal normalized totally positive basis (B-basis) and a new class of DT BB-like basis functions over a triangular domain with three denominator shape parameters.The presented DT Bspline-like curves and surfaces are  2 continuous with respect to a nonuniform knot vector.The corresponding curves and surfaces are  2−1 ( = 1, 2, . ..) continuous for the shape parameters, which select a special value with respect to a uniform knot vector.The denominator parameter introduced in the basis function has a tension effect, and the parameters can be used to predictably adjust the corresponding curves and surfaces generated.
The remainder of this work is organized as follows.Section 2 provides the definition and properties of the DTBlike basis functions and shows the corresponding curves.Section 3 presents a class of DT B-spline-like basis with two denominator shape parameters.The properties of the proposed basis are analyzed, and the associated DT B-like curves are shown.Section 4 proposes a class of DT BB-like basis over a trigonometric domain with three denominator shape parameters.We provided the definition and properties of related DT BB-like patches on the basis of the presented basis functions.Then, we developed a de Casteljau algorithm to calculate the proposed patch.Finally,  1 connecting conditions of the two proposed patches are given.Section 5 presents the conclusion.

Trigonometric Bernstein-Like Basis Functions
2.1.Preliminaries.For a good understanding of this study, related background knowledge about the extended Chebyshev (EC) space and the extended completed Chebyshev (ECC) space is provided in this subsection.Additional details are available in [37][38][39]. <  for any closed bounded interval [, ], which can be denoted by .The function space ( 0 , . . .,   ) is called  + 1-dimension ECC-space, which is generated by the positive weight function   ∈  − () in canonical form.The weight function shows that The necessary and sufficient condition of +1-dimension function space ( 0 , . . .,   ) ⊂   () is called an ECC-space on  that is for arbitrary , 0 ≤  ≤ , and an arbitrary nontrivial linear combination of the elements of the subspace ( 0 , . . .,   ) with the most  zeros (counting multiplicities).
If the collocation matrix (  (  )) 0≤,≤ related to the basis ( 0 , . . .,   ) for an arbitrary sequence of points  ≤  0 <  1 < ⋅⋅⋅ <   ≤  is totally positive, then the basis is deemed totally positive on [, ].If a function space has a totally positive basis, then the other totally positive basis can be formed by multiplying the optimal normalized totally positive basis (B-basis) by a totally positive matrix.Moreover, this basis is unique in space and has optimal shape preservation properties [40][41][42].
Proof that  , is an EC-space on [0, /2] is provided.Thus, we must verify that the arbitrary nonzero element of  , has, at most, two zeros (counting multiplicities) on [0, /2].

DTB-Like Curve with Denominator Shape Parameters Definition 4. Given control points 𝑃
are called a cubic DTB-like curve with two denominator shape parameters  and .
Thus, the corresponding DTB-like curve given in ( 26) has the properties of affine invariance, convex hull, and variation diminishing, which are crucial properties in curve design, given that (22) possesses the properties of partition of unity, nonnegativity, and total positivity.Moreover, we have the following end-point property: For arbitrary ,  ∈ [2, +∞), the curve given in ( 26) has the end-point interpolation property, and  0  1 and  2  3 are the tangent lines of the curve at points  0 and  3 , respectively.From these properties, we can easily find that the curve given in (26) has similar geometric properties to the classical cubic Bézier curve.
The corner cutting algorithm is a steady and highefficiency algorithm for generating the presented DTB-like curves.We rewrite (26) into the following matrix to develop the algorithm: ) .(28) By rewriting the curve expression into matrix form, we can rapidly obtain this algorithm.Figure 2 shows an example of this algorithm.
In addition, for  ∈ [0, /2], we can rewrite (26) into the following form: For arbitrary fixed  ∈ (0, /2),  0 (; ) monotonically decreases with respect to the shape parameter .This phenomenon also means that, as shape parameter  increases, the generated curve moves in the same direction as the edge  0 −  1 .By contrast, as shape parameter  decreases, the opposite is true for the generated curve.On the edge  3 −  2 , parameter  has similar influences.When  = , as the shape parameters increase or decrease, the generated curve moves to the edge  2 −  1 in the same or opposite direction, respectively.Thus, the two denominator shape parameters have a tension effect.Figure 3 shows the generated curves for different shape parameter values.
The discussion indicates that any arc of an ellipse or parabola can be exactly represented by using the proposed DTB-like curves.Figure 4 shows the elliptic and parabolic segments generated by using the cubic DTB-like curves (marked with solid black lines).
Proof.Without loss of generality, we consider the knot  +1 .For arbitrary   ,   ∈ [2, +∞), we have From the aforementioned calculations and Lemma 7, we can easily find that the theorem is established at knot  +1 .The case of the continuity at other knots can be discussed similarly.
Proof.We use mathematical induction to prove that the (2 − 1)-order derivative of basis function (22) has the following form: When  = 1, the derivative of the basis functions ( 22) is These forms are satisfied when  = 1.
We assume that the aforementioned forms are also satisfied when  = .Therefore, the (2 − 1)-order derivative of the basis functions (22) is (57) By direct computing, we have These forms are satisfied when  =  + 1.In summary, the 2 − 1-order derivative of the basis functions ( 22) has the form of (55).Finally, we prove that the basis function   () is  2−1 ( = 1, 2, . ..) continuous at each knot.Without loss of generality, we first consider the continuity at the knot  +1 .From here and Remark 6, direct computation gives Thus, we can immediately conclude that  (2−1)  ( + +1 ) =  (2−1)  ( − +1 ) for a uniform knot vector and all   =   = 2.In summary, the theorem is established at knot  +1 .The continuity of the basis functions with respect to other knots can be discussed similarly.

DT B-Spline-Like Curves
Definition 14.Given control points   ( = 0, 1, . . ., ) in  2 or  3 and a knot vector , for arbitrary real numbers is called a DT B-spline-like curve with two denominator shape parameters   ,   .
Proof.Without loss of generality, we consider the knot  +1 .From Theorem 15 and Remark 6, for all   =   = 2, we have 3 3 We can immediately conclude that  (2−1) ) for a uniform knot vector and all   =   = 2.In summary, the theorem is established at knot  +1 .The continuity of () with respect to other knots can be similarly discussed.
Figure 6 shows () with different denominator shape parameters.The left figure shows (), which is generated by setting all   =   = 2 (black lines); the blue line is generated by changing one   to 4; and the green line is generated by changing one   to 4. The right figure shows that () is generated by setting all   = 2 or 5 and   = 2 or 5.
Theorem 18.Given two nonuniform knot vectors  and , the DT B-spline-like surface (, V) possesses  2 continuity at each knot.
Proof.Without loss of generality, we consider the continuity at the region From Definition 17, we have We prove that the continuity of the  direction and the continuity of the V direction can be similarly discussed.
Considering the continuity at the knot ( +1 , V +1 ), we have From Theorem 16, for all   =   = 2, we have In summary, the theorem is established at knot ( +1 , V +1 ).
These findings imply the theorem.Figure 8 shows four images of DT BB-like basis functions with the parameter values  =  =  = 3.
Next, we provide the properties of the DT BB-like patch given in (73).
(a) Affine invariance and convex hull property: the related patch (73) has affine invariance and convex hull property because the basis functions (69) possess the properties of partition of unity and nonnegativity.
(89) Thus, we can summarize the following theorems.The aforementioned theorem shows that the two DT BBlike patches that connect the conditions are similar to those of the two triangular Bernstein-Bézier-like patches.Detailed content is available in [44].The only difference is that we can obtain different  1 continuous surfaces by changing the value of the denominator shape parameter.Figure 11 shows the  1 continuous surface under different shape parameters, where  = 1,  = −1.

Conclusion
In this study, the proposed DT B-like basis function forms a set of optimal normalized totally positive bases under the framework of the ECC-space, which leads to the DT BBspline basis function and the DT-BB-like basis function.Curve and surface construction and related discussions are based on these kinds of basis functions.Compared with the traditional Bézier method and the B-spline technique, the proposed method not only retains all the remarkable properties of the traditional method, such as variation diminishing, but can also accurately represent special industrial curves, such as parabolic and elliptical arcs.In special cases, the B-spline curves and surfaces constructed in this study can automatically reach  (2−1) ( = 1, 2, 3, . ..) continuity, thereby satisfying the geometric design requirements of highorder continuity, which is not possible in the traditional literature.In addition, this study introduces a new method for the construction of triangular domain patches.This technique can flexibly adjust the patch with parameters and can accurately represent the boundary as a parabolic arc, an elliptical arc, or even an arc surface.Meanwhile, a de Casteljau algorithm is given to efficiently generate triangular domain patches and the  1 connecting conditions of the patch.Although the method in this study solves the problem of the traditional method and has many advantages, the construction of the basis function is only the first step.In the design of a curve or surface that is closely in-line with the requirements of the geometric industry, then many problems still need to be addressed.These issues include the accurate quantitative analysis of the influence of the denominator parameter on the DT B-like curve and the DT BB-spline curve; shape analysis of the DT B-like curve and the DT BBspline curve (convexity, cusp, inflection point, monotonicity, heavy node, etc.); and the higher-order continuous problem analysis of the DT BB-like patch on the triangular domain, except for the  1 continuity.These issues will be the focus of future research.

Figure 4 :
Figure 4: Representation of elliptic and parabolic arcs.
Figure 7  shows (, V) with different shape parameters.The figure on the left shows (, V) generated when all shape parameters are set  1 =  1 =  2 =  2 = 2.The figure on the right shows (, V) generated when all shape parameters are set  1 =  1 =  2 =  2 = 5.

Figure 9 :
Figure 9: DT BB-like patches whose boundaries are arcs of elliptic, circle, and parabola.