Research Article | Open Access
Kai Wang, Guicang Zhang, "New Trigonometric Basis Possessing Denominator Shape Parameters", Mathematical Problems in Engineering, vol. 2018, Article ID 9569834, 25 pages, 2018. https://doi.org/10.1155/2018/9569834
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 continuous for a non-uniform knot vector. For a special value, the generated curves can be 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 continuous for a nonuniform knot vector. When given a special value, the related surfaces can be 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 patch with three denominator shape parameters (DT BB-like patch). The condition for continuous jointing two DT BB-like patches over the triangular domain is deduced.
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 [1–3], 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 [4–7] and the corresponding references therein. For example, researchers have used shape parameters to propose quadratic and cubic trigonometric polynomial splines [8, 9]. In , the extended cubic trigonometric spline curve of  was given. In , a class of C-Bézier curves was constructed in the space span , where the length of the interval serves as shape parameter. The sine and ellipse curves can be represented by the C-Bézier curves.
Many basis functions with tension effect, which possess one or more parameters, have been proposed to predictably adjust the positions and shapes of generated curves. For example, Nieson  proposed a cubic rational spline curve that can generate an interactive curve. The proposed curve can be continuous with different tension parameters. Barsky [13, 14] provided a class of beta-splines that can locally modify the shape of the generated curves. In a special case, the curve proposed by Barsky contains a kind of uniform B-spline. In , Gregory proposed a class of cubic continuous rational spline, which possesses tension shape parameters so that the shape of the generated curve can be modified. A class of spline curves was proposed in  by using tension shape parameters. According to  and the related references therein, curve design has attracted widespread interest due to the appearance of plenty of exponential splines. In the space , Costantini constructed a kind of continuous polynomial splines . In [19, 20], this space was reported to be a quasi-extended Chebyshev space. In [21–25], different points of view about variable degree spaces were proposed. In , a class of -Bernstein-like basis was proposed in the space span . In the space span , four trigonometric Bernstein-like basis functions were constructed . In the space , researchers in [28–30] discussed the subdivision scheme with respect to a constructed class of basis functions. In the space span , a cubic rational basis was constructed . In [32, 33], variable degree polynomial splines exhibited enormous potential in the geometric industry in view of the problem of shape preservation and approximation.
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 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 , 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 , 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 . 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 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 B-spline-like curves and surfaces are continuous with respect to a nonuniform knot vector. The corresponding curves and surfaces are 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 DTB-like 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, connecting conditions of the two proposed patches are given. Section 5 presents the conclusion.
2. Trigonometric Bernstein-Like Basis Functions
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 [37–39].
for any closed bounded interval , which can be denoted by . The function space is called -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 -dimension function space is called an ECC-space on that is for arbitrary , , and an arbitrary nontrivial linear combination of the elements of the subspace with the most zeros (counting multiplicities).
If the collocation matrix related to the basis for an arbitrary sequence of points 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 [40–42].
Assume that -dimensional is an -dimensional EC-space on closed bounded interval , which possesses constants, where . Select functions in so that forms a basis of . We set a mother function . This function being on , for arbitrary nonnegative integer , we can consider its osculating flat of order at , defined as the affine space passing through and the direction of which is spanned by , that is,
For arbitrary positive integers , with , and arbitrary pairwise distinct in , if ever the osculating flats , , possess a unique common point, then we define the blossom of as the function such thatThe arbitrary -tuple is equivalent to up to permutation, where
Reference  showed that the blossom that exists in space deduces three important properties, namely, pseudo-affinity, diagonal, and symmetry. In addition, we must emphasize a useful conclusion that the blossom that exists in space is equal to that in space , which is an EC- space on . Additional details can be seen in Theorem 3.1 in .
For arbitrary , the points , , are defined by the Chebyshev–Bézier points of about . When , , for all . However, when , the Chebyshev–Bézier points are obtained asMoreover, when , a de Casteljau algorithm starting from points can be developed by using the three important properties of the blossom. At the nth step of this algorithm, the values of can be obtained asGiven that is a set of bases in the function of , the affine flat of the mother function spans the whole space . Therefore, points are affinely independent. Furthermore, , , from a basis of .
2.2. Construction of Trigonometric Basis Function
For arbitrary real numbers , , we consider constructing a basis function in the trigonometric function space . We can easily obtain the corresponding mother function, which is defined as First, let us prove that the spaceis a three-dimensional EC-space on . Therefore, according to Theorem 2.1 of , possesses a blossom, indicating that is suitable for curve and surface designs.
Theorem 1. For arbitrary real numbers , is a three-dimensional EC-space on .
Proof. For arbitrary , we consider a linear combinationFor , from (9), we can easily obtain . In the same way, for , from (9), we can obtain . Thus, . Therefore, is a three-dimensional space.
Below, we prove that is an ECC-space in . For arbitrary , let andDirect computation indicates thatThus, for the Wronskian of and , For , we obtain the definition of the following weight functions:where are three arbitrary positive real numbers. All of the three weight functions are bounded, positive, and on closed bounded . Next, we discuss the following ECC-spaces: We can easily verify that these functions are linear combinations of Thus, is an ECC-space on . Furthermore, are any subintervals of ; hence, is also an ECC-space on .
Proof that is an EC-space on is provided. Thus, we must verify that the arbitrary nonzero element of has, at most, two zeros (counting multiplicities) on .
Consider the arbitrary nonzero functionwhere . has, at most, two zeros in because is an ECC-space in . Initially, is assumed to vanish at 0; then, we can obtain . Under these circumstances, if , then has two singular zeros at 0 and . If , then 0 is a double zero of . If , then has a singular zero at 0 but does not have a zero on . If , then has a singular zero at 0 but not at . In addition, for the function,direct computation yields where is a monotonic function on . From these values, together with , we can see that has exactly one zero in . Thus, we can immediately conclude that has one zero in . Similarly, if vanishes at , then function has, at most, two zeros on (counting multiplicities as far as possible up to 2). In summary, the space is an EC-space on .
Therefore, is an EC-space on . According to Theorem 3.1 in , a blossom exists in . This theorem also means that is suitable for curve and surface designs. According to Theorem 2.18 in , we can also infer that has a B-basis on .
Theorem 2. The four Chebyshev–Bézier points of the mother function defined in (7) are given by Moreover, the related new cubic trigonometric Bernstein-like basis of are given by
Proof. Through the definition of given in (7), we can easily obtain Thus, by simply computing, we have Thus, for any , from together with , we obtain the expressions of , .
Remark 3. From the expressions of the basis function given in (22), we can infer that (22) has important properties, such as linear independence, nonnegativity, and partition of unity. In addition, for all , we have the following end-point properties:Thus, system is precisely the B-basis of , which implies that (22) possesses totally positive and optimal shape preservation properties .
We define a cubic trigonometric Bernstein-like basis with denominator shape parameters given in (22) as , , or , and , to facilitate the following discussion and distinguish the traditional literature. Figure 1 shows images of the DTB-like basis under different shape parameters.
2.3. DTB-Like Curve with Denominator Shape Parameters
Definition 4. Given control points in or , 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 , the curve given in (26) has the end-point interpolation property, and and are the tangent lines of the curve at points and , 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 high-efficiency algorithm for generating the presented DTB-like curves. We rewrite (26) into the following matrix to develop the algorithm: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 , we can rewrite (26) into the following form:
For arbitrary fixed , 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 . By contrast, as shape parameter decreases, the opposite is true for the generated curve. On the edge , parameter has similar influences. When , as the shape parameters increase or decrease, the generated curve moves to the edge 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.
2.4. Representation of Elliptic and Parabolic Arcs
For , if the control points are , , , and , then the coordinates of the generated curve arewhere is a quarter of an elliptic arc whose center is located at . By constraining the parameter on the desired interval , we can obtain an arc of an ellipse whose starting and ending angles are and , respectively.
Furthermore, for , , if the control points , and with respective coordinates , and , then we obtain the following from (26):which presents a segment of the parabola , .
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).
3. DT B-Spline-Like Basis Function
3.1. Construction of Trigonometric B-Spline-Like Basis Functions
Given a sequence of knots , we refer to as a knot vector. Let , and , , for arbitrary real numbers , , we construct trigonometric B-spline-like basis functions with the following forms: where , are the DTB-like basis functions given in (22).
To determine coefficient values of the basis functions (32), we add two restricting conditions: (1) including continuity at each knot and (2) forming a partition of unity on . Direct computation shows that
We will provide the following definition to facilitate the following discussion and distinguish the traditional literature.
Definition 5. For arbitrary real numbers , given a knot vector , with the coefficients , (32) expressions are defined as DT B-spline-like basis with two denominator shape parameters.
Remark 6. The DT B-spline-like basis functions are constructed in the spacewhere
In particular, for , and , direct computation yieldsfrom which we can immediately obtain the following explicit expressions of :
Given an equidistant knot vector, we call a uniform DT B-spline-like basis function, and the related knot vector is called a uniform knot vector. On the contrary, given a nonequidistant knot vector, we call and a nonuniform DT B-spline-like basis function and a nonuniform knot vector, respectively. Figure 5 illustrates the DT B-spline-like basis function under different shape parameter values. Before discussing the properties of the DT B-spline-like basis function, we prove the following lemma, which is crucial for the discussion of the curves and surfaces.
Lemma 7. For all possible , , the coefficients have the following properties: