# New Trigonometric Basis Possessing Denominator Shape Parameters

Authors: Kai Wang; Guicang Zhang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

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 beC2 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 patch with three denominator shape parameters (DT BB-like patch). The condition for G1 continuous jointing two DT BB-like patches over the triangular domain is deduced.


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## Body

## 1. 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 [1–3], 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 [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,t,sin⁡t,cos⁡t}, 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 [12] proposed a cubic rational spline curve that can generate an interactive curve. The proposed curve can be G2continuous 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 [15], Gregory proposed a class of cubic C2 continuous rational spline, which possesses tension shape parameters so that the shape of the generated curve can be modified. A class of C2∩FC3 spline curves was proposed in [16] by using tension shape parameters. According to [17] and the related references therein, curve design has attracted widespread interest due to the appearance of plenty of exponential splines. In the space span{1,t,(1-t)p,tq}, Costantini constructed a kind of C2 continuous polynomial splines [18]. 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 [26], a class of αβ-Bernstein-like basis was proposed in the space span {1,3t2-2t3,(1-t)α,tβ}. In the space span {1,sin2,(1-sin⁡t)α,(1-cos⁡t)β}, four trigonometric Bernstein-like basis functions were constructed [27]. In the space span⁡{1,t,(1-t)3/[1+(α-3)t(1-t)],t3/[1+(β-3)t(1-t)}, researchers in [28–30] discussed the subdivision scheme with respect to a constructed class of basis functions. In the space span {1,3t2-2t3,(1-t)3/[1+(α-3)t],t3/[1+(β-3)(1-t)]}, a cubic rational basis was constructed [31]. 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 haveC2 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 αβγ-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 spacespan1,sin2t,(1-sin⁡t)2/[1+(α-2)sin⁡t],(1-cos⁡t)2/[1+(β-2)cos⁡t] 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 C2 continuous with respect to a nonuniform knot vector. The corresponding curves and surfaces are C2n-1(n=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. Section2 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, G1 connecting conditions of the two proposed patches are given. Section 5 presents the conclusion.


## 2. 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 [37–39].a < b for any closed bounded interval [a,b], which can be denoted by I. The function space (u0,…,un) is called n+1-dimension ECC-space, which is generated by the positive weight function wj∈Cn-j(I) in canonical form. The weight function shows that(1)u0t=w0t,u1t=w0t∫atw1t1dt1,u2t=w0t∫atw1t1∫at1w2t2dt2dt1,⋮unt=w0t∫atw1t1∫at1w2t2⋯∫atn-1wntndtn⋯dt1.The necessary and sufficient condition ofn+1-dimension function space (u0,…,un)⊂Cn(I) is called an ECC-space on I that is for arbitrary k, 0≤k≤n, and an arbitrary nontrivial linear combination of the elements of the subspace (u0,…,uk) with the most k zeros (counting multiplicities).If the collocation matrix(uj(ti))0≤i,j≤n related to the basis (u0,…,un) for an arbitrary sequence of points a≤t0<t1<⋯<tn≤b is totally positive, then the basis is deemed totally positive on [a,b]. 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 thatn+1-dimensional ε⊂Cn is an n+1-dimensional EC-space on closed bounded interval I, which possesses constants, where n≥2. Select n functions Φ1,…,Φn in ε so that (1,Φ1,…,Φn) forms a basis of ε. We set a mother function Φ≔(1,Φ1,…,Φn):I→Rn. This function being Cn on I, for arbitrary nonnegative integer i≤n-1, we can consider its osculating flat of order i at x∈I, defined as the affine space passing through Φ(x) and the direction of which is spanned by Φ′(x),…,Φ(i)(x), that is,(2)OsciΦx≔Φx+λ1Φ′x+⋯+λiΦix∣λ1,…λi∈R.For arbitrary positive integersμ1,…,μr, with ∑i=1rμi=n, and arbitrary pairwise distinct a1,…,ar in I, if ever the r osculating flats Oscn-μiΦ(ai), 1≤i≤r, possess a unique common point, then we define the blossom of Φ as the function φ≔(φ1,…φn):In→Rn such that(3)φx1,…,xn≔⋂i=0nOscn-μiΦai.The arbitrary n-tuple (x1,…,xn) is equivalent to (a1[μ1],…,a1[μn]) up to permutation, where(4)a1μ1,…,arur≔a1,…,a1︸μ1,…a1,…,a1︸μr.Reference [37] showed that the blossom that exists in space ε 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 ε is equal to that in space Dε, which is an EC- space on I. Additional details can be seen in Theorem 3.1 in [38].For arbitrary(a,b)∈I2, the n+1 points Pi(a,b)≔φ(a[n-i],b[i]), i=0,…,n, are defined by the Chebyshev–Bézier points of φ about (a,b). When a=b, Pi(a,b)=φ(a), for all i=0,…,n. However, when a≠b, the Chebyshev–Bézier points are obtained as(5)Π0a,b=Φa,Πna,b=Φb,Πia,b=OsciΦa∩Oscn-iΦb,1≤i≤n-1.Moreover, when a≠b, a de Casteljau algorithm starting from points Πi(a,b) can be developed by using the three important properties of the blossom. At the nth step of this algorithm, the values of φ can be obtained as(6)Φx=∑i=0nBia,bxΠia,b,∑i=0nBia,bx=1,x∈I.Given that (1,Φ1,…,Φn) is a set of bases in the function of ε, the affine flat of the mother function Φ spans the whole space Rn. Therefore, points Π0(a,b),…,Πn(a,b) are affinely independent. Furthermore, Bi(a,b), i=0,…,n, from a basis of ε [37].


### 2.2. Construction of Trigonometric Basis Function

For arbitrary real numbersα,β∈[2,+∞), t∈[0,π/2], we consider constructing a basis function in the trigonometric function space Tα,β≔span1,sin2t,(1-sin⁡t)2/(1+α-2sin⁡t),(1-cos⁡t)2/(1+β-2cos⁡t). We can easily obtain the corresponding mother function, which is defined as (7)Φt≔sin2t,1-sin⁡t21+α-2sin⁡t,1-cos⁡t21+β-2cos⁡t,t∈0,π2.First, let us prove that the space(8)DTα,β=span2sin⁡tcos⁡t,-cos⁡t1-sin⁡tα+α-2sin⁡t1+α-2sin⁡t2,sin⁡t1-cos⁡tβ+β-2cos⁡t1+β-2cos⁡t2is a three-dimensional EC-space on [0,π/2]. Therefore, according to Theorem 2.1 of [38], Tα,β possesses a blossom, indicating that Tα,β is suitable for curve and surface designs.Theorem 1.
For arbitrary real numbersα,β∈[2,+∞), DTα,β is a three-dimensional EC-space on [0,π/2].Proof.
For arbitraryξi∈R(i=0,1,2),t∈[0,π/2], we consider a linear combination(9)ξ02sin⁡tcos⁡t+ξ1-cos⁡t1-sin⁡tα+α-2sin⁡t1+α-2sin⁡t2+ξ2sin⁡t1-cos⁡tβ+β-2cos⁡t1+β-2cos⁡t2=0.
Fort=0, from (9), we can easily obtain ξ1=0. In the same way, for t=π/2, from (9), we can obtain ξ2=0. Thus, ξ0=0. Therefore, DTα,β is a three-dimensional space.
Below, we prove thatDTα,β is an ECC-space in [0,π/2]. For arbitrary t∈[a,b]⊆(0,π/2), let (10)ut=-1-sin⁡tα+α-2sin⁡tsin⁡t1+α-2sin⁡t2′=cos⁡t3sin⁡t2-sin⁡t+1+α-23-sin2tsin⁡tα-2+2sin2t1+α-2sin⁡t3>0and(11)vt=1-cos⁡tβ+β-2cos⁡t1+β-2cos⁡t2′=sin⁡t3cos⁡t2-cos⁡t+1+β-23-cos2tcos⁡tβ-2+2cos2t1+β-2cos⁡t3>0.Direct computation indicates that(12)u′t=-sin⁡t3sin⁡t2-sin⁡t+1+α-23-sin2tsin⁡tα-2+2sin2t1+α-2sin⁡t3-2cos2t2+α-21+8sin⁡t+α-23sin2t5+1+sin⁡t1-sin⁡t+4α-22sin⁡t2-sin⁡tsin⁡t+sin⁡t+1sin3t1+α-2sin⁡t4<0,(13)v′t=sin⁡t3cos⁡t2-cos⁡t+1+β-23-cos2tcos⁡tβ-2+2cos2t1+β-2cos⁡t3+sin2t2+β-21+8cos⁡t+β-23cos3t5+1+cos⁡t1-cos⁡t+4β-22cos⁡t2-cos⁡tcos⁡t+cos⁡t+1cos3t1+β-2cos⁡t4>0.Thus, for the Wronskian of u(t) and v(t), (14)Wu,vt=utv′t-u′tvt>0,∀t∈0,π2.For t∈[a,b], we obtain the definition of the following weight functions:(15)ω0t=2sin⁡tcos⁡t,ω1t=Aut+Bvt,ω2t=CWu,vtAut+Bvt2,where A,B,C are three arbitrary positive real numbers. All of the three weight functions ωi(t)(i=0,1,2) are bounded, positive, and C∞ on closed bounded [a,b]. Next, we discuss the following ECC-spaces: (16)u0t=ω0t,u1t=ω0t∫atω1t1dt1,u2t=ω0t∫atω1t1∫at1ω2t2dt2dt1,We can easily verify that these functions ui(t)(i=0,1,2) are linear combinations of (17)2sin⁡tcos⁡t,-cos⁡t1-sin⁡tα+α-2sin⁡t1+α-2sin⁡t2,sin⁡t1-cos⁡tβ+β-2cos⁡t1+β-2cos⁡t2.Thus, DTα,β is an ECC-space on [a,b]. Furthermore, [a,b] are any subintervals of (0,π/2); hence, DTα,β is also an ECC-space on (0,π/2).
Proof thatDTα,β is an EC-space on [0,π/2] is provided. Thus, we must verify that the arbitrary nonzero element of DTα,β has, at most, two zeros (counting multiplicities) on [0,π/2].
Consider the arbitrary nonzero function(18)Gt=C02sin⁡tcos⁡t+C1-cos⁡t1-sin⁡tα+α-2sin⁡t1+α-2sin⁡t2+C2sin⁡t1-cos⁡tβ+β-2cos⁡t1+β-2cos⁡t2,where t∈[0,π/2]. G(t) has, at most, two zeros in (0,π/2) because DTα,β is an ECC-space in (0,π/2). Initially, G(t) is assumed to vanish at 0; then, we can obtain C1=0. Under these circumstances, if C2=0, then G(t) has two singular zeros at 0 and π/2. If C0=0, then 0 is a double zero of G(t). If C0C2>0, then G(t) has a singular zero at 0 but does not have a zero on (0,π/2]. If C0C2<0, then G(t) has a singular zero at 0 but not at π/2. In addition, for the function,(19)Kt≔2C0cos⁡t+C21-cos⁡tβ+β-2cos⁡t1+β-2cos⁡t2,direct computation yields (20)K′t=-2C0cos⁡t+C22b-12sin⁡t1+b-2cos⁡t3,where K(t) is a monotonic function on [0,π/2]. From these values, together with K(0)K(π/2)=2βC0C2<0, we can see that K(t) has exactly one zero in (0,π/2). Thus, we can immediately conclude that G(t)=sin⁡tK(t) has one zero in (0,π/2). Similarly, if G(t) vanishes at π/2, then function G(t) has, at most, two zeros on [0,π/2] (counting multiplicities as far as possible up to 2). In summary, the space DTα,β is an EC-space on [0,π/2].
Therefore,DTα,β is an EC-space on [0,π/2]. According to Theorem 3.1 in [43], a blossom exists in Tα,β. This theorem also means that Tα,β is suitable for curve and surface designs. According to Theorem 2.18 in [43], we can also infer that Tα,β has a B-basis on [0,π/2].Theorem 2.
The four Chebyshev–Bézier pointsPi≔Pi(0,π/2) of the mother function Φ(t) defined in (7) are given by (21)Π0=0,1,0,Π1=0,0,0,Π0=1,0,0,Π0=1,0,1.Moreover, the related new cubic trigonometric Bernstein-like basis Ai≔Ai(0,π/2) of Tα,β are given by (22)A0t=1-sin⁡t21+α-2sin⁡t,A1t=1-sin2t-1-sin⁡t21+α-2sin⁡t,A2t=1-cos2t-1-cos⁡t21+β-2cos⁡t,A3t=1-cos⁡t21+β-2cos⁡t.Proof.
Through the definition ofΦ(t) given in (7), we can easily obtain (23)Φ0=0,1,0,Φπ2=1,0,1,Φ′0=0,-α,0,Φ′π2=0,0,β,Φ′′0=2,2α-12,0,Φ′′π2=-2,0,2β-12.Thus, by simply computing, we have (24)Π0=Φ0=0,1,0,Π3=Φπ2=1,0,1,Π1=Osc1Φ0∩Osc2Φπ2=0,0,0,Π2=Osc2Φ0∩Osc1Φπ2=1,0,0.Thus, for any t∈[0,π/2], from Φ(t)=∑i=03Ai(t)Πi together with ∑i=03Ai(t)=1, we obtain the expressions of Ai(t), i=0,1,2,3.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 i=0,1,2,3, we have the following end-point properties:(25)Aij0=0,j=0,…,i-1,Aii0>0,Aijπ2=0,j=0,…,2-i.Thus, system (A0(t),A1(t),A2(t),A3(t)) is precisely the B-basis of Tα,β, which implies that (22) possesses totally positive and optimal shape preservation properties [33].We define a cubic trigonometric Bernstein-like basis with denominator shape parameters given in (22) as Ai(t;α,β), i=0,1,2,3, or Ai(t;α), i=0,1 and Ai(t;β), i=2,3 to facilitate the following discussion and distinguish the traditional literature. Figure 1 shows images of the DTB-like basis under different shape parameters.Figure 1
DTB-like basis under different shape parameters.


### 2.3. DTB-Like Curve with Denominator Shape Parameters

Definition 4.
Given control pointsPi(i=0,1,2,3) in R2 or R3, (26)Qt;α,β=∑i=03Ait;α,βPi,t∈0,π2,α,β∈2,+∞,are called a cubic DTB-like curve with two denominator shape parameters α and β.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:(27)Q0;α,β=P0,Qπ2;α,β=P3,Q′0;α,β=αP1-P0,Q′π2;α,β=βP3-P2,Q′′0;α,β=2α-12P0-P1+2P2-P1,Q′′π2;α,β=2β-12P3-P2+2P1-P2.For arbitraryα,β∈[2,+∞), the curve given in (26) has the end-point interpolation property, and P0P1 and P2P3 are the tangent lines of the curve at points P0 and P3, 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:(28)Qt;α,β=1-sin2t1-cos2t×1-sin⁡tsin⁡t00cos⁡t1-cos⁡t×11+sin⁡t1+α-2sin⁡t1+sin⁡t1+α-2sin⁡t-11+sin⁡t1+α-2sin⁡t000cos⁡tsin⁡t+cos⁡tsin⁡tsin⁡t+cos⁡t0001+cos⁡t1+β-2cos⁡t-11+cos⁡t1+β-2cos⁡t11+cos⁡t1+β-2cos⁡t×P0P1P2P3.By rewriting the curve expression into matrix form, we can rapidly obtain this algorithm. Figure 2 shows an example of this algorithm.Figure 2
Corner cutting algorithm.In addition, fort∈[0,π/2], we can rewrite (26) into the following form:(29)Qt;α,β=sin2P1+cos2P2+A0t;αP0-P1+A3t;βP3-P2.For arbitrary fixedt∈(0,π/2), A0(t;α) monotonically decreases with respect to the shape parameter α. This phenomenon also means that, as shape parameter α increases, the generated curve moves in the same direction as the edge P0-P1. By contrast, as shape parameter α decreases, the opposite is true for the generated curve. On the edge P3-P2, parameter β has similar influences. When α=β, as the shape parameters increase or decrease, the generated curve moves to the edge P2-P1 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.Figure 3
DTB-like curve with different shape parameters.


### 2.4. Representation of Elliptic and Parabolic Arcs

Forα=β=2, if the control points are P0=(x0+a,y0), P1=(x0+a,y0+b/2), P2=(x0+a/2,y0+b), and P3=(x0,y0+b), then the coordinates of the generated curve T(t;2,2) are(30)xt=x0+acos⁡t,yt=y0+bsin⁡t,t∈0,π2,where T(t;2,2) is a quarter of an elliptic arc whose center is located at (x0,y0). By constraining the parameter t on the desired interval [θ1,θ2], we can obtain an arc of an ellipse whose starting and ending angles are θ1 and θ2, respectively.Furthermore, forα=β=2, b-a>0, if the control points P0,P1,P2, and P3 with respective coordinates (b,c2b2+c1b+c0),(b,c2b2+c1b+c0),((a+b)/2,c2ab+c1(a+b)/2+c0), and (a,c2a2+c1a+c0), then we obtain the following from (26):(31)xt=b-acos⁡t+a,yt=c2b-acos⁡t+a2+c1b-acos⁡t+a+c0,t∈0,π2,which presents a segment of the parabola y=c2x2+c1x+c0, x∈[a,b].The discussion indicates that any arc of an ellipse or parabola can be exactly represented by using the proposed DTB-like curves. Figure4 shows the elliptic and parabolic segments generated by using the cubic DTB-like curves (marked with solid black lines).Figure 4
Representation of elliptic and parabolic arcs.


## 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 [37–39].a < b for any closed bounded interval [a,b], which can be denoted by I. The function space (u0,…,un) is called n+1-dimension ECC-space, which is generated by the positive weight function wj∈Cn-j(I) in canonical form. The weight function shows that(1)u0t=w0t,u1t=w0t∫atw1t1dt1,u2t=w0t∫atw1t1∫at1w2t2dt2dt1,⋮unt=w0t∫atw1t1∫at1w2t2⋯∫atn-1wntndtn⋯dt1.The necessary and sufficient condition ofn+1-dimension function space (u0,…,un)⊂Cn(I) is called an ECC-space on I that is for arbitrary k, 0≤k≤n, and an arbitrary nontrivial linear combination of the elements of the subspace (u0,…,uk) with the most k zeros (counting multiplicities).If the collocation matrix(uj(ti))0≤i,j≤n related to the basis (u0,…,un) for an arbitrary sequence of points a≤t0<t1<⋯<tn≤b is totally positive, then the basis is deemed totally positive on [a,b]. 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 thatn+1-dimensional ε⊂Cn is an n+1-dimensional EC-space on closed bounded interval I, which possesses constants, where n≥2. Select n functions Φ1,…,Φn in ε so that (1,Φ1,…,Φn) forms a basis of ε. We set a mother function Φ≔(1,Φ1,…,Φn):I→Rn. This function being Cn on I, for arbitrary nonnegative integer i≤n-1, we can consider its osculating flat of order i at x∈I, defined as the affine space passing through Φ(x) and the direction of which is spanned by Φ′(x),…,Φ(i)(x), that is,(2)OsciΦx≔Φx+λ1Φ′x+⋯+λiΦix∣λ1,…λi∈R.For arbitrary positive integersμ1,…,μr, with ∑i=1rμi=n, and arbitrary pairwise distinct a1,…,ar in I, if ever the r osculating flats Oscn-μiΦ(ai), 1≤i≤r, possess a unique common point, then we define the blossom of Φ as the function φ≔(φ1,…φn):In→Rn such that(3)φx1,…,xn≔⋂i=0nOscn-μiΦai.The arbitrary n-tuple (x1,…,xn) is equivalent to (a1[μ1],…,a1[μn]) up to permutation, where(4)a1μ1,…,arur≔a1,…,a1︸μ1,…a1,…,a1︸μr.Reference [37] showed that the blossom that exists in space ε 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 ε is equal to that in space Dε, which is an EC- space on I. Additional details can be seen in Theorem 3.1 in [38].For arbitrary(a,b)∈I2, the n+1 points Pi(a,b)≔φ(a[n-i],b[i]), i=0,…,n, are defined by the Chebyshev–Bézier points of φ about (a,b). When a=b, Pi(a,b)=φ(a), for all i=0,…,n. However, when a≠b, the Chebyshev–Bézier points are obtained as(5)Π0a,b=Φa,Πna,b=Φb,Πia,b=OsciΦa∩Oscn-iΦb,1≤i≤n-1.Moreover, when a≠b, a de Casteljau algorithm starting from points Πi(a,b) can be developed by using the three important properties of the blossom. At the nth step of this algorithm, the values of φ can be obtained as(6)Φx=∑i=0nBia,bxΠia,b,∑i=0nBia,bx=1,x∈I.Given that (1,Φ1,…,Φn) is a set of bases in the function of ε, the affine flat of the mother function Φ spans the whole space Rn. Therefore, points Π0(a,b),…,Πn(a,b) are affinely independent. Furthermore, Bi(a,b), i=0,…,n, from a basis of ε [37].


## 2.2. Construction of Trigonometric Basis Function

For arbitrary real numbersα,β∈[2,+∞), t∈[0,π/2], we consider constructing a basis function in the trigonometric function space Tα,β≔span1,sin2t,(1-sin⁡t)2/(1+α-2sin⁡t),(1-cos⁡t)2/(1+β-2cos⁡t). We can easily obtain the corresponding mother function, which is defined as (7)Φt≔sin2t,1-sin⁡t21+α-2sin⁡t,1-cos⁡t21+β-2cos⁡t,t∈0,π2.First, let us prove that the space(8)DTα,β=span2sin⁡tcos⁡t,-cos⁡t1-sin⁡tα+α-2sin⁡t1+α-2sin⁡t2,sin⁡t1-cos⁡tβ+β-2cos⁡t1+β-2cos⁡t2is a three-dimensional EC-space on [0,π/2]. Therefore, according to Theorem 2.1 of [38], Tα,β possesses a blossom, indicating that Tα,β is suitable for curve and surface designs.Theorem 1.
For arbitrary real numbersα,β∈[2,+∞), DTα,β is a three-dimensional EC-space on [0,π/2].Proof.
For arbitraryξi∈R(i=0,1,2),t∈[0,π/2], we consider a linear combination(9)ξ02sin⁡tcos⁡t+ξ1-cos⁡t1-sin⁡tα+α-2sin⁡t1+α-2sin⁡t2+ξ2sin⁡t1-cos⁡tβ+β-2cos⁡t1+β-2cos⁡t2=0.
Fort=0, from (9), we can easily obtain ξ1=0. In the same way, for t=π/2, from (9), we can obtain ξ2=0. Thus, ξ0=0. Therefore, DTα,β is a three-dimensional space.
Below, we prove thatDTα,β is an ECC-space in [0,π/2]. For arbitrary t∈[a,b]⊆(0,π/2), let (10)ut=-1-sin⁡tα+α-2sin⁡tsin⁡t1+α-2sin⁡t2′=cos⁡t3sin⁡t2-sin⁡t+1+α-23-sin2tsin⁡tα-2+2sin2t1+α-2sin⁡t3>0and(11)vt=1-cos⁡tβ+β-2cos⁡t1+β-2cos⁡t2′=sin⁡t3cos⁡t2-cos⁡t+1+β-23-cos2tcos⁡tβ-2+2cos2t1+β-2cos⁡t3>0.Direct computation indicates that(12)u′t=-sin⁡t3sin⁡t2-sin⁡t+1+α-23-sin2tsin⁡tα-2+2sin2t1+α-2sin⁡t3-2cos2t2+α-21+8sin⁡t+α-23sin2t5+1+sin⁡t1-sin⁡t+4α-22sin⁡t2-sin⁡tsin⁡t+sin⁡t+1sin3t1+α-2sin⁡t4<0,(13)v′t=sin⁡t3cos⁡t2-cos⁡t+1+β-23-cos2tcos⁡tβ-2+2cos2t1+β-2cos⁡t3+sin2t2+β-21+8cos⁡t+β-23cos3t5+1+cos⁡t1-cos⁡t+4β-22cos⁡t2-cos⁡tcos⁡t+cos⁡t+1cos3t1+β-2cos⁡t4>0.Thus, for the Wronskian of u(t) and v(t), (14)Wu,vt=utv′t-u′tvt>0,∀t∈0,π2.For t∈[a,b], we obtain the definition of the following weight functions:(15)ω0t=2sin⁡tcos⁡t,ω1t=Aut+Bvt,ω2t=CWu,vtAut+Bvt2,where A,B,C are three arbitrary positive real numbers. All of the three weight functions ωi(t)(i=0,1,2) are bounded, positive, and C∞ on closed bounded [a,b]. Next, we discuss the following ECC-spaces: (16)u0t=ω0t,u1t=ω0t∫atω1t1dt1,u2t=ω0t∫atω1t1∫at1ω2t2dt2dt1,We can easily verify that these functions ui(t)(i=0,1,2) are linear combinations of (17)2sin⁡tcos⁡t,-cos⁡t1-sin⁡tα+α-2sin⁡t1+α-2sin⁡t2,sin⁡t1-cos⁡tβ+β-2cos⁡t1+β-2cos⁡t2.Thus, DTα,β is an ECC-space on [a,b]. Furthermore, [a,b] are any subintervals of (0,π/2); hence, DTα,β is also an ECC-space on (0,π/2).
Proof thatDTα,β is an EC-space on [0,π/2] is provided. Thus, we must verify that the arbitrary nonzero element of DTα,β has, at most, two zeros (counting multiplicities) on [0,π/2].
Consider the arbitrary nonzero function(18)Gt=C02sin⁡tcos⁡t+C1-cos⁡t1-sin⁡tα+α-2sin⁡t1+α-2sin⁡t2+C2sin⁡t1-cos⁡tβ+β-2cos⁡t1+β-2cos⁡t2,where t∈[0,π/2]. G(t) has, at most, two zeros in (0,π/2) because DTα,β is an ECC-space in (0,π/2). Initially, G(t) is assumed to vanish at 0; then, we can obtain C1=0. Under these circumstances, if C2=0, then G(t) has two singular zeros at 0 and π/2. If C0=0, then 0 is a double zero of G(t). If C0C2>0, then G(t) has a singular zero at 0 but does not have a zero on (0,π/2]. If C0C2<0, then G(t) has a singular zero at 0 but not at π/2. In addition, for the function,(19)Kt≔2C0cos⁡t+C21-cos⁡tβ+β-2cos⁡t1+β-2cos⁡t2,direct computation yields (20)K′t=-2C0cos⁡t+C22b-12sin⁡t1+b-2cos⁡t3,where K(t) is a monotonic function on [0,π/2]. From these values, together with K(0)K(π/2)=2βC0C2<0, we can see that K(t) has exactly one zero in (0,π/2). Thus, we can immediately conclude that G(t)=sin⁡tK(t) has one zero in (0,π/2). Similarly, if G(t) vanishes at π/2, then function G(t) has, at most, two zeros on [0,π/2] (counting multiplicities as far as possible up to 2). In summary, the space DTα,β is an EC-space on [0,π/2].
Therefore,DTα,β is an EC-space on [0,π/2]. According to Theorem 3.1 in [43], a blossom exists in Tα,β. This theorem also means that Tα,β is suitable for curve and surface designs. According to Theorem 2.18 in [43], we can also infer that Tα,β has a B-basis on [0,π/2].Theorem 2.
The four Chebyshev–Bézier pointsPi≔Pi(0,π/2) of the mother function Φ(t) defined in (7) are given by (21)Π0=0,1,0,Π1=0,0,0,Π0=1,0,0,Π0=1,0,1.Moreover, the related new cubic trigonometric Bernstein-like basis Ai≔Ai(0,π/2) of Tα,β are given by (22)A0t=1-sin⁡t21+α-2sin⁡t,A1t=1-sin2t-1-sin⁡t21+α-2sin⁡t,A2t=1-cos2t-1-cos⁡t21+β-2cos⁡t,A3t=1-cos⁡t21+β-2cos⁡t.Proof.
Through the definition ofΦ(t) given in (7), we can easily obtain (23)Φ0=0,1,0,Φπ2=1,0,1,Φ′0=0,-α,0,Φ′π2=0,0,β,Φ′′0=2,2α-12,0,Φ′′π2=-2,0,2β-12.Thus, by simply computing, we have (24)Π0=Φ0=0,1,0,Π3=Φπ2=1,0,1,Π1=Osc1Φ0∩Osc2Φπ2=0,0,0,Π2=Osc2Φ0∩Osc1Φπ2=1,0,0.Thus, for any t∈[0,π/2], from Φ(t)=∑i=03Ai(t)Πi together with ∑i=03Ai(t)=1, we obtain the expressions of Ai(t), i=0,1,2,3.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 i=0,1,2,3, we have the following end-point properties:(25)Aij0=0,j=0,…,i-1,Aii0>0,Aijπ2=0,j=0,…,2-i.Thus, system (A0(t),A1(t),A2(t),A3(t)) is precisely the B-basis of Tα,β, which implies that (22) possesses totally positive and optimal shape preservation properties [33].We define a cubic trigonometric Bernstein-like basis with denominator shape parameters given in (22) as Ai(t;α,β), i=0,1,2,3, or Ai(t;α), i=0,1 and Ai(t;β), i=2,3 to facilitate the following discussion and distinguish the traditional literature. Figure 1 shows images of the DTB-like basis under different shape parameters.Figure 1
DTB-like basis under different shape parameters.


## 2.3. DTB-Like Curve with Denominator Shape Parameters

Definition 4.
Given control pointsPi(i=0,1,2,3) in R2 or R3, (26)Qt;α,β=∑i=03Ait;α,βPi,t∈0,π2,α,β∈2,+∞,are called a cubic DTB-like curve with two denominator shape parameters α and β.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:(27)Q0;α,β=P0,Qπ2;α,β=P3,Q′0;α,β=αP1-P0,Q′π2;α,β=βP3-P2,Q′′0;α,β=2α-12P0-P1+2P2-P1,Q′′π2;α,β=2β-12P3-P2+2P1-P2.For arbitraryα,β∈[2,+∞), the curve given in (26) has the end-point interpolation property, and P0P1 and P2P3 are the tangent lines of the curve at points P0 and P3, 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:(28)Qt;α,β=1-sin2t1-cos2t×1-sin⁡tsin⁡t00cos⁡t1-cos⁡t×11+sin⁡t1+α-2sin⁡t1+sin⁡t1+α-2sin⁡t-11+sin⁡t1+α-2sin⁡t000cos⁡tsin⁡t+cos⁡tsin⁡tsin⁡t+cos⁡t0001+cos⁡t1+β-2cos⁡t-11+cos⁡t1+β-2cos⁡t11+cos⁡t1+β-2cos⁡t×P0P1P2P3.By rewriting the curve expression into matrix form, we can rapidly obtain this algorithm. Figure 2 shows an example of this algorithm.Figure 2
Corner cutting algorithm.In addition, fort∈[0,π/2], we can rewrite (26) into the following form:(29)Qt;α,β=sin2P1+cos2P2+A0t;αP0-P1+A3t;βP3-P2.For arbitrary fixedt∈(0,π/2), A0(t;α) monotonically decreases with respect to the shape parameter α. This phenomenon also means that, as shape parameter α increases, the generated curve moves in the same direction as the edge P0-P1. By contrast, as shape parameter α decreases, the opposite is true for the generated curve. On the edge P3-P2, parameter β has similar influences. When α=β, as the shape parameters increase or decrease, the generated curve moves to the edge P2-P1 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.Figure 3
DTB-like curve with different shape parameters.


## 2.4. Representation of Elliptic and Parabolic Arcs

Forα=β=2, if the control points are P0=(x0+a,y0), P1=(x0+a,y0+b/2), P2=(x0+a/2,y0+b), and P3=(x0,y0+b), then the coordinates of the generated curve T(t;2,2) are(30)xt=x0+acos⁡t,yt=y0+bsin⁡t,t∈0,π2,where T(t;2,2) is a quarter of an elliptic arc whose center is located at (x0,y0). By constraining the parameter t on the desired interval [θ1,θ2], we can obtain an arc of an ellipse whose starting and ending angles are θ1 and θ2, respectively.Furthermore, forα=β=2, b-a>0, if the control points P0,P1,P2, and P3 with respective coordinates (b,c2b2+c1b+c0),(b,c2b2+c1b+c0),((a+b)/2,c2ab+c1(a+b)/2+c0), and (a,c2a2+c1a+c0), then we obtain the following from (26):(31)xt=b-acos⁡t+a,yt=c2b-acos⁡t+a2+c1b-acos⁡t+a+c0,t∈0,π2,which presents a segment of the parabola y=c2x2+c1x+c0, x∈[a,b].The discussion indicates that any arc of an ellipse or parabola can be exactly represented by using the proposed DTB-like curves. Figure4 shows the elliptic and parabolic segments generated by using the cubic DTB-like curves (marked with solid black lines).Figure 4
Representation of elliptic and parabolic arcs.


## 3. DT B-Spline-Like Basis Function

### 3.1. Construction of Trigonometric B-Spline-Like Basis Functions

Given a sequence of knotsu0<u1<⋯<un+4, we refer to U=(u0,u1,…,un+4) as a knot vector. Let hj=uj+1-uj, and tj(u)=π(u-uj)/2hj, j=0,1,…,n+3, for arbitrary real numbers αi,βi∈[2,+∞), i=0,1,…,n, we construct trigonometric B-spline-like basis functions with the following forms: (32)Biu=Bi,0ti=diA3ti;βi,u∈ui,ui+1,Bi,1ti+1=∑j=03ci+1,jAjti+1,αi+1,βi+1,u∈ui+1,ui+2,Bi,2ti+2=∑j=03bi+2,jAjti+2,αi+2,βi+2,u∈ui+2,ui+3,Bi,3ti+3=ai+3A0ti+3;αi+3,u∈ui+3,ui+4,0,u∉ui,ui+4,where Aj(ti;αi,βi), j=0,1,2,3 are the DTB-like basis functions given in (22).To determineai,bi,j,ci,j,di coefficient values of the basis functions (32), we add two restricting conditions: (1) including C2 continuity at each knot and (2) forming a partition of unity on [a,b]. Direct computation shows that(33)λi=αi+1-12βihi+αi+1βi-12hi+1,μi=αi+1hi+βihi+1,ϕi=λihi+μihi+1βihi+12,φi=λi-1hi+μi-1hi-1αihi-12ai=βi-1λi-2hi2λi-1μi-2hi-2+λi-2λi-1hi-1+λi-2μi-1hi,di=αi+1λi+1hi2λi+1μihi+λiλi+1hi+1+λiμi+1hi+2,bi,0=αiϕihi-1μi-1ai+1+βi-1φi-1hiμi-1di-2,ci,0=di-1,bi,1=ϕiai+1,ci,1=μi-1αihi-1di-1,bi,2=μiβihi+1ai+1,ci,2=φidi-1,bi,3=ai+1,ci,3=αi+1ϕi+1hiμiai+2+βiφihi+1μidi-1.We will provide the following definition to facilitate the following discussion and distinguish the traditional literature.Definition 5.
For arbitrary real numbersαi,βi∈[2,+∞), given a knot vector U, with the coefficients ai,bi,j,ci,j,di, (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 space(34)S≔s∈C2u0,un+4s.t..sui,ui+1∈Tαi,βi,αi,βi∈2,+∞,i=0,1,…,n+3,where(35)Tαi,βi≔span1,sin2t,1-sin⁡t21+αi-2sin⁡t,1-cos⁡t21+βi-2cos⁡t.In particular, forui+j=ui+jh,h>0, j=1,2,3,4 and αi+1=αi+2=αi+3=βi=βi+1=βi+2=2, direct computation yields(36)di=16,ci+1,0=16,ci+1,1=13,ci+1,2=23,ci+1,3=23,bi+2,0=23,bi+2,1=23,bi+2,2=13,bi+2,3=16,ai+3=16,from which we can immediately obtain the following explicit expressions of Bi(u):(37)Biu=161-cos⁡ti2u∈ui,ui+h,161-sin⁡ti+12+132sin⁡ti+11-sin⁡ti+1+232cos⁡ti+11-cos⁡ti+1+231-sin⁡ti+12u∈ui+h,ui+2h,231-sin⁡ti+22+232sin⁡ti+21-sin⁡ti+2+132cos⁡ti+21-cos⁡ti+2+161-sin⁡ti+22u∈ui+2h,ui+3h,161-sin⁡ti+32u∈ui+3h,ui+4h,0,u∉ui,ui+4h.Given an equidistant knot vector, we callBi(u) a uniform DT B-spline-like basis function, and the related knot vector U is called a uniform knot vector. On the contrary, given a nonequidistant knot vector, we call Bi(u) and U 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.Figure 5
Image of DT B-spline-like basis function.Lemma 7.
For all possiblei∈Z+, αi,βi∈[2,+∞), the coefficients ai,bi.j,ci,j,di have the following properties:(38)ai+bi0+ci0=1,bi1+ci1=1,bi2+ci2=1,bi3+ci3+di=1,di=ci+1,0,bi+2,0=ci+1,3,bi+2,3=ai+3,π2hiβidi=π2hi+1αi+1ci+1,1-ci+1,0,π2hi+1βi+1ci+1,3-ci+1,2=π2hi+2αi+2bi+2,1-bi+2,0,π2hi+2βi+2bi+2,3-bi+2,2=-π2hi+3αi+3ai+3,π2hi22βi-12di=π2hi+122αi+1-12ci+1,0+-2-2αi+1-12ci+1,1+2ci+1,2,π2hi+122ci+1,1+ci+1,2-2-2βi+1-12+2βi+1-12ci+1,3,=π2hi+222αi+2-12bi+2,0+-2-2αi+2-12bi+2,1+2bi+2,2,π2hi+222bi+2,1+bi+2,2-2-2βi+2-12+bi+2,32βi+2-12=π2hi+322αi+3-12,Proof.
The aforementioned expressions of the coefficientsai,bi.j,ci,j,di show that di=ci+1,0,bi+2,0=ci+1,3 and bi+2,3=ai+3. Straightforward computation yields(39)bi0=αiϕihi-1μi-1ai+1+βi-1φi-1hiμi-1di-2=αihi-1μi-1λihi+μihi+1βihi+12βiλi-1hi+12λiμi-1hi-1+λi-1λihi+λi-1μihi+1+βi-1hiμi-1λi-2hi-1+μi-2hi-2αi-1hi-22αi-1λi-1hi-22λi-1μi-2hi-2+λi-2λi-1hi-1+λi-2μi-1hi=αihi-1μi-11-λiμi-1hi-1λiμi-1hi-1+λi-1λihi+λi-1μihi+1+βi-1hiμi-11-λi-2μi-1hiλi-1μi-2hi-2+λi-2λi-1hi-1+λi-2μi-1hi=1-ai-di-1.bi0=αiϕihi-1μi-1ai+1+βi-1φi-1hiμi-1di-2=αihi-1μi-1λihi+μihi+1βihi+12βiλi-1hi+12λiμi-1hi-1+λi-1λihi+λi-1μihi+1+βi-1hiμi-1λi-2hi-1+μi-2hi-2αi-1hi-22αi-1λi-1hi-22λi-1μi-2hi-2+λi-2λi-1hi-1+λi-2μi-1hi=αihi-1μi-11-λiμi-1hi-1λiμi-1hi-1+λi-1λihi+λi-1μihi+1+βi-1hiμi-11-λi-2μi-1hiλi-1μi-2hi-2+λi-2λi-1hi-1+λi-2μi-1hi=1-ai-di-1.Similarly, ci,3=1-ai+1-di. Thus, we have ai+bi0+ci0=1 and bi3+ci3+di=1. For bi1+ci1 and bi2+ci2, we have (40)bi1+ci1=ϕiai+1+μi-1αihi-1di-1=λihi+μihi+1βihi+12βiλi-1hi+12λiμi-1hi-1+λi-1λihi+λi-1μihi+1+αihi-1+βi-1hiαihi-1αiλihi-12λiμi-1hi-1+λi-1λihi+λi-1μihi+1=λiλi-1hi+μiλi-1hi+1λiμi-1hi-1+λi-1λihi+λi-1μihi+1+αiλihi-12+βi-1λihi-1hiλiμi-1hi-1+λi-1λihi+λi-1μihi+1=1,(41)bi2+ci2=μiβihi+1ai+1+φidi-1=μiβihi+1βiλi-1hi+12λiμi-1hi-1+λi-1λihi+λi-1μihi+1+λi-1hi+μi-1hi-1αihi-12αiλihi-12λiμi-1hi-1+λi-1λihi+λi-1μihi+1=μiλi-1hi+1λiμi-1hi-1+λi-1λihi+λi-1μihi+1+λiλi-1hi+λiμi-1hi-1λiμi-1hi-1+λi-1λihi+λi-1μihi+1=1.Direct computation provides (42)π2hi+1αi+1ci+1,1-ci+1,0=π2hi+1αi+1μiαi+1hidi-di=π2hi+1αi+1αi+1λi+1hi2λi+1μihi+λiλi+1hi+1+λiμi+1hi+2μiαi+1hi-1=π2hi+1αi+1αi+1λi+1hi2λi+1μihi+λiλi+1hi+1+λiμi+1hi+2βihi+1αi+1hi=π2hiβidi,π2hi+1βi+1ci+1,3-ci+1,2=π2hi+1βi+1αi+2ϕi+2hi+1μi+1ai+3+βi+1φi+1hi+2μi+1di-φi+1di=π2hi+1βi+1αi+2ϕi+2hi+1μi+1ai+3-αi+2hi+1μi+1φi+1di=παi+2βi+12μi+1ϕi+2ai+3-φi+1di=π2hi+2αi+2ϕi+2ai+3-αi+2ϕi+2hi+1μi+1ai+3-βi+1φi+1hi+2μi+1di=π2hi+2αi+2bi+2,1-bi+2,0.π2hi+2βi+2bi+2,3-bi+2,2=π2hi+2βi+2ai+3-μi+2βi+2hi+3ai+3=-π2hi+3αi+3ai+3.Furthermore, (43)π2hi+122αi+1-12ci+1,0+-2-2αi+1-12ci+1,1+2ci+1,2=π2hi+122αi+1-12ci+1,0-ci+1,1+2ci+1,2-ci+1,1=π2hi+122αi+1-12-hi+1βidihiαi+1+2λihi+1αi+1hi2di=π2hi22βi-12di,π2hi+222bi+2,1+bi+2,2-2-2βi+2-12+bi+2,32βi+2-12=π2hi+222bi+2,1-bi+2,2+2βi+2-12bi+2,3-bi+2,2=π2hi+222ai+3λi+2hi+2βi+2hi+32-2βi+2-12hi+2αi+3hi+3βi+2ai+3=π2hi+332αi+3-12.Finally, (44)π2hi+122ci+1,1+ci+1,2-2-2βi+1-12+2βi+1-12ci+1,3=π2hi+122ci+1,1-ci+1,2+2βi+1-12ci+1,3-ci+1,2=π2hi+122ci+1,1-ci+1,2+2βi+1-12αi+2hi+1hi+2βi+1bi+2,1-bi+2,0=2π2hi+12ci+1,1-ci+1,2+2π2hi+12λi+1hi+22βi+1-αi+2-12hi+1hi+22hi+1bi+2,1-bi+2,0=2π2hi+12ci+1,1-ci+1,2+2π2hi+12λi+1hi+22βi+1hi+1bi+2,1-bi+2,0+2π2hi+22αi+2-12bi+2,0-bi+2,1=-2λi+2hi+2βi+2hi+32ai+3π2hi+22+2π2hi+22αi+2-12bi+2,0-bi+2,1=π2hi+222αi+2-12bi+2,0-bi+2,1+2bi+2,2-bi+2,1=π2hi+222αi+2-12bi+2,0+-2-2αi+2-12bi+2,1+2bi+2,2.These expressions simplify the lemma.


### 3.2. Properties of the DT B-Spline-Like Basis Functions

Theorem 8.
Foru∈[u3,un+1], we have ∑i=0nBi(u)=1.Proof.
Foru∈[ui,ui+1), i=3,4,…,n, Bj(u)=0 for j≠i-3,i-2,i-1,i. (45)Bi-3u=aiA0ti;αi,Bi-2u=∑j=03bi,jAjti;αi,βi,Bi-1u=∑j=03ci,jAjti;αi,βi,Biu=diA3ti,βi,By using Lemma 7, we have (46)∑i=0nBiu=aiA0ti;αi+∑j=03bi,jAjti;αi,βi+∑j=03ci,jAjti;αi,βi+diA3ti;βi=∑j=03Ajti;βi=1.Theorem 9.
Ifαi,βi∈[2,+∞), then Bi(u)>0 for ui<u<ui+4.Proof.
Ifαi,βi∈[2,+∞), then ai,bi,j,ci,j,di>0 for all possible i∈Z+,j=0,1,2,3. Thus, from the totally positive values of Aj(ti;αi,βi), we can immediately induce that Bi(u)>0 for ui<u<ui+4.Theorem 10.
For all possiblei∈Z+, αi,βi∈[2,+∞), the system B0(u),B1(u),…,Bn(u) is linearly independent from [u3,un+1].Proof.
Forξi∈R(i=0,1,…,n), u∈[u3,un+1], let(47)Bu=∑i=0nξiBiu=0.For αi,βi∈[2,+∞), direct computation yields (48)Bui=aiξi-3+bi,0ξi-2+ci,0ξi-1=0,B′ui=π2hiαiaiξi-2-ξi-3+βi-1di-1hihi-1ξi-1-ξi-2=0,B′′ui=π2hi22αi-12aiξi-3-ξi-2+2βi-1-12di-1hi2hi-12ξi-1-ξi-2=0,where i=3,4,…,n+1. Thus, we have(49)aiξi-3-ξi-2+ai+bi,0+ci,0ξi-2+ci,0ξi-1-ξi-2=0,-αiaiξi-3-ξi-2+βi-1di-1hihi-1ξi-1-ξi-2=0,αi-12aiξi-3-ξi-2+βi-1-12di-1hi2hi-12ξi-1-ξi-2=0.Considering that ai+bi0+ci0=1, we can obtain the following determinant with respect to the aforementioned linear system of equations:(50)
M
i
=
a
i
1
c
i
,
0
-
α
i
a
i
0
β
i
-
1
d
i
-
1
h
i
h
i
-
1
α
i
-
1
2
a
i
0
β
i
-
1
-
1
2
d
i
-
1
h
i
2
h
i
-
1
2
=
a
i
d
i
-
1
h
i
h
i
-
1
2
β
i
-
1
-
1
2
h
i
α
i
+
α
i
-
1
2
β
i
-
1
h
i
-
1
>
0
.Therefore, ξi-3=ξi-2=ξi-1=0 for i=3,4,…,n+1.Theorem 11.
Foru∈[ui,ui+1], αi,βi∈[2,+∞), i=3,4,…,n, the system (Bi-3(u),Bi-2(u),Bi-1(u),Bi(u) is a totally positive basis of the space span Tαi,βi.Proof.
Foru∈[ui,ui+1], i=3,4,…,n, we can obtain(51)Bi-3u,Bi-2u,Bi-1u,Biu=A0ti;αi,A1ti;αi,A2ti;βi,A3ti;βiHi,where (52)Hi=aibi,0ci,000bi,1ci,100bi,2ci,200bi,3ci,3diand αi,βi∈[2,+∞), ti(u)=(u-ui)/hi. Given that the system (A0(ti;αi),A1(ti;αi),A2(ti;βi),A3(ti;βi)) is the B-basis of the space Tαi,βi, according to Theorem 4.2 in [41], we can conclude that Hi is a nonsingular stochastic and totally positive matrix.
For arbitraryαi,βi∈[2,+∞), we have ai,bi,j,ci,j,di>0 for all possible i∈Z+, j=0,1,2,3. Moreover, we can conclude that Hi is stochastic from Lemma 7. Straightforward computation provides (53)bi,0ci,0bi,1ci,1=βi-1φi-1hiαihi-1di-2di-1>0,bi,0ci,0bi,2ci,2=hiλi-1μihi+1+λi-1λihi+λiμi-1hi-1βiμi-1hi-1hi+12ai+1di-1+βi-1φi-1φihiμi-1di-2di-1>0,bi,0ci,0bi,3ci,3=αiαi+1ϕiϕi+1hi-1hiμi-1μiai+1ai+2+hiλi-1μihi+1+λi-1λihi+λiμi-1hi-1μi-1μihi-1hi+1ai+1di-1+αi+1βi-1ϕi+1φi-1hi2μi-1μiai+2di-2+βi-1βiφi-1φihihi+1μi-1μidi-2di-1>0,bi,1ci,1bi,2ci,2=hiλi-1μihi+1+λi-1λihi+λiμi-1hi-1αiβiμi-1hi-12hi+12ai+1di-1>0,bi,1ci,1bi,3ci,3=hiλi-1μihi+1+λi-1λihi+λiμi-1hi-1αiμihi-12hi+1ai+1di-1+αi+1ϕiϕi+1hiμiai+1ai+2>0,bi,2ci,2bi,3ci,3=αi+1ϕi+1hiβihi+1ai+1ai+2>0.This verification shows that Hi is also a totally positive matrix and thus proves the theorem.Theorem 12.
For arbitraryαi,βi∈[2,+∞), given a nonuniform knot vector, Bi(u) has C2 continuity at each knot.Proof.
Without loss of generality, we consider the knotui+1. For arbitrary αi,βi∈[2,+∞), we have (54)Biui+1-=di,Biui+1+=ci+1,0,Bi′ui+1-=π2·βidihi,Bi′ui+1+=π2·αi+1ci+1,1-ci+1.0hi+1,B′′ui+1-=π222βi-12dihi2,B′′ui+1+=π222αi+1-12ci+1,0-ci+1,1+2ci+1,2-ci+1,1hi+12.From the aforementioned calculations and Lemma 7, we can easily find that the theorem is established at knot ui+1. The case of the continuity at other knots can be discussed similarly.Theorem 13.
For arbitraryαi=βi=2, given a nonuniform knot vector, Bi(u) has C2n-1(n=1,2,3,…) continuity at each knot.Proof.
We use mathematical induction to prove that the(2n-1)-order derivative of basis function (22) has the following form:(55)A02n-1=-1n(2cos⁡t+22n-2sin⁡2t,A12n-1=-1n2-cos⁡t+22n-2sin⁡2t,A22n-1=-1n2sin⁡t-22n-2sin⁡2t,A32n-1=-1n-2sin⁡t+22n-2sin⁡2t.When n=1, the derivative of the basis functions (22) is(56)A0′=-2cos⁡t-sin⁡2t,A1′=2cos⁡t-sin⁡2t,A2′=2-sin⁡t+sin⁡2t,A3′=2sin⁡t-sin⁡2t.These forms are satisfied when n=1.
We assume that the aforementioned forms are also satisfied whenn=k. Therefore, the (2k-1)-order derivative of the basis functions (22) is(57)A02k-1=-1k2cos⁡t+22k-2sin⁡2t,A12k-1=-1k2-cos⁡t+22k-2sin⁡2t,A22k-1=-1k2sin⁡t-22k-2sin⁡2t,A32k-1=-1k-2sin⁡t+22k-2sin⁡2t.By direct computing, we have(58)A02k-1′′=-1k+1(2cos⁡t+22k+1-2sin⁡2t,A12k-1′′=-1k+12-cos⁡t+22k+1-2sin⁡2t,A22k-1′′=-1k+12sin⁡t-22k+1-2sin⁡2t,A32k-1′′=-1k+1-2sin⁡t+22k+1-2sin⁡2t.These forms are satisfied when n=k+1. In summary, the 2n-1-order derivative of the basis functions (22) has the form of (55). Finally, we prove that the basis function Bi(u) is C2n-1(n=1,2,…) continuous at each knot.
Without loss of generality, we first consider the continuity at the knotui+1. From here and Remark 6, direct computation gives (59)Bi2n-1ui+1-=-12·-2·16,Bi2n-1ui+1+=-12·2·16+-12·2·13·-1=-12·-2·16.
Thus, we can immediately conclude thatBi(2n-1)(ui+1+)=Bi(2n-1)(ui+1-) for a uniform knot vector and all αi=βi=2. In summary, the theorem is established at knot ui+1. The continuity of the basis functions with respect to other knots can be discussed similarly.


### 3.3. DT B-Spline-Like Curves

Definition 14.
Given control pointsPi(i=0,1,…,n) in R2 or R3and a knot vector U, for arbitrary real numbers αi,βi∈[2,+∞),n≥3, u∈[u3,un+1],(60)Bu=∑i=0nBiuPiis called a DT B-spline-like curve with two denominator shape parameters αi,βi.
Foru∈[ui,ui+1], i=3,4,…,n, we rewrite (60) as the following curve segment:(61)Bu=∑j=i-3iBjuPj=aiPi-3+bi0Pi-2+ci0Pi-1A0ti;αi+bi1Pi-2+ci1Pi-1A1ti;αi+bi2Pi-2+ci2Pi-1A2ti;βi+bi3Pi-2+ci3Pi-1+diPiA3ti;βi.
From Theorems9 and 10, B(u) possesses affine invariance and lies in the convex hull of the points Pi-3,Pi-2,Pi-1,Pi for u∈[ui,ui+1]. Theorem 11 shows that B(u) possesses the variation diminishing property, indicating that B(u) is suitable for curve design. From Theorem 12, we can introduce the following theorem.Theorem 15.
Given a non-uniform knot vector,B(u) has C2 continuity at each knot. Furthermore, considering the simple knot ui(i=3,4,…,n+1), we have (62)Bui=aiPi-3+bi0Pi-2+ci0Pi-1,B′ui=π21hiαiaiPi-2-Pi-3+βi-1di-1hi-1Pi-1-Pi-2,B′′ui=π222hi2αi-12aiPi-3-Pi-2+2βi-1-12di-1hi-12Pi-1-Pi-2.Theorem 16.
Given a uniform knot vector,B(u) has C2n-1(n=1,2,3,…) continuity for all ai=βi=2 at each knot.Proof.
Without loss of generality, we consider the knotui+1. From Theorem 15 and Remark 6, for all ai=βi=2, we have (63)Biui+1-=16A3ti,Biui+1+=16A0ti+1+13A1ti+1+23A2ti+1+23A3ti+1,Bi2n-1ui+1+=16A32n-1ti=16-1n-2sin⁡t+22n-2sin⁡2t=-13-1n,Bi2n-1ui+1-=16A02n-1ti+13A12n-1ti+23A22n-1ti+23A32n-1ti=16-1n(2cos⁡t+22n-2sin⁡2t+13-1n2-cos⁡t+22n-2sin⁡2t+23-1n2sin⁡t-22n-2sin⁡2t+23-1n-2sin⁡t+22n-2sin⁡2t=-13-1n.
We can immediately conclude thatBi(2n-1)(ui+1+)=Bi(2n-1)(ui+1+) for a uniform knot vector and all ai=βi=2. In summary, the theorem is established at knot ui+1. The continuity of B(u) with respect to other knots can be similarly discussed.
The DT B-spline-like curve proposed in this study possesses two denominator shape parametersαi,βi. Therefore, when the control points are unchanged, we can easily adjust the proposed curve shape by changing the parameter values. From (61), the shape parameters αi-1,αi,αi+1,αi+2,βi-2,βi-1,βi,βi+1 affect the curve segment B(u), u∈[ui,ui+1]. Therefore, shape parameters αi affect four curve segments [ui-2,ui+2], and βi affects four curve segments [ui-1,ui+3]. Furthermore, the shape of the proposed curve can be predicted on the basis of the shape parameter value. As αi and βi increase, the coefficients of control points Pi-3 and Pi decrease while the coefficients of Pi-2 and Pi-1 increase. Therefore, as αi and βi increase simultaneously, the curve segments B(u)(u∈[ui,ui+1]) tend to Pi-2Pi-1. If αi and βi increase, the curve segments B(u)(u∈[ui,ui+1]) will tend to control point Pi-2 or Pi-1, respectively.
Figure6 shows B(u) with different denominator shape parameters. The left figure shows B(u), which is generated by setting all αi=βi=2 (black lines); the blue line is generated by changing one αi to 4; and the green line is generated by changing one βi to 4. The right figure shows that B(u) is generated by setting all αi=2or5 and βi=2or5.Figure 6
DT B-spline-like curves.


### 3.4. DT B-Spline-Like Surface

Definition 17.
Given control pointsPij(i=0,1,…,m,j=0,1,…,n) in R3 and two knot vectors U=(u0,u1,…,un+4), V=(v0,v1,…,vn+4), for arbitrary real numbers αij,βij∈[2,+∞),m,n≥3,u∈[u3,um+1], v∈[v3,vn+3],(64)Bu,v=∑i=0m∑j=0nBiuBjvPijis called a DT B-spline-like surface with four denominator shape parameters, namely, αi1,βi1,αj2,βj2, where αi1,βi1 belong to basis function Bi(u) and αj2,βj2 belong to basis function Bj(u).Theorem 18.
Given two nonuniform knot vectorsU and V, the DT B-spline-like surface B(u,v) possesses C2 continuity at each knot.Theorem 19.
Foru∈[u3,um+1], v∈[v3,vn+1], the DT B-spline-like surface B(u,v) possesses C2n-1(n=1,2,3,…) continuity for all aij=βij=2 at each knot.Proof.
Without loss of generality, we consider the continuity at the region[ui,ui+1)×[vj,vj+1). For u∈ui,ui+1, v∈vj,vj+1, i=3,4,…,m, j=3,4,…,n, Bk(u)=0, Bl(v)=0 for k≠i-3,i-2,i-1, l≠j-3,j-2,j-1,j. (65)Biu=aiA0ti;αi,Bi-1u=∑k=03bi,kAkti;αi,βi,Bi-2u=∑k=03ci,kAkti;αi,βi,Bi-3u=diA3ti,βi,Bju=ajA0tj;αj,Bj-1u=∑l=03bj,lAltj;αj,βj,Bj-2u=∑l=03cj,lAltj;αj,βj,Bj-3u=djA3tj,βj.From Definition 17, we have(66)Biju,v=∑s=i-3i∑t=j-3jBsuBtvPst.We prove that the continuity of the u direction and the continuity of the v direction can be similarly discussed. Considering the continuity at the knot (ui+1,vi+1), we have(67)δ2n-1Bijui+1-,vj+1δu2n-1=∑s=i-3i∑t=j-3jδ2n-1Bsui+1-δu2n-1Btvj+1Pst,δ2n-1Bijui+1+,vj+1δu2n-1=∑s=i-3i∑t=j-3jδ2n-1Bsui+1+δu2n-1Btvj+1Pst.From Theorem 16, for all aij=βij=2, we have(68)δ2n-1Bijui+1-,vj+1δu2n-1=∑s=i-3i∑t=j-3j-13-1nBtvj+1Pst=δ2n-1Bijui+1+,vj+1δu2n-1.In summary, the theorem is established at knot (ui+1,vi+1). The continuity of B(u,v) given in (64) with respect to other knots can be similarly discussed.
The given spline surfaceB(u,v) possesses four local denominator shape parameters, namely, αi1,βi1,αj2,βj2. By changing the value of the shape parameter, we can adjust the shape of B(u,v) flexibly. According to (66), shape parameters αi1-1,αi1,αi1+1,αi1+2, βi1-2,βi1-1,βi1,βi1+1,αi2-1, αi2,αi2+1,αi2+2,βi2-2,βi2-1, βi2,βi2+1 affect the surface patch R(u,v),u∈[ui,ui+1],v∈[vj,vj+1]. Therefore, shape parameters αi1,αj2 affect surface patch [ui-2,ui+2]×[vj-2,vj+2], shape parameters βi1,βj2 affect surface patch [ui-1,ui+3]×[vj-1,vj+3], shape parameters αi1,βj2 affect the surface patch [ui-2,ui+2]×[vj-1,vj+3], and shape parameters αj2,βi1 affect the surface patch [ui-1,ui+3]×[vj-2,vj+2]. These results show that parameters αi1,βi1,αj2,βj2 serve to local control tension in the surface. Increasing αi1,βi1,αj2,βj2 locally moves the surface R(u,v),u∈[ui,ui+1], v∈[vj,vj+1] toward the control polygon. Figure 7 shows B(u,v) with different shape parameters. The figure on the left shows B(u,v) generated when all shape parameters are set αi1=βi1=αj2=βj2=2. The figure on the right shows B(u,v) generated when all shape parameters are set αi1=βi1=αj2=βj2=5.Figure 7
DT B-spline-like surface.


## 3.1. Construction of Trigonometric B-Spline-Like Basis Functions

Given a sequence of knotsu0<u1<⋯<un+4, we refer to U=(u0,u1,…,un+4) as a knot vector. Let hj=uj+1-uj, and tj(u)=π(u-uj)/2hj, j=0,1,…,n+3, for arbitrary real numbers αi,βi∈[2,+∞), i=0,1,…,n, we construct trigonometric B-spline-like basis functions with the following forms: (32)Biu=Bi,0ti=diA3ti;βi,u∈ui,ui+1,Bi,1ti+1=∑j=03ci+1,jAjti+1,αi+1,βi+1,u∈ui+1,ui+2,Bi,2ti+2=∑j=03bi+2,jAjti+2,αi+2,βi+2,u∈ui+2,ui+3,Bi,3ti+3=ai+3A0ti+3;αi+3,u∈ui+3,ui+4,0,u∉ui,ui+4,where Aj(ti;αi,βi), j=0,1,2,3 are the DTB-like basis functions given in (22).To determineai,bi,j,ci,j,di coefficient values of the basis functions (32), we add two restricting conditions: (1) including C2 continuity at each knot and (2) forming a partition of unity on [a,b]. Direct computation shows that(33)λi=αi+1-12βihi+αi+1βi-12hi+1,μi=αi+1hi+βihi+1,ϕi=λihi+μihi+1βihi+12,φi=λi-1hi+μi-1hi-1αihi-12ai=βi-1λi-2hi2λi-1μi-2hi-2+λi-2λi-1hi-1+λi-2μi-1hi,di=αi+1λi+1hi2λi+1μihi+λiλi+1hi+1+λiμi+1hi+2,bi,0=αiϕihi-1μi-1ai+1+βi-1φi-1hiμi-1di-2,ci,0=di-1,bi,1=ϕiai+1,ci,1=μi-1αihi-1di-1,bi,2=μiβihi+1ai+1,ci,2=φidi-1,bi,3=ai+1,ci,3=αi+1ϕi+1hiμiai+2+βiφihi+1μidi-1.We will provide the following definition to facilitate the following discussion and distinguish the traditional literature.Definition 5.
For arbitrary real numbersαi,βi∈[2,+∞), given a knot vector U, with the coefficients ai,bi,j,ci,j,di, (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 space(34)S≔s∈C2u0,un+4s.t..sui,ui+1∈Tαi,βi,αi,βi∈2,+∞,i=0,1,…,n+3,where(35)Tαi,βi≔span1,sin2t,1-sin⁡t21+αi-2sin⁡t,1-cos⁡t21+βi-2cos⁡t.In particular, forui+j=ui+jh,h>0, j=1,2,3,4 and αi+1=αi+2=αi+3=βi=βi+1=βi+2=2, direct computation yields(36)di=16,ci+1,0=16,ci+1,1=13,ci+1,2=23,ci+1,3=23,bi+2,0=23,bi+2,1=23,bi+2,2=13,bi+2,3=16,ai+3=16,from which we can immediately obtain the following explicit expressions of Bi(u):(37)Biu=161-cos⁡ti2u∈ui,ui+h,161-sin⁡ti+12+132sin⁡ti+11-sin⁡ti+1+232cos⁡ti+11-cos⁡ti+1+231-sin⁡ti+12u∈ui+h,ui+2h,231-sin⁡ti+22+232sin⁡ti+21-sin⁡ti+2+132cos⁡ti+21-cos⁡ti+2+161-sin⁡ti+22u∈ui+2h,ui+3h,161-sin⁡ti+32u∈ui+3h,ui+4h,0,u∉ui,ui+4h.Given an equidistant knot vector, we callBi(u) a uniform DT B-spline-like basis function, and the related knot vector U is called a uniform knot vector. On the contrary, given a nonequidistant knot vector, we call Bi(u) and U 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.Figure 5
Image of DT B-spline-like basis function.Lemma 7.
For all possiblei∈Z+, αi,βi∈[2,+∞), the coefficients ai,bi.j,ci,j,di have the following properties:(38)ai+bi0+ci0=1,bi1+ci1=1,bi2+ci2=1,bi3+ci3+di=1,di=ci+1,0,bi+2,0=ci+1,3,bi+2,3=ai+3,π2hiβidi=π2hi+1αi+1ci+1,1-ci+1,0,π2hi+1βi+1ci+1,3-ci+1,2=π2hi+2αi+2bi+2,1-bi+2,0,π2hi+2βi+2bi+2,3-bi+2,2=-π2hi+3αi+3ai+3,π2hi22βi-12di=π2hi+122αi+1-12ci+1,0+-2-2αi+1-12ci+1,1+2ci+1,2,π2hi+122ci+1,1+ci+1,2-2-2βi+1-12+2βi+1-12ci+1,3,=π2hi+222αi+2-12bi+2,0+-2-2αi+2-12bi+2,1+2bi+2,2,π2hi+222bi+2,1+bi+2,2-2-2βi+2-12+bi+2,32βi+2-12=π2hi+322αi+3-12,Proof.
The aforementioned expressions of the coefficientsai,bi.j,ci,j,di show that di=ci+1,0,bi+2,0=ci+1,3 and bi+2,3=ai+3. Straightforward computation yields(39)bi0=αiϕihi-1μi-1ai+1+βi-1φi-1hiμi-1di-2=αihi-1μi-1λihi+μihi+1βihi+12βiλi-1hi+12λiμi-1hi-1+λi-1λihi+λi-1μihi+1+βi-1hiμi-1λi-2hi-1+μi-2hi-2αi-1hi-22αi-1λi-1hi-22λi-1μi-2hi-2+λi-2λi-1hi-1+λi-2μi-1hi=αihi-1μi-11-λiμi-1hi-1λiμi-1hi-1+λi-1λihi+λi-1μihi+1+βi-1hiμi-11-λi-2μi-1hiλi-1μi-2hi-2+λi-2λi-1hi-1+λi-2μi-1hi=1-ai-di-1.bi0=αiϕihi-1μi-1ai+1+βi-1φi-1hiμi-1di-2=αihi-1μi-1λihi+μihi+1βihi+12βiλi-1hi+12λiμi-1hi-1+λi-1λihi+λi-1μihi+1+βi-1hiμi-1λi-2hi-1+μi-2hi-2αi-1hi-22αi-1λi-1hi-22λi-1μi-2hi-2+λi-2λi-1hi-1+λi-2μi-1hi=αihi-1μi-11-λiμi-1hi-1λiμi-1hi-1+λi-1λihi+λi-1μihi+1+βi-1hiμi-11-λi-2μi-1hiλi-1μi-2hi-2+λi-2λi-1hi-1+λi-2μi-1hi=1-ai-di-1.Similarly, ci,3=1-ai+1-di. Thus, we have ai+bi0+ci0=1 and bi3+ci3+di=1. For bi1+ci1 and bi2+ci2, we have (40)bi1+ci1=ϕiai+1+μi-1αihi-1di-1=λihi+μihi+1βihi+12βiλi-1hi+12λiμi-1hi-1+λi-1λihi+λi-1μihi+1+αihi-1+βi-1hiαihi-1αiλihi-12λiμi-1hi-1+λi-1λihi+λi-1μihi+1=λiλi-1hi+μiλi-1hi+1λiμi-1hi-1+λi-1λihi+λi-1μihi+1+αiλihi-12+βi-1λihi-1hiλiμi-1hi-1+λi-1λihi+λi-1μihi+1=1,(41)bi2+ci2=μiβihi+1ai+1+φidi-1=μiβihi+1βiλi-1hi+12λiμi-1hi-1+λi-1λihi+λi-1μihi+1+λi-1hi+μi-1hi-1αihi-12αiλihi-12λiμi-1hi-1+λi-1λihi+λi-1μihi+1=μiλi-1hi+1λiμi-1hi-1+λi-1λihi+λi-1μihi+1+λiλi-1hi+λiμi-1hi-1λiμi-1hi-1+λi-1λihi+λi-1μihi+1=1.Direct computation provides (42)π2hi+1αi+1ci+1,1-ci+1,0=π2hi+1αi+1μiαi+1hidi-di=π2hi+1αi+1αi+1λi+1hi2λi+1μihi+λiλi+1hi+1+λiμi+1hi+2μiαi+1hi-1=π2hi+1αi+1αi+1λi+1hi2λi+1μihi+λiλi+1hi+1+λiμi+1hi+2βihi+1αi+1hi=π2hiβidi,π2hi+1βi+1ci+1,3-ci+1,2=π2hi+1βi+1αi+2ϕi+2hi+1μi+1ai+3+βi+1φi+1hi+2μi+1di-φi+1di=π2hi+1βi+1αi+2ϕi+2hi+1μi+1ai+3-αi+2hi+1μi+1φi+1di=παi+2βi+12μi+1ϕi+2ai+3-φi+1di=π2hi+2αi+2ϕi+2ai+3-αi+2ϕi+2hi+1μi+1ai+3-βi+1φi+1hi+2μi+1di=π2hi+2αi+2bi+2,1-bi+2,0.π2hi+2βi+2bi+2,3-bi+2,2=π2hi+2βi+2ai+3-μi+2βi+2hi+3ai+3=-π2hi+3αi+3ai+3.Furthermore, (43)π2hi+122αi+1-12ci+1,0+-2-2αi+1-12ci+1,1+2ci+1,2=π2hi+122αi+1-12ci+1,0-ci+1,1+2ci+1,2-ci+1,1=π2hi+122αi+1-12-hi+1βidihiαi+1+2λihi+1αi+1hi2di=π2hi22βi-12di,π2hi+222bi+2,1+bi+2,2-2-2βi+2-12+bi+2,32βi+2-12=π2hi+222bi+2,1-bi+2,2+2βi+2-12bi+2,3-bi+2,2=π2hi+222ai+3λi+2hi+2βi+2hi+32-2βi+2-12hi+2αi+3hi+3βi+2ai+3=π2hi+332αi+3-12.Finally, (44)π2hi+122ci+1,1+ci+1,2-2-2βi+1-12+2βi+1-12ci+1,3=π2hi+122ci+1,1-ci+1,2+2βi+1-12ci+1,3-ci+1,2=π2hi+122ci+1,1-ci+1,2+2βi+1-12αi+2hi+1hi+2βi+1bi+2,1-bi+2,0=2π2hi+12ci+1,1-ci+1,2+2π2hi+12λi+1hi+22βi+1-αi+2-12hi+1hi+22hi+1bi+2,1-bi+2,0=2π2hi+12ci+1,1-ci+1,2+2π2hi+12λi+1hi+22βi+1hi+1bi+2,1-bi+2,0+2π2hi+22αi+2-12bi+2,0-bi+2,1=-2λi+2hi+2βi+2hi+32ai+3π2hi+22+2π2hi+22αi+2-12bi+2,0-bi+2,1=π2hi+222αi+2-12bi+2,0-bi+2,1+2bi+2,2-bi+2,1=π2hi+222αi+2-12bi+2,0+-2-2αi+2-12bi+2,1+2bi+2,2.These expressions simplify the lemma.


## 3.2. Properties of the DT B-Spline-Like Basis Functions

Theorem 8.
Foru∈[u3,un+1], we have ∑i=0nBi(u)=1.Proof.
Foru∈[ui,ui+1), i=3,4,…,n, Bj(u)=0 for j≠i-3,i-2,i-1,i. (45)Bi-3u=aiA0ti;αi,Bi-2u=∑j=03bi,jAjti;αi,βi,Bi-1u=∑j=03ci,jAjti;αi,βi,Biu=diA3ti,βi,By using Lemma 7, we have (46)∑i=0nBiu=aiA0ti;αi+∑j=03bi,jAjti;αi,βi+∑j=03ci,jAjti;αi,βi+diA3ti;βi=∑j=03Ajti;βi=1.Theorem 9.
Ifαi,βi∈[2,+∞), then Bi(u)>0 for ui<u<ui+4.Proof.
Ifαi,βi∈[2,+∞), then ai,bi,j,ci,j,di>0 for all possible i∈Z+,j=0,1,2,3. Thus, from the totally positive values of Aj(ti;αi,βi), we can immediately induce that Bi(u)>0 for ui<u<ui+4.Theorem 10.
For all possiblei∈Z+, αi,βi∈[2,+∞), the system B0(u),B1(u),…,Bn(u) is linearly independent from [u3,un+1].Proof.
Forξi∈R(i=0,1,…,n), u∈[u3,un+1], let(47)Bu=∑i=0nξiBiu=0.For αi,βi∈[2,+∞), direct computation yields (48)Bui=aiξi-3+bi,0ξi-2+ci,0ξi-1=0,B′ui=π2hiαiaiξi-2-ξi-3+βi-1di-1hihi-1ξi-1-ξi-2=0,B′′ui=π2hi22αi-12aiξi-3-ξi-2+2βi-1-12di-1hi2hi-12ξi-1-ξi-2=0,where i=3,4,…,n+1. Thus, we have(49)aiξi-3-ξi-2+ai+bi,0+ci,0ξi-2+ci,0ξi-1-ξi-2=0,-αiaiξi-3-ξi-2+βi-1di-1hihi-1ξi-1-ξi-2=0,αi-12aiξi-3-ξi-2+βi-1-12di-1hi2hi-12ξi-1-ξi-2=0.Considering that ai+bi0+ci0=1, we can obtain the following determinant with respect to the aforementioned linear system of equations:(50)
M
i
=
a
i
1
c
i
,
0
-
α
i
a
i
0
β
i
-
1
d
i
-
1
h
i
h
i
-
1
α
i
-
1
2
a
i
0
β
i
-
1
-
1
2
d
i
-
1
h
i
2
h
i
-
1
2
=
a
i
d
i
-
1
h
i
h
i
-
1
2
β
i
-
1
-
1
2
h
i
α
i
+
α
i
-
1
2
β
i
-
1
h
i
-
1
>
0
.Therefore, ξi-3=ξi-2=ξi-1=0 for i=3,4,…,n+1.Theorem 11.
Foru∈[ui,ui+1], αi,βi∈[2,+∞), i=3,4,…,n, the system (Bi-3(u),Bi-2(u),Bi-1(u),Bi(u) is a totally positive basis of the space span Tαi,βi.Proof.
Foru∈[ui,ui+1], i=3,4,…,n, we can obtain(51)Bi-3u,Bi-2u,Bi-1u,Biu=A0ti;αi,A1ti;αi,A2ti;βi,A3ti;βiHi,where (52)Hi=aibi,0ci,000bi,1ci,100bi,2ci,200bi,3ci,3diand αi,βi∈[2,+∞), ti(u)=(u-ui)/hi. Given that the system (A0(ti;αi),A1(ti;αi),A2(ti;βi),A3(ti;βi)) is the B-basis of the space Tαi,βi, according to Theorem 4.2 in [41], we can conclude that Hi is a nonsingular stochastic and totally positive matrix.
For arbitraryαi,βi∈[2,+∞), we have ai,bi,j,ci,j,di>0 for all possible i∈Z+, j=0,1,2,3. Moreover, we can conclude that Hi is stochastic from Lemma 7. Straightforward computation provides (53)bi,0ci,0bi,1ci,1=βi-1φi-1hiαihi-1di-2di-1>0,bi,0ci,0bi,2ci,2=hiλi-1μihi+1+λi-1λihi+λiμi-1hi-1βiμi-1hi-1hi+12ai+1di-1+βi-1φi-1φihiμi-1di-2di-1>0,bi,0ci,0bi,3ci,3=αiαi+1ϕiϕi+1hi-1hiμi-1μiai+1ai+2+hiλi-1μihi+1+λi-1λihi+λiμi-1hi-1μi-1μihi-1hi+1ai+1di-1+αi+1βi-1ϕi+1φi-1hi2μi-1μiai+2di-2+βi-1βiφi-1φihihi+1μi-1μidi-2di-1>0,bi,1ci,1bi,2ci,2=hiλi-1μihi+1+λi-1λihi+λiμi-1hi-1αiβiμi-1hi-12hi+12ai+1di-1>0,bi,1ci,1bi,3ci,3=hiλi-1μihi+1+λi-1λihi+λiμi-1hi-1αiμihi-12hi+1ai+1di-1+αi+1ϕiϕi+1hiμiai+1ai+2>0,bi,2ci,2bi,3ci,3=αi+1ϕi+1hiβihi+1ai+1ai+2>0.This verification shows that Hi is also a totally positive matrix and thus proves the theorem.Theorem 12.
For arbitraryαi,βi∈[2,+∞), given a nonuniform knot vector, Bi(u) has C2 continuity at each knot.Proof.
Without loss of generality, we consider the knotui+1. For arbitrary αi,βi∈[2,+∞), we have (54)Biui+1-=di,Biui+1+=ci+1,0,Bi′ui+1-=π2·βidihi,Bi′ui+1+=π2·αi+1ci+1,1-ci+1.0hi+1,B′′ui+1-=π222βi-12dihi2,B′′ui+1+=π222αi+1-12ci+1,0-ci+1,1+2ci+1,2-ci+1,1hi+12.From the aforementioned calculations and Lemma 7, we can easily find that the theorem is established at knot ui+1. The case of the continuity at other knots can be discussed similarly.Theorem 13.
For arbitraryαi=βi=2, given a nonuniform knot vector, Bi(u) has C2n-1(n=1,2,3,…) continuity at each knot.Proof.
We use mathematical induction to prove that the(2n-1)-order derivative of basis function (22) has the following form:(55)A02n-1=-1n(2cos⁡t+22n-2sin⁡2t,A12n-1=-1n2-cos⁡t+22n-2sin⁡2t,A22n-1=-1n2sin⁡t-22n-2sin⁡2t,A32n-1=-1n-2sin⁡t+22n-2sin⁡2t.When n=1, the derivative of the basis functions (22) is(56)A0′=-2cos⁡t-sin⁡2t,A1′=2cos⁡t-sin⁡2t,A2′=2-sin⁡t+sin⁡2t,A3′=2sin⁡t-sin⁡2t.These forms are satisfied when n=1.
We assume that the aforementioned forms are also satisfied whenn=k. Therefore, the (2k-1)-order derivative of the basis functions (22) is(57)A02k-1=-1k2cos⁡t+22k-2sin⁡2t,A12k-1=-1k2-cos⁡t+22k-2sin⁡2t,A22k-1=-1k2sin⁡t-22k-2sin⁡2t,A32k-1=-1k-2sin⁡t+22k-2sin⁡2t.By direct computing, we have(58)A02k-1′′=-1k+1(2cos⁡t+22k+1-2sin⁡2t,A12k-1′′=-1k+12-cos⁡t+22k+1-2sin⁡2t,A22k-1′′=-1k+12sin⁡t-22k+1-2sin⁡2t,A32k-1′′=-1k+1-2sin⁡t+22k+1-2sin⁡2t.These forms are satisfied when n=k+1. In summary, the 2n-1-order derivative of the basis functions (22) has the form of (55). Finally, we prove that the basis function Bi(u) is C2n-1(n=1,2,…) continuous at each knot.
Without loss of generality, we first consider the continuity at the knotui+1. From here and Remark 6, direct computation gives (59)Bi2n-1ui+1-=-12·-2·16,Bi2n-1ui+1+=-12·2·16+-12·2·13·-1=-12·-2·16.
Thus, we can immediately conclude thatBi(2n-1)(ui+1+)=Bi(2n-1)(ui+1-) for a uniform knot vector and all αi=βi=2. In summary, the theorem is established at knot ui+1. The continuity of the basis functions with respect to other knots can be discussed similarly.


## 3.3. DT B-Spline-Like Curves

Definition 14.
Given control pointsPi(i=0,1,…,n) in R2 or R3and a knot vector U, for arbitrary real numbers αi,βi∈[2,+∞),n≥3, u∈[u3,un+1],(60)Bu=∑i=0nBiuPiis called a DT B-spline-like curve with two denominator shape parameters αi,βi.
Foru∈[ui,ui+1], i=3,4,…,n, we rewrite (60) as the following curve segment:(61)Bu=∑j=i-3iBjuPj=aiPi-3+bi0Pi-2+ci0Pi-1A0ti;αi+bi1Pi-2+ci1Pi-1A1ti;αi+bi2Pi-2+ci2Pi-1A2ti;βi+bi3Pi-2+ci3Pi-1+diPiA3ti;βi.
From Theorems9 and 10, B(u) possesses affine invariance and lies in the convex hull of the points Pi-3,Pi-2,Pi-1,Pi for u∈[ui,ui+1]. Theorem 11 shows that B(u) possesses the variation diminishing property, indicating that B(u) is suitable for curve design. From Theorem 12, we can introduce the following theorem.Theorem 15.
Given a non-uniform knot vector,B(u) has C2 continuity at each knot. Furthermore, considering the simple knot ui(i=3,4,…,n+1), we have (62)Bui=aiPi-3+bi0Pi-2+ci0Pi-1,B′ui=π21hiαiaiPi-2-Pi-3+βi-1di-1hi-1Pi-1-Pi-2,B′′ui=π222hi2αi-12aiPi-3-Pi-2+2βi-1-12di-1hi-12Pi-1-Pi-2.Theorem 16.
Given a uniform knot vector,B(u) has C2n-1(n=1,2,3,…) continuity for all ai=βi=2 at each knot.Proof.
Without loss of generality, we consider the knotui+1. From Theorem 15 and Remark 6, for all ai=βi=2, we have (63)Biui+1-=16A3ti,Biui+1+=16A0ti+1+13A1ti+1+23A2ti+1+23A3ti+1,Bi2n-1ui+1+=16A32n-1ti=16-1n-2sin⁡t+22n-2sin⁡2t=-13-1n,Bi2n-1ui+1-=16A02n-1ti+13A12n-1ti+23A22n-1ti+23A32n-1ti=16-1n(2cos⁡t+22n-2sin⁡2t+13-1n2-cos⁡t+22n-2sin⁡2t+23-1n2sin⁡t-22n-2sin⁡2t+23-1n-2sin⁡t+22n-2sin⁡2t=-13-1n.
We can immediately conclude thatBi(2n-1)(ui+1+)=Bi(2n-1)(ui+1+) for a uniform knot vector and all ai=βi=2. In summary, the theorem is established at knot ui+1. The continuity of B(u) with respect to other knots can be similarly discussed.
The DT B-spline-like curve proposed in this study possesses two denominator shape parametersαi,βi. Therefore, when the control points are unchanged, we can easily adjust the proposed curve shape by changing the parameter values. From (61), the shape parameters αi-1,αi,αi+1,αi+2,βi-2,βi-1,βi,βi+1 affect the curve segment B(u), u∈[ui,ui+1]. Therefore, shape parameters αi affect four curve segments [ui-2,ui+2], and βi affects four curve segments [ui-1,ui+3]. Furthermore, the shape of the proposed curve can be predicted on the basis of the shape parameter value. As αi and βi increase, the coefficients of control points Pi-3 and Pi decrease while the coefficients of Pi-2 and Pi-1 increase. Therefore, as αi and βi increase simultaneously, the curve segments B(u)(u∈[ui,ui+1]) tend to Pi-2Pi-1. If αi and βi increase, the curve segments B(u)(u∈[ui,ui+1]) will tend to control point Pi-2 or Pi-1, respectively.
Figure6 shows B(u) with different denominator shape parameters. The left figure shows B(u), which is generated by setting all αi=βi=2 (black lines); the blue line is generated by changing one αi to 4; and the green line is generated by changing one βi to 4. The right figure shows that B(u) is generated by setting all αi=2or5 and βi=2or5.Figure 6
DT B-spline-like curves.


## 3.4. DT B-Spline-Like Surface

Definition 17.
Given control pointsPij(i=0,1,…,m,j=0,1,…,n) in R3 and two knot vectors U=(u0,u1,…,un+4), V=(v0,v1,…,vn+4), for arbitrary real numbers αij,βij∈[2,+∞),m,n≥3,u∈[u3,um+1], v∈[v3,vn+3],(64)Bu,v=∑i=0m∑j=0nBiuBjvPijis called a DT B-spline-like surface with four denominator shape parameters, namely, αi1,βi1,αj2,βj2, where αi1,βi1 belong to basis function Bi(u) and αj2,βj2 belong to basis function Bj(u).Theorem 18.
Given two nonuniform knot vectorsU and V, the DT B-spline-like surface B(u,v) possesses C2 continuity at each knot.Theorem 19.
Foru∈[u3,um+1], v∈[v3,vn+1], the DT B-spline-like surface B(u,v) possesses C2n-1(n=1,2,3,…) continuity for all aij=βij=2 at each knot.Proof.
Without loss of generality, we consider the continuity at the region[ui,ui+1)×[vj,vj+1). For u∈ui,ui+1, v∈vj,vj+1, i=3,4,…,m, j=3,4,…,n, Bk(u)=0, Bl(v)=0 for k≠i-3,i-2,i-1, l≠j-3,j-2,j-1,j. (65)Biu=aiA0ti;αi,Bi-1u=∑k=03bi,kAkti;αi,βi,Bi-2u=∑k=03ci,kAkti;αi,βi,Bi-3u=diA3ti,βi,Bju=ajA0tj;αj,Bj-1u=∑l=03bj,lAltj;αj,βj,Bj-2u=∑l=03cj,lAltj;αj,βj,Bj-3u=djA3tj,βj.From Definition 17, we have(66)Biju,v=∑s=i-3i∑t=j-3jBsuBtvPst.We prove that the continuity of the u direction and the continuity of the v direction can be similarly discussed. Considering the continuity at the knot (ui+1,vi+1), we have(67)δ2n-1Bijui+1-,vj+1δu2n-1=∑s=i-3i∑t=j-3jδ2n-1Bsui+1-δu2n-1Btvj+1Pst,δ2n-1Bijui+1+,vj+1δu2n-1=∑s=i-3i∑t=j-3jδ2n-1Bsui+1+δu2n-1Btvj+1Pst.From Theorem 16, for all aij=βij=2, we have(68)δ2n-1Bijui+1-,vj+1δu2n-1=∑s=i-3i∑t=j-3j-13-1nBtvj+1Pst=δ2n-1Bijui+1+,vj+1δu2n-1.In summary, the theorem is established at knot (ui+1,vi+1). The continuity of B(u,v) given in (64) with respect to other knots can be similarly discussed.
The given spline surfaceB(u,v) possesses four local denominator shape parameters, namely, αi1,βi1,αj2,βj2. By changing the value of the shape parameter, we can adjust the shape of B(u,v) flexibly. According to (66), shape parameters αi1-1,αi1,αi1+1,αi1+2, βi1-2,βi1-1,βi1,βi1+1,αi2-1, αi2,αi2+1,αi2+2,βi2-2,βi2-1, βi2,βi2+1 affect the surface patch R(u,v),u∈[ui,ui+1],v∈[vj,vj+1]. Therefore, shape parameters αi1,αj2 affect surface patch [ui-2,ui+2]×[vj-2,vj+2], shape parameters βi1,βj2 affect surface patch [ui-1,ui+3]×[vj-1,vj+3], shape parameters αi1,βj2 affect the surface patch [ui-2,ui+2]×[vj-1,vj+3], and shape parameters αj2,βi1 affect the surface patch [ui-1,ui+3]×[vj-2,vj+2]. These results show that parameters αi1,βi1,αj2,βj2 serve to local control tension in the surface. Increasing αi1,βi1,αj2,βj2 locally moves the surface R(u,v),u∈[ui,ui+1], v∈[vj,vj+1] toward the control polygon. Figure 7 shows B(u,v) with different shape parameters. The figure on the left shows B(u,v) generated when all shape parameters are set αi1=βi1=αj2=βj2=2. The figure on the right shows B(u,v) generated when all shape parameters are set αi1=βi1=αj2=βj2=5.Figure 7
DT B-spline-like surface.


## 4. DT BB-Like Basis with Three Denominator Shape Parameters

### 4.1. DT BB-Like Basis

Definition 20.
Forα,β,γ∈[2,+∞), D=u,v,w∣u+v+w=π/2,u≥0,v≥0,w≥0, the 10 functions given below are considered new cubic DT BB-like basis functions with three denominator shape parameters α, β, and γ over the triangular domain D.(69)T3,0,03u,v,w;α,β,γ=1-cos⁡u21+α-2cos⁡u,T0,3,03u,v,w;α,β,γ=1-cos⁡v21+β-2cos⁡v,T0,0,33u,v,w;α,β,γ=1-cos⁡w21+γ-2cos⁡w,T2,1,03u,v,w;α,β,γ=cos⁡wsin⁡v1-cos⁡uα+α-2cos⁡u1+α-2cos⁡uT2,0,13u,v,w;α,β,γ=cos⁡vsin⁡w1-cos⁡uα+α-2cos⁡u1+α-2cos⁡uT1,2,03u,v,w;α,β,γ=cos⁡wsin⁡u1-cos⁡vβ+β-2cos⁡v1+β-2cos⁡vT0,2,13u,v,w;α,β,γ=cos⁡usin⁡w1-cos⁡vβ+β-2cos⁡v1+β-2cos⁡vT1,0,23u,v,w;α,β,γ=cos⁡vsin⁡u1-cos⁡wγ+γ-2cos⁡w1+γ-2cos⁡wT0,1,23u,v,w;α,β,γ=cos⁡usin⁡v1-cos⁡wγ+γ-2cos⁡w1+γ-2cos⁡wT1,1,13u,v,w;α,β,γ=1-∑i+j+k=3,i·j·k=1Ti,j,k3u,v,w;α,β,γ.
Whenw=0, the 10 DT BB-like basis functions Ti,j,k3(u,v,w;α,β,γ) degenerate to the DT B-like basis functions Ai(t;α,β) (notice v=π/2-u) given in (22). Thus, the DT BB-like basis Ti,j,k3(u,v,w;α,β,γ) is the extension of Ai(t;α,β) over the triangular domain.


### 4.2. Properties of the DT BB-Like Basis

In this subsection, we will give important properties of DT BB-like basis functions given in (69).Theorem 21.
The basis functions given in (69) have the following properties.
(a) Nonnegativity:Ti,j,k3(u,v,w;α,β,γ)≥0, i,j,k∈N, i+j+k=3.
(b) Partition of unity:∑i+j+k=3Ti,j,k3(u,v,w;α,β,γ)=1.
(c) Symmetry: for alli,j,k∈N,i+j+k=3, we have (70)Ti,j,k3u,v,w;α,β,γ=Tj,i,k3v,u,w;β,α,γ=Tj,k,i3v,w,u;β,γ,α=Ti,k,j3u,w,v;α,γ,β=Tk,i,j3w,u,v;γ,α,β=Tk,j,i3w,v,u;γ,β,α(d) Linear independence: Ti,j,k3(u,v,w;α,β,γ),i+j+k=3 are linearly independent.Proof.
We include the proofs of (a) and (d) only. The remaining cases can be directly computed.
(a) For anyα,β,γ∈[2,+∞),i,j,k∈N,i+j+k=3 and i·j·k≠1, Ti,j,k3(u,v,w;α,β,γ)≥0 is apparent. Furthermore, for T1,1,13(u,v,w;α,β,γ), by directly computing, we have(71)T1,1,13u,v,w;α,β,γ=1-∑i+j+k=3,i·j·k≠1Ti,j,k3u,v,w;α,β,γ=1-sin2u+sin2v+sin2w=12cos⁡2u+cos⁡2v+cos⁡2w-1=2sin⁡usin⁡vsin⁡w≥0.(d) For α,β,γ∈[2,+∞),ξi,j,k∈R(i+j+k=3), we consider a linear combination(72)∑i+j+k=3ξi,j,kTi,j,k3u,v,w;α,β,γ=0.
Forw=0, we can immediately induce that ξi,3-i,0,i=0,1,2,3 because the DT B-like basis given in (22) possesses the property of linear independence. In the same way, for s=t=0, we have ξi,0,3-i=ξ0,i,3-i=0, i=0,1,2,3. Finally, we have ξ1,1,1=0.
These findings imply the theorem.
Figure8 shows four images of DT BB-like basis functions with the parameter values α=β=γ=3.Figure 8
Plots of DT BB-like basis functions.


### 4.3. DT BB-Like Patch

Definition 22.
Letα,β,γ∈[2,+∞), given control points Qij∈R3(i,j,k∈N,i+j+k=3) and a domain triangle D=u,v,w∣u+v+w=π/2,u≥0,v≥0,w≥0. We call(73)Ru,v,w=∑i+j+k=3Ti,j,k3u,v,w;α,β,γPi,j,k,u,v,w∈Dthe DT BB-like patch over the triangular domain with three shape denominator parameters α,β,γ.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.(b) Geometric property at the corner points: straightforward computation gives(74)Rπ2,0,0=P3,0,0,R0,π2,0=P0,3,0,R0,0,π2=P0,0,3.(c) Corner point tangent plane: let w=π/2-u-v. We have(75)δRu,v,wδuπ/2,0,0=αP3,0,0-P2,0,1,δRu,v,wδuπ/2,0,0=αP2,1,0-P2,0,1,δRu,v,wδu0,π/2,0=βP1,2,0-P0,2,1,δRu,v,wδv0,π/2,0=βP0,3,0-P0,2,1,δRu,v,wδu0,0,π/2=γP1,0,2-P0,0,3,δRu,v,wδw0,0,π/2=γP0,1,2-P0,0,3.(d) Boundary property: For w=0, R(u,v,w) is a DTB-like curve given in (26), which can be rewritten as (76)Ru,v,0=∑i=03Pi,3-i,0Ait;β,α.This discussion suggests thatR(u,v,w) possesses two shape parameters, namely, α and β, when w=0. In the same way, when u=0 and v=0, R(0,v,w) and R(u,0,w) possess shape parameters β,γ and α,γ, respectively. The DTB-like curve can represent elliptic and parabolic arcs when the shape parameters assume a special value (Section 2). Thus, for α=β=γ=2, the three boundaries of the DT BB-like patch can be arcs of an ellipse, circle, and parabola, respectively. Figure 9 shows DT BB-like patches generated by setting α=β=γ=2. The patch whose boundaries are elliptic, circular, and parabolic arcs is shown on the left figure. Its control points are {P3,0,0=(0,-4,0),P0,3,0=(2,0,0),P0,0,3=(0,0,2),P2,1,0=(1,-4,0),P2,0,1=(0,-4,1),P1,2,0=(2,0,0),P0,2,1=(2,0,1),P1,0,2=(1,0,2),P0,1,2=(0,-2,2),andP1,1,1=(1,-2,1)}.The related boundary parametric equations are u=0,v=-4sin⁡x,w=2cos⁡x;u=2sin⁡x,v=0,w=2cos⁡x; and u=2cos⁡x,v=-4+4cos2x,w=0 where x∈[0,π/2]. The right figure shows a patch that possesses three same boundaries as a quarter of the unit circle. The related control points are {P3,0,0=(0,2,0), P0,3,0=(2,0,0), P0,0,3=(0,0,2), P2,1,0=(1,2,0), P2,0,1=(0,2,1), P1,2,0=(2,1,0), P0,2,1=(2,0,1), P1,0,2=(0,1,2), P0,1,2=(1,0,2), P1,1,1=(2,-4,2)}. The three related parametric equations of the boundaries are u=0,v=2sin⁡x,w=2cos⁡x; u=2sin⁡x,v=0,w=2cos⁡x; and u=2cos⁡x, v=2sin⁡x, v=2sin⁡x, w=0, where x∈[0,π/2].Figure 9
DT BB-like patches whose boundaries are arcs of elliptic, circle, and parabola.(e) Shape adjustable property: by changing the value without changing the control net, we can easily adjust the DT BB-like patch flexibly. As the three denominator shape parameters increase, the patch approaches the control net. Therefore, the three denominator shape parameters are three tension parameters. The three boundaries of the DT BB-like patch are determined by the control pointsPi,j,k,i+j+k=3,i·j·k≠1, independent of P1,1,1. Therefore, the proposed DT BB-like patch (73) can still change the shape by adjusting the control point P1,1,1 without changing the three boundaries. In addition, according to the properties of the boundary curve given in (26) of the proposed DT BB-like patch, the three denominator shape parameters α,β, and γ are independent of the boundary curves R(0,v,w),R(u,0,w), and R(u,v,0), respectively. This phenomenon also means that when changing the value of a certain denominator parameter, one of the three boundary curves will not be affected. Figure 10 shows the DT BB-like patch under different denominator shape parameters.Figure 10
DT BB-like patch with different shape parameters.


### 4.4. De Casteljau-Type Algorithm

We develop a de Casteljau algorithm that can calculate the DT BB-like patch given in (73). For arbitrary (u,v,w)∈D, let (77)f1u,v,w≔sin⁡ucos⁡wsin2u+sin2v+sin2wcos⁡wsin⁡u+sin⁡vsin2u+sin2v+sin2w+sin⁡wsin2u+sin2v,f2u,v,w≔sin⁡vcos⁡wsin2u+sin2v+sin2wcos⁡wsin⁡u+sin⁡vsin2u+sin2v+sin2w+sin⁡wsin2u+sin2v,f3u,v,w≔sin⁡wsin2u+sin2vcos⁡wsin⁡u+sin⁡vsin2u+sin2v+sin2w+sin⁡wsin2u+sin2v,g1u,v,w≔1-cos⁡usin2u+sin2v+sin2w,g2u,v,w≔sin⁡vcos⁡wsin2u+sin2v+sin2w+sin⁡usin⁡vsin⁡w,g3u,v,w≔cos⁡vsin⁡wsin2u+sin2v+sin2w+sin⁡usin⁡vsin⁡w,and(78)P2,0,01≔1-cos⁡u1+cos⁡u1+α-2cos⁡uP3,0,0+α-1+α-2cos⁡usin⁡vcos⁡w1+cos⁡u1+α-2cos⁡uP2,1,0+α-1+α-2cos⁡usin⁡wcos⁡v1+cos⁡u1+α-2cos⁡uP2,0,1,P0,2,01≔1-cos⁡v1+cos⁡v1+β-2cos⁡vP0,3,0+β-1+β-2cos⁡vsin⁡ucos⁡w1+cos⁡v1+β-2cos⁡vP1,2,0+β-1+β-2cos⁡vsin⁡wcos⁡u1+cos⁡v1+β-2cos⁡vP0,2,1,P0,0,21≔1-cos⁡u1+cos⁡u1+γ-2cos⁡uP0,0,3+α-1+α-2cos⁡wsin⁡ucos⁡v1+cos⁡w1+α-2cos⁡wP1,0,2+γ-1+γ-2cos⁡wsin⁡vcos⁡u1+cos⁡w1+γ-2cos⁡wP0,1,2,P1,1,01≔f1u,v,wP2,1,0+f2u,v,wP1,2,0+f3u,v,wP1,1,1,P1,1,01≔f1u,w,vP2,1,0+f2u,w,vP1,0,2+f3u,w,vP1,1,1,P1,1,01≔f1v,w,uP0,2,1+f2v,w,uP0,1,2+f3v,w,uP1,1,1.Then we rewrite (73) as (79)Ru,v,w=1-cos2usin2u+sin2v+sin2wg1u,v,wP2,0,01+g2u,v,wP1,1,01+g3u,v,wP1,0,11+1-cos2vsin2u+sin2v+sin2wg1v,u,wP0,2,01+g2v,u,wP1,1,01+g3v,u,wP0,1,11+1-cos2wsin2u+sin2v+sin2wg1w,v,uP0,0,21+g2w,v,uP0,1,11+g3w,v,uP1,0,11.Furthermore, by setting(80)P1,0,02≔g1u,v,wP2,0,01+g2u,v,wP1,1,01+g3u,v,wP1,0,11P0,1,02≔g2v,u,wP1,1,01+g1v,u,wP0,2,01+g3v,u,wP0,1,11P0,0,12≔g3w,v,uP1,0,11+g2w,v,uP0,1,11+g1w,v,uP0,0,21,we have(81)Ru,v,w=1-cos2usin2u+sin2v+sin2wP1,0,02+1-cos2vsin2u+sin2v+sin2wP0,1,02+1-cos2wsin2u+sin2v+sin2wP0,0,12≔P0,0,03.Foru+v+w=π/2, checking that f1(u,v,w)+f2(u,v,w)+f3(u,v,w)=1 and g1(u,v,w)+g2(u,v,w)+g3(u,v,w)=1 (by using sin2u+sin2v+sin2w+2sin⁡usin⁡vsin⁡w=1) is easy. Thus (79) and (81) describe the de Casteljau-type algorithm for calculating the DT BB-like patch given in (73).


### 4.5. Composite Triangular DT BB-Like Patches

Let two DT BB-like patches be(82)R1u,v,w=∑i+j+k=3Ti,j,k3u,v,w;α1,β,γPi,j,k,u,v,w∈D,(83)R2u,v,w=∑i+j+k=3Ti,j,k3u,v,w;α2,β,γQi,j,k,u,v,w∈D.If the control points satisfy(84)P0,j,k=Q0,j,k,j,k∈N,j+k=3,then (82) and (83) possess the common boundary curve R1(0,v,w)=R2(0,v,w),v+w=π/2. Therefore, the two patches form a surface with C0 continuity.DifferentiatingR1(0,v,π/2-v) with respect to v yields (85)dR10,v,π/2-vdv=cos⁡v1-sin⁡vβ+β-2sin⁡v1+β-2sin⁡v2P0,3,0-P0,2,1+2sin⁡vcos⁡vP0,2,1-P0,1,2+sin⁡v1-cos⁡vγ+γ-2cos⁡v1+γ-2cos⁡v2P0,1,2-P0,0,3.ForR1(u,v,π/2-u-v) and R2(u,v,π/2-u-v), we differentiate about u. Thus, we have (86)δR1u,v,π/2-vδuu=0=cos⁡v1-sin⁡vβ+β-2sin⁡v1+β-2sin⁡v2P1,2,0-P0,2,1+2sin⁡vcos⁡vP1,1,1-P0,1,2+sin⁡v1-cos⁡vγ+γ-2cos⁡v1+γ-2cos⁡v2P1,0,2-P0,0,3,(87)δR2u,v,π/2-vδuu=0=cos⁡v1-sin⁡vβ+β-2sin⁡v1+β-2sin⁡v2Q1,2,0-Q0,2,1+2sin⁡vcos⁡vQ1,1,1-Q0,1,2+sin⁡v1-cos⁡vγ+γ-2cos⁡v1+γ-2cos⁡v2Q1,0,2-Q0,0,3.For the arbitrary value ofv, refer to [44]. If the vectors defined by (85) to (87) are coplanar, then the conditions for G1 continuous smooth jointing can be achieved. Thus, it can be expressed as(88)δR2u,v,π/2-u-vδuu=0=ϕdR10,v,π/2-vdv+φδR1u,v,π/2-u-vδuu=0,where ϕ and φ are constants. In summary, we can obtain the following rules:(89)Q1,2,0-Q0,2,1=ϕP0,3,0-P0,2,1+φP1,2,0-P0,2,1,Q1,1,1-Q0,1,2=ϕP0,2,1-P0,1,2+φP1,1,1-P0,1,2,Q1,0,2-Q0,0,3=ϕP0,1,2-P0,0,3+φP1,0,2-P0,0,3.Thus, we can summarize the following theorems.Theorem 23.
Forαk,β,γ(k=1,2), if conditions (84) and (89) hold, then the surface connecting (82) with (83) is G1 continuous.
The aforementioned theorem shows that the two DT BB-like 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 G1 continuous surfaces by changing the value of the denominator shape parameter. Figure 11 shows the G1continuous surface under different shape parameters, where ϕ=1,φ=-1.Figure 11
G 1 continuous surfaces under different shape parameters.


## 4.1. DT BB-Like Basis

Definition 20.
Forα,β,γ∈[2,+∞), D=u,v,w∣u+v+w=π/2,u≥0,v≥0,w≥0, the 10 functions given below are considered new cubic DT BB-like basis functions with three denominator shape parameters α, β, and γ over the triangular domain D.(69)T3,0,03u,v,w;α,β,γ=1-cos⁡u21+α-2cos⁡u,T0,3,03u,v,w;α,β,γ=1-cos⁡v21+β-2cos⁡v,T0,0,33u,v,w;α,β,γ=1-cos⁡w21+γ-2cos⁡w,T2,1,03u,v,w;α,β,γ=cos⁡wsin⁡v1-cos⁡uα+α-2cos⁡u1+α-2cos⁡uT2,0,13u,v,w;α,β,γ=cos⁡vsin⁡w1-cos⁡uα+α-2cos⁡u1+α-2cos⁡uT1,2,03u,v,w;α,β,γ=cos⁡wsin⁡u1-cos⁡vβ+β-2cos⁡v1+β-2cos⁡vT0,2,13u,v,w;α,β,γ=cos⁡usin⁡w1-cos⁡vβ+β-2cos⁡v1+β-2cos⁡vT1,0,23u,v,w;α,β,γ=cos⁡vsin⁡u1-cos⁡wγ+γ-2cos⁡w1+γ-2cos⁡wT0,1,23u,v,w;α,β,γ=cos⁡usin⁡v1-cos⁡wγ+γ-2cos⁡w1+γ-2cos⁡wT1,1,13u,v,w;α,β,γ=1-∑i+j+k=3,i·j·k=1Ti,j,k3u,v,w;α,β,γ.
Whenw=0, the 10 DT BB-like basis functions Ti,j,k3(u,v,w;α,β,γ) degenerate to the DT B-like basis functions Ai(t;α,β) (notice v=π/2-u) given in (22). Thus, the DT BB-like basis Ti,j,k3(u,v,w;α,β,γ) is the extension of Ai(t;α,β) over the triangular domain.


## 4.2. Properties of the DT BB-Like Basis

In this subsection, we will give important properties of DT BB-like basis functions given in (69).Theorem 21.
The basis functions given in (69) have the following properties.
(a) Nonnegativity:Ti,j,k3(u,v,w;α,β,γ)≥0, i,j,k∈N, i+j+k=3.
(b) Partition of unity:∑i+j+k=3Ti,j,k3(u,v,w;α,β,γ)=1.
(c) Symmetry: for alli,j,k∈N,i+j+k=3, we have (70)Ti,j,k3u,v,w;α,β,γ=Tj,i,k3v,u,w;β,α,γ=Tj,k,i3v,w,u;β,γ,α=Ti,k,j3u,w,v;α,γ,β=Tk,i,j3w,u,v;γ,α,β=Tk,j,i3w,v,u;γ,β,α(d) Linear independence: Ti,j,k3(u,v,w;α,β,γ),i+j+k=3 are linearly independent.Proof.
We include the proofs of (a) and (d) only. The remaining cases can be directly computed.
(a) For anyα,β,γ∈[2,+∞),i,j,k∈N,i+j+k=3 and i·j·k≠1, Ti,j,k3(u,v,w;α,β,γ)≥0 is apparent. Furthermore, for T1,1,13(u,v,w;α,β,γ), by directly computing, we have(71)T1,1,13u,v,w;α,β,γ=1-∑i+j+k=3,i·j·k≠1Ti,j,k3u,v,w;α,β,γ=1-sin2u+sin2v+sin2w=12cos⁡2u+cos⁡2v+cos⁡2w-1=2sin⁡usin⁡vsin⁡w≥0.(d) For α,β,γ∈[2,+∞),ξi,j,k∈R(i+j+k=3), we consider a linear combination(72)∑i+j+k=3ξi,j,kTi,j,k3u,v,w;α,β,γ=0.
Forw=0, we can immediately induce that ξi,3-i,0,i=0,1,2,3 because the DT B-like basis given in (22) possesses the property of linear independence. In the same way, for s=t=0, we have ξi,0,3-i=ξ0,i,3-i=0, i=0,1,2,3. Finally, we have ξ1,1,1=0.
These findings imply the theorem.
Figure8 shows four images of DT BB-like basis functions with the parameter values α=β=γ=3.Figure 8
Plots of DT BB-like basis functions.


## 4.3. DT BB-Like Patch

Definition 22.
Letα,β,γ∈[2,+∞), given control points Qij∈R3(i,j,k∈N,i+j+k=3) and a domain triangle D=u,v,w∣u+v+w=π/2,u≥0,v≥0,w≥0. We call(73)Ru,v,w=∑i+j+k=3Ti,j,k3u,v,w;α,β,γPi,j,k,u,v,w∈Dthe DT BB-like patch over the triangular domain with three shape denominator parameters α,β,γ.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.(b) Geometric property at the corner points: straightforward computation gives(74)Rπ2,0,0=P3,0,0,R0,π2,0=P0,3,0,R0,0,π2=P0,0,3.(c) Corner point tangent plane: let w=π/2-u-v. We have(75)δRu,v,wδuπ/2,0,0=αP3,0,0-P2,0,1,δRu,v,wδuπ/2,0,0=αP2,1,0-P2,0,1,δRu,v,wδu0,π/2,0=βP1,2,0-P0,2,1,δRu,v,wδv0,π/2,0=βP0,3,0-P0,2,1,δRu,v,wδu0,0,π/2=γP1,0,2-P0,0,3,δRu,v,wδw0,0,π/2=γP0,1,2-P0,0,3.(d) Boundary property: For w=0, R(u,v,w) is a DTB-like curve given in (26), which can be rewritten as (76)Ru,v,0=∑i=03Pi,3-i,0Ait;β,α.This discussion suggests thatR(u,v,w) possesses two shape parameters, namely, α and β, when w=0. In the same way, when u=0 and v=0, R(0,v,w) and R(u,0,w) possess shape parameters β,γ and α,γ, respectively. The DTB-like curve can represent elliptic and parabolic arcs when the shape parameters assume a special value (Section 2). Thus, for α=β=γ=2, the three boundaries of the DT BB-like patch can be arcs of an ellipse, circle, and parabola, respectively. Figure 9 shows DT BB-like patches generated by setting α=β=γ=2. The patch whose boundaries are elliptic, circular, and parabolic arcs is shown on the left figure. Its control points are {P3,0,0=(0,-4,0),P0,3,0=(2,0,0),P0,0,3=(0,0,2),P2,1,0=(1,-4,0),P2,0,1=(0,-4,1),P1,2,0=(2,0,0),P0,2,1=(2,0,1),P1,0,2=(1,0,2),P0,1,2=(0,-2,2),andP1,1,1=(1,-2,1)}.The related boundary parametric equations are u=0,v=-4sin⁡x,w=2cos⁡x;u=2sin⁡x,v=0,w=2cos⁡x; and u=2cos⁡x,v=-4+4cos2x,w=0 where x∈[0,π/2]. The right figure shows a patch that possesses three same boundaries as a quarter of the unit circle. The related control points are {P3,0,0=(0,2,0), P0,3,0=(2,0,0), P0,0,3=(0,0,2), P2,1,0=(1,2,0), P2,0,1=(0,2,1), P1,2,0=(2,1,0), P0,2,1=(2,0,1), P1,0,2=(0,1,2), P0,1,2=(1,0,2), P1,1,1=(2,-4,2)}. The three related parametric equations of the boundaries are u=0,v=2sin⁡x,w=2cos⁡x; u=2sin⁡x,v=0,w=2cos⁡x; and u=2cos⁡x, v=2sin⁡x, v=2sin⁡x, w=0, where x∈[0,π/2].Figure 9
DT BB-like patches whose boundaries are arcs of elliptic, circle, and parabola.(e) Shape adjustable property: by changing the value without changing the control net, we can easily adjust the DT BB-like patch flexibly. As the three denominator shape parameters increase, the patch approaches the control net. Therefore, the three denominator shape parameters are three tension parameters. The three boundaries of the DT BB-like patch are determined by the control pointsPi,j,k,i+j+k=3,i·j·k≠1, independent of P1,1,1. Therefore, the proposed DT BB-like patch (73) can still change the shape by adjusting the control point P1,1,1 without changing the three boundaries. In addition, according to the properties of the boundary curve given in (26) of the proposed DT BB-like patch, the three denominator shape parameters α,β, and γ are independent of the boundary curves R(0,v,w),R(u,0,w), and R(u,v,0), respectively. This phenomenon also means that when changing the value of a certain denominator parameter, one of the three boundary curves will not be affected. Figure 10 shows the DT BB-like patch under different denominator shape parameters.Figure 10
DT BB-like patch with different shape parameters.


## 4.4. De Casteljau-Type Algorithm

We develop a de Casteljau algorithm that can calculate the DT BB-like patch given in (73). For arbitrary (u,v,w)∈D, let (77)f1u,v,w≔sin⁡ucos⁡wsin2u+sin2v+sin2wcos⁡wsin⁡u+sin⁡vsin2u+sin2v+sin2w+sin⁡wsin2u+sin2v,f2u,v,w≔sin⁡vcos⁡wsin2u+sin2v+sin2wcos⁡wsin⁡u+sin⁡vsin2u+sin2v+sin2w+sin⁡wsin2u+sin2v,f3u,v,w≔sin⁡wsin2u+sin2vcos⁡wsin⁡u+sin⁡vsin2u+sin2v+sin2w+sin⁡wsin2u+sin2v,g1u,v,w≔1-cos⁡usin2u+sin2v+sin2w,g2u,v,w≔sin⁡vcos⁡wsin2u+sin2v+sin2w+sin⁡usin⁡vsin⁡w,g3u,v,w≔cos⁡vsin⁡wsin2u+sin2v+sin2w+sin⁡usin⁡vsin⁡w,and(78)P2,0,01≔1-cos⁡u1+cos⁡u1+α-2cos⁡uP3,0,0+α-1+α-2cos⁡usin⁡vcos⁡w1+cos⁡u1+α-2cos⁡uP2,1,0+α-1+α-2cos⁡usin⁡wcos⁡v1+cos⁡u1+α-2cos⁡uP2,0,1,P0,2,01≔1-cos⁡v1+cos⁡v1+β-2cos⁡vP0,3,0+β-1+β-2cos⁡vsin⁡ucos⁡w1+cos⁡v1+β-2cos⁡vP1,2,0+β-1+β-2cos⁡vsin⁡wcos⁡u1+cos⁡v1+β-2cos⁡vP0,2,1,P0,0,21≔1-cos⁡u1+cos⁡u1+γ-2cos⁡uP0,0,3+α-1+α-2cos⁡wsin⁡ucos⁡v1+cos⁡w1+α-2cos⁡wP1,0,2+γ-1+γ-2cos⁡wsin⁡vcos⁡u1+cos⁡w1+γ-2cos⁡wP0,1,2,P1,1,01≔f1u,v,wP2,1,0+f2u,v,wP1,2,0+f3u,v,wP1,1,1,P1,1,01≔f1u,w,vP2,1,0+f2u,w,vP1,0,2+f3u,w,vP1,1,1,P1,1,01≔f1v,w,uP0,2,1+f2v,w,uP0,1,2+f3v,w,uP1,1,1.Then we rewrite (73) as (79)Ru,v,w=1-cos2usin2u+sin2v+sin2wg1u,v,wP2,0,01+g2u,v,wP1,1,01+g3u,v,wP1,0,11+1-cos2vsin2u+sin2v+sin2wg1v,u,wP0,2,01+g2v,u,wP1,1,01+g3v,u,wP0,1,11+1-cos2wsin2u+sin2v+sin2wg1w,v,uP0,0,21+g2w,v,uP0,1,11+g3w,v,uP1,0,11.Furthermore, by setting(80)P1,0,02≔g1u,v,wP2,0,01+g2u,v,wP1,1,01+g3u,v,wP1,0,11P0,1,02≔g2v,u,wP1,1,01+g1v,u,wP0,2,01+g3v,u,wP0,1,11P0,0,12≔g3w,v,uP1,0,11+g2w,v,uP0,1,11+g1w,v,uP0,0,21,we have(81)Ru,v,w=1-cos2usin2u+sin2v+sin2wP1,0,02+1-cos2vsin2u+sin2v+sin2wP0,1,02+1-cos2wsin2u+sin2v+sin2wP0,0,12≔P0,0,03.Foru+v+w=π/2, checking that f1(u,v,w)+f2(u,v,w)+f3(u,v,w)=1 and g1(u,v,w)+g2(u,v,w)+g3(u,v,w)=1 (by using sin2u+sin2v+sin2w+2sin⁡usin⁡vsin⁡w=1) is easy. Thus (79) and (81) describe the de Casteljau-type algorithm for calculating the DT BB-like patch given in (73).


## 4.5. Composite Triangular DT BB-Like Patches

Let two DT BB-like patches be(82)R1u,v,w=∑i+j+k=3Ti,j,k3u,v,w;α1,β,γPi,j,k,u,v,w∈D,(83)R2u,v,w=∑i+j+k=3Ti,j,k3u,v,w;α2,β,γQi,j,k,u,v,w∈D.If the control points satisfy(84)P0,j,k=Q0,j,k,j,k∈N,j+k=3,then (82) and (83) possess the common boundary curve R1(0,v,w)=R2(0,v,w),v+w=π/2. Therefore, the two patches form a surface with C0 continuity.DifferentiatingR1(0,v,π/2-v) with respect to v yields (85)dR10,v,π/2-vdv=cos⁡v1-sin⁡vβ+β-2sin⁡v1+β-2sin⁡v2P0,3,0-P0,2,1+2sin⁡vcos⁡vP0,2,1-P0,1,2+sin⁡v1-cos⁡vγ+γ-2cos⁡v1+γ-2cos⁡v2P0,1,2-P0,0,3.ForR1(u,v,π/2-u-v) and R2(u,v,π/2-u-v), we differentiate about u. Thus, we have (86)δR1u,v,π/2-vδuu=0=cos⁡v1-sin⁡vβ+β-2sin⁡v1+β-2sin⁡v2P1,2,0-P0,2,1+2sin⁡vcos⁡vP1,1,1-P0,1,2+sin⁡v1-cos⁡vγ+γ-2cos⁡v1+γ-2cos⁡v2P1,0,2-P0,0,3,(87)δR2u,v,π/2-vδuu=0=cos⁡v1-sin⁡vβ+β-2sin⁡v1+β-2sin⁡v2Q1,2,0-Q0,2,1+2sin⁡vcos⁡vQ1,1,1-Q0,1,2+sin⁡v1-cos⁡vγ+γ-2cos⁡v1+γ-2cos⁡v2Q1,0,2-Q0,0,3.For the arbitrary value ofv, refer to [44]. If the vectors defined by (85) to (87) are coplanar, then the conditions for G1 continuous smooth jointing can be achieved. Thus, it can be expressed as(88)δR2u,v,π/2-u-vδuu=0=ϕdR10,v,π/2-vdv+φδR1u,v,π/2-u-vδuu=0,where ϕ and φ are constants. In summary, we can obtain the following rules:(89)Q1,2,0-Q0,2,1=ϕP0,3,0-P0,2,1+φP1,2,0-P0,2,1,Q1,1,1-Q0,1,2=ϕP0,2,1-P0,1,2+φP1,1,1-P0,1,2,Q1,0,2-Q0,0,3=ϕP0,1,2-P0,0,3+φP1,0,2-P0,0,3.Thus, we can summarize the following theorems.Theorem 23.
Forαk,β,γ(k=1,2), if conditions (84) and (89) hold, then the surface connecting (82) with (83) is G1 continuous.
The aforementioned theorem shows that the two DT BB-like 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 G1 continuous surfaces by changing the value of the denominator shape parameter. Figure 11 shows the G1continuous surface under different shape parameters, where ϕ=1,φ=-1.Figure 11
G 1 continuous surfaces under different shape parameters.


## 5. 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 BB-spline 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 reachC(2n-1)(n=1,2,3,…) continuity, thereby satisfying the geometric design requirements of high-order 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 G1 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 BB-spline 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 G1 continuity. These issues will be the focus of future research.



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*Source: 9569834-2018-10-24.xml*