MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2019/3057134 3057134 Research Article A Generalized Cubic Exponential B-Spline Scheme with Shape Control http://orcid.org/0000-0001-9307-4372 Zhang Baoxing 1 Zheng Hongchan 1 Pan Lulu 1 Nguyen-Thanh Nhon Department of Applied Mathematics Northwestern Polytechnical University Xi’an Shaanxi 710072 China nwpu.edu.cn 2019 362019 2019 30 01 2019 02 05 2019 362019 2019 Copyright © 2019 Baoxing Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, a generalized cubic exponential B-spline scheme is presented, which can generate different kinds of curves, including the conics. Such a scheme is obtained by generalizing the cubic exponential B-spline scheme based on an iteration from the generation of exponential polynomials and a suitable function with two parameters μ and ν. By changing the values of μ and ν, the sensitivity of the shape of the subdivision curve to the initial control value v0 can be changed and different kinds of curves can then be obtained by adjusting the value of v0. For this new scheme, we show that, with any admissible choice of μ and ν, it owns the same smoothness order and support as the cubic exponential B-spline scheme. Besides, based on a different iteration and another suitable function, we construct a similar nonstationary scheme to generate more curves with different shapes and show the role of iterations and suitably chosen functions in the construction and analysis of such schemes. Several examples are given to illustrate the performance of our new schemes.

National Science Foundation for Young Scientists of China 61801389
1. Introduction

Subdivision schemes are efficient tools to generate smooth curves/surfaces from a set of initial control points and they play an important role in computer graphics, wavelets, and other fields like biomedical imaging . According to whether the refinement rules depend on the recursion level, subdivision schemes can be divided into stationary and nonstationary ones. Stationary schemes can generate algebraic polynomials while nonstationary ones can generate richer function spaces, such as the exponential polynomial spaces. As a result, nonstationary schemes can generate curves like hyperbolas or surfaces like spheres and other ones with different shapes, which can not be done using stationary schemes.

Since the nonstationary schemes can generate richer function spaces and more kinds of curves, there have been continuous works on the construction and analysis of nonstationary schemes. The exponential B-spline schemes  are such typical examples, which can generate exponential polynomials. Besides, Conti & Romani  gave conditions on the symbols of nonstationary schemes to reproduce exponential polynomials. Siddiqi et al.  presented ternary nonstationary schemes generating hyperbolas and parabolas. Zheng & Zhang  applied the push-back operation in the nonstationary case and constructed a combined non-stationary scheme generating different exponential polynomials. Asghar & Mustafa  constructed p-ary nonstationary schemes which are new versions of the Lane-Riesenfeld algorithms. For other nonstationary schemes generating exponential polynomials, see  and the references therein. In fact, there are other nonstationary schemes generating curves with different kinds of shapes. Beccari et al.  proposed a ternary 4-point nonstationary interpolatory scheme, whose nonstationarity can be seen as the result of an iteration, and this scheme can generate curves with considerable variations of shapes. Similarly, Tan et al.  proposed a 3-point nonstationary approximating subdivision, which can be seen as constructed based on a different iteration and can also generate a wide variety of curves.

Due to nonstationary schemes’s ability in curve design, as illustrated above, in this paper, we aim to propose a new nonstationary subdivision scheme, which can generate curves with considerable variations of shapes and exponential polynomials as well. The inspiration comes from the works in [10, 11]. In fact, we start from the C2-convergent cubic exponential B-spline scheme , which generates the conics. To our purpose, we see the cubic exponential B-spline scheme as obtained based on an iteration coming from the generation of exponential polynomials and treat the coefficients of the subdivision rules as functions of this iteration. Then, together with a function with two parameters μ and ν, we can obtain a generalized cubic exponential B-spline scheme, which is just the one we want. For this new scheme, by changing the values of μ and ν, we can change the sensitivity of the shape of the obtained curve to the initial control value v0, and different kinds of curves, including the conics, can then be obtained by adjusting v0. We point out that compared with the cubic exponential B-spline scheme, this newly obtained one can generate curves with more kinds of shapes and, compared with the schemes in [10, 11], this new one enjoys the advantages like shorter support and generation of exponential polynomials. For such a new scheme, we show that, with any admissible choice of μ and ν, it keeps the same smoothness order and the support as the cubic exponential B-spline scheme. Besides, based on a different iteration and another suitable function, we also present a similar nonstationary scheme to generate curves with more kinds of shapes. This also shows the role of iterations and suitably chosen functions in the construction and analysis of such schemes. The performance of our schemes is illustrated by several numerical examples.

The rest of this paper is organized as follows. In Section 2, we review some known definitions and results about subdivision schemes and iterations. Section 3 is devoted to the construction and analysis of the generalized cubic exponential B-spline scheme. In Section 4, we present several examples and compare the new scheme with several existing nonstationary ones. In Section 5, we move a further step and construct a similar scheme based on a different iteration and a different suitable function. Section 6 concludes this paper.

2. Preliminary

In this section, we recall some basic definitions and known results about subdivision schemes and iterations to form the basis of the rest of this paper. Let l0(Z) denote the linear space of real sequences with finite support. For a sequence λl0(Z), its support is the finite set {iZ,λi0}. Given an initial data sequence q0=qi0,iZl0(Z), we consider the nonstationary subdivision scheme (1)qk+1iSakqki=jZai-2jkqjk,iZ,where Sak is the k-level subdivision operator and the sequence ak=aik,iZ is the k-level mask with finite support. We denote this scheme by {Sak}k0 and the so-called k-level symbol of the scheme {Sak}k0 is the Laurent polynomial ak(z)=iZaikzi.

By attaching qik,iZ to the parameter values i/2k, k0, we say the subdivision scheme {Sak}k0 is C0-convergent if, for the initial data sequence q0l0(Z), there exists a function fq0C0(R) satisfying (2)limkfq0·2k-Sakqk=0,where fq0·/2k denotes the sequence {fq0(i/2k)}iZ. If fq0Cl(R), we say the subdivision {Sak}k0 is Cl-convergent.

In order to investigate the convergence and smoothness of nonstationary subdivision schemes, let us recall some definitions and results as follows.

Definition 1 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

A nonstationary subdivision scheme {Suk}k0 with the k-level mask uk is said to be asymptotically similar to the stationary subdivision scheme Su with the mask u, if the k-level mask uk and the mask u have the same support D (i.e. aik=ai=0 for iD) and satisfy (3)limkuik=ui,iD.

Definition 2 (see [<xref ref-type="bibr" rid="B14">14</xref>]).

let Dn be the nth order differentiation operator. A nonstationary subdivision scheme {Suk}k0 with the k-level symbol uk(z) is said to satisfy the approximate sum rules of order r+1 if (4)μk=uk1-2,δk=maxηr2-kηDηuk-1,satisfy (5)kμk<,k2krδk<.

Theorem 3 (see [<xref ref-type="bibr" rid="B14">14</xref>]).

Assume that the nonstationary subdivision scheme {Suk}k0 satisfies approximate sum rules of order r+1 and is asymptotically similar to a convergent stationary subdivision scheme Su who is Cr-convergent. Then the nonstationary scheme {Suk}k0 is Cr-convergent.

Now we recall some knowledge about the generation of exponential polynomials.

Definition 4 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

Let TZ+ and γ=(γ0,,γT) with γT0 a finite set of real or imaginary numbers. The space of exponential polynomials VT,γ is (6)VT,γf:RC,fCTR:j=0TγjDjf=0.

The exponential polynomial space VT,γ can be characterized as in the following lemma.

Lemma 5 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

Let γ(z)=j=0Tγjzj and denote by {θl,τl}l=1,,N the set of zeros with multiplicity satisfying (7)Drγθl=0,r=0,,τl-1,l=1,,N.Then (8)T=l=1Nτl,VT,γspanxreθlx,r=0,,τl-1,l=1,,N.

Definition 6 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

We say the subdivision scheme {Sak}k0 is VT,γ-generating if it is convergent and for fVT,γ, there exists an initial sequence f0 uniformly sampled from f~VT,γ, such that (9)limkSam+kSam+k-1Samf0=f,m0.

Now let us now recall some known definitions and results about fixed point iterations.

Definition 7 (see [<xref ref-type="bibr" rid="B15">15</xref>]).

We say xR is a fixed point for a given function φ(·) if (10)φx=x.

The following result gives sufficient conditions for the existence and uniqueness of a fixed point and how to approximate it.

Theorem 8 (see [<xref ref-type="bibr" rid="B15">15</xref>]).

Let φ(·)C[a,b] be such that φ(x)[a,b] for all x[a,b]. Suppose, in addition, that D1φ exists on (a,b) and that there exists a constant 0<L<1 such that (11)D1φxL,xa,b.Then, there exists a unique fixed point x for φ(·) on [a,b]. Besides, for any x0[a,b], the sequence defined by (12)xn=φxn-1,n1,converges to the unique fixed point x in [a,b].

3. The Generalized Cubic Exponential B-Spline Scheme

In this section, we construct the generalized cubic exponential B-spline scheme and then investigate its convergence and smoothness.

3.1. Construction of the Generalized Cubic Exponential B-Spline Scheme

Before we construct the generalized cubic exponential B-spline scheme, we briefly review the cubic exponential B-spline scheme , which can be written down as(13)p2ik+1=141+vk+1pi-1k+pi+1k+1-121+vk+1pik,p2i+1k+1=12pik+pi+1k,where(14)tk=t2k,vk=12etk+e-tk,with t being a nonnegative or pure imaginary constant. From the definition of vk in (14), we see that vk+1 and vk satisfies the iteration (15)vk+1=1+vk2,with v0(-1,).

As is known, the cubic exponential B-spline scheme (13) is C2-convergent and can generate the function space Espan{1,x,e±tx} in the sense of Definition 6 . Thus, it can generate conic sections.

In fact, from the cubic exponential B-spline scheme (13), we can see that it is just the iteration in (15) that forces the subdivision rules of this scheme to depend on the recursive level k. This means that the nonstationarity of this scheme results from the iteration in (15). Besides, we can see the coefficients in the corresponding subdivision rules as functions of this iteration. That is to say, there exists a function h¯(x)=1/1+x such that the cubic exponential B-spline scheme (13) can be rewritten as(16)p2ik+1=h¯vk+14pi-1k+pi+1k+1-h¯vk+12pik,p2i+1k+1=12pik+pi+1k.In this way, if we replace the function h¯(·) or the iteration (15) in (16) by a different one, we can obtain a different nonstationary subdivision scheme.

Now based on this observation, let us construct the new generalized cubic exponential B-spline scheme. In fact, to obtain our new scheme, we replace the function h¯(·) by a more generalized one as follows: (17)hx=1xμ+xν,μ,νN0N0,xΩμ,ν,with Ωμ,ν{x:xμ+xν0,μ,νN0}. Then, from (16), we can obtain a new scheme, which can be written as(18)p2ik+1=hvk+14pi-1k+pi+1k+1-hvk+12pik,p2i+1k+1=12pik+pi+1k.

Note that when one of the parameters μ and ν is 0 and the other is 1, h(·) becomes h¯(·) and the scheme (18) reduces to the cubic exponential B-spline scheme (13). We point out that the scheme (18) is just the desired generalized cubic exponential B-spline scheme and we denote it by {Saμ,νk}k0. From the definition of vk in (14), we see that limkvk=1. As a result, it can be seen that limkh(vk+1)=h(1)=1/2 and the limit stationary scheme of {Saμ,νk}k0 is thus the cubic B-spline scheme.

In fact, we only modified the existing elements in the masks of the cubic exponential B-spline scheme. Thus, the support of the new scheme {Saμ,νk}k0 is the same as the cubic exponential B-spline scheme, i.e., [-2,2].

3.2. Convergence and Smoothness

Now let us investigate the convergence and smoothness of the generalized cubic exponential B-spline scheme {Saμ,νk}k0. In fact, we have the following result.

Theorem 9.

For μ,νN0, the generalized cubic exponential B-spline scheme {Saμ,νk}k0 is C2-convergent.

Proof.

We show that the generalized cubic exponential B-spline scheme {Saμ,νk}k0 satisfies approximate sum rules of order 3. Then by Theorem 3, we can conclude that the scheme {Saμ,νk}k0 is C2-convergent.

To show that the scheme {Saμ,νk}k0 satisfies approximate sum rules of order 3, we denote by aμ,νk(z) the k-level symbol of the scheme {Saμ,νk}k0 and then aμ,νk(z) can be written down as (19)aμ,νkz=hvk+14z2+z-2+12z+z-1+1-hvk+12.It can be computed that aμ,νk(z) contains the factor (1+z)2/2 and that (20)aμ,νk1=2,aμ,νk-1=D1aμ,νk-1=0,D2aμ,νk-1=2hvk+1-1.Therefore, from Definition 2, for μk and δk, we have (21)μk=aμ,νk1-2=0,δk=maxη22-kηDηaμ,νk-1=21-2khvk+1-12.Together with the definition of h(·), it can be seen that there exists a constant c1 independent of k such that (22)k=022kδk=k=02hvk+1-122c1k=0vk+1-1.Therefore, to show that the scheme {Saμ,νk}k0 satisfies approximate sum rules of order 3, we only have to show k=0vk+1-1<.

Let vk+1=φ(vk)=vk/2+1/2, with φ(x)=x/2+1/2, x(-1,). When v0>1, choose M>v0; then for x[1,M], we see that there exists a constant L=1/22<1 satisfying (23)1φxM,dφxdxL.Therefore, from Theorem 8, there exists a unique fixed point v for φ(·) on [1,M] and that the sequence {vk}k0 converges to v. In fact, the fixed point is v=1. In this way, we have(24)vk-1=φvk-1-φ1Lvk-1-1Lk-1v1-1Lkv0-1.

When v0(-1,1], notice that vk[0,1] for k1. For x[0,1], we still have (25)0φx1,dφxdxL.Therefore, from Theorem 8, there exists a unique fixed point for φ(·) on [0,1], which, in fact, is just v=1. In this case, similar to (24), we have(26)vk-1=φvk-1-φ1Lvk-1-1Lk-1v1-1c2Lkv0-1,where c2 is a positive constant independent of k.

Therefore, by combining (24) and (26), we see that, for v0(-1,), (27)k=0vk+1-1k=0Lk+1v0-1<,and thus the scheme {Saμ,νk}k0 satisfies approximate sum rules of order 3. Then, from Theorem 3, the scheme {Saμ,νk}k0 is C2-convergent.

Remark 10.

Note that k=0vk-1< can also be derived from the fact that vk-1c32-k with c3 being a constant independent of k. Here, we used the technique of fixed point iteration, which, we point out that, can also be used in the analysis of other nonstationary subdivision schemes, such as the ones in [10, 11]. This will be shown in Section 5.

4. Examples and Comparison

In this section, we present several numerical examples and compare it with some existing subdivision schemes to illustrate the performance of the scheme {Saμ,νk}k0.

Figure 1 shows the curves generated by the scheme {Saμ,νk}k0 with μ=2, ν=1, and different values of v0. From Figure 1, we can see that, for v0(-1,), with a suitable choice of μ and ν, the scheme {Saμ,νk}k0 can indeed generate curves with a wide variety of shapes. Figure 2 shows how the parameters μ and ν affect the shape of the obtained curve and gives some hints on how to choose them to generate the curve we want.

Curves generated by the scheme {Sa2,1k}k0 with v0=-0.88,-0.7,-0.5,-0.2,1.5 (left to right).

Curves generated by the scheme {Saμ,νk}k0 with different values of μ, ν, and v0=-0.95,-0.9,-0.7,-0.3,cos(π/3),5.

Curves generated by {Sa0,1k}k0

Curves generated by {Sa1,1k}k0

Curves generated by {Sa1,2k}k0

Curves generated by {Sa2,2k}k0

Figure 2 shows the curves generated by the scheme {Saμ,νk}k0 with different values of μ, ν, and v0 starting from the initial control points uniformly sampled from the unit circle. From Figure 2, we see that when both μ and ν are nonzero, the shape of the obtained curve is sensitive to the change of v0 near v0=-1. But, this is not the case if only one of μ and ν is 0. In particular, when μ=0 and ν=1, the scheme {Saμ,νk}k0 reduces to the cubic exponential B-spline scheme and the curve generated with v0=cos(π/3) is exactly a circle. Similarly, the parabola and hyperbola can also be generated in this case. Besides, when μ=ν=0, the scheme {Saμ,νk}k0 reduces to the cubic B-spline scheme, and the shape of the obtained curve can not be changed. From Figure 2, we can also see that when both μ and ν are nonzero, the scheme {Saμ,νk}k0 will get more sensitive to v0 near -1 if μ+ν becomes bigger and thus can generate more kinds of curves with the change of v0 near -1. Besides, with the increasing of v0, the obtained curve will tend to the initial control polygon.

Figure 3 shows the curves generated by the cubic B-spline scheme, the cubic exponential B-spline scheme (13) and the scheme {Sa2,3k}k0 with v0=4. We can see from Figure 3 that, for v0=4>1, when μ and ν are chosen large enough, the scheme {Saμ,νk}k0 performs better than the cubic B-spline scheme and the cubic exponential B-spline scheme.

Curves generated by the cubic B-spline scheme (left column) cubic exponential B-spline (middle column), and the scheme {Sa2,3k}k0 (right column) with v0=4.

Recall that the nonstationary subdivision schemes in [10, 11] can also generate different kinds of curves and we denote them by {Sbk}k0 and {Sck}k0, respectively. The scheme {Sbk}k0 is a ternary interpolatory one while the schemes {Sck}k0 and {Saμ,νk}k0 are binary approximating ones. Table 1 shows the comparison between these schemes. From Table 1, we see that the new scheme {Saμ,νk}k0 outperforms the other two in terms of support.

Comparison between different schemes.

Schemes { S a μ , ν k } k 0 { S b k } k 0 { S c k } k 0
support [ - 2,2 ] [-2.5,2.5] [ - 3,3 ]
smoothness order 2 2 3
type approximating interpolatory approximating
p-ary binary ternary binary
5. Further Discussion

Note that the scheme {Saμ,νk}k0 is obtained based on the iteration (15) and the function h(·). In this section, based on a different iteration and another suitable function, we move a further step and try to obtain a similar scheme, which can also generate different kinds of curves. This will show us the role of fixed point iterations and suitably chosen functions in the construction and analysis of such schemes.

Recall that the iteration (15) can be written as vk+1=φ(vk) with φ(x)=1+x/2 for x(-1,). Now we replace φ(·) by a different one ψ(·), which maps its domain I onto I. Meanwhile, we also replace the function h(·) by a different one g(·). Recall from Section 4 that when both μ and ν are nonzero, the shape of the obtained curve is sensitive to the change of v0 near v0=-1. Note that v0=-1 leads to v1=0 and that limx0h(x)=. Therefore, to obtain a scheme similar to {Saμ,νk}k0, we assume that there exists a point xI such that limxxg(x)=.

Specifically speaking, we take ψ(x)=x+2 with I=(-2,) and the function g(x)=2/x2 with x0. In this way, from {Saμ,νk}k0, we can derive a new scheme, which can be written as(28)p2ik+1=gxk+14pi-1k+pi+1k+1-gxk+12pik,p2i+1k+1=12pik+pi+1k,where the sequence {xk}k0 is generated by xk+1=ψ(xk) with x0(-2,). We denote the scheme (28) by {Sek}k0.

Similar to the iteration vk+1=φ(vk), it can be shown that, for x0(-2,), we have limkxk=2x, which, in fact, is the fixed point of ψ(·). Besides, notice that g(x)=1/2. Thus, the limit stationary scheme of the scheme {Sek}k0 is also the cubic B-spline scheme.

Now we investigate the convergence and smoothness of the newly obtained scheme {Sek}k0. In fact, similar to Theorem 8, we have the following result.

Theorem 11.

The new scheme {Sek}k0 is C2-convergent.

Proof.

Denote by ek(z) the k-level symbol of the scheme {Sek}k0. Then we have (29)ekz=1+z24gxk+1z2+1+21-gxk+1zz-2.Similar to the proof of Theorem 8, from Definition 6, for μk and δk, we have (30)μk=ak1-1=0,δk=maxη22-kηDηak-1=21-2kgxk+1-12.Therefore, since g(x)=1/2, we have (31)k=022kδ=k=02gxk+1-12=k=02gxk+1-gxk=02D1gθxk+1-xc4k=0xk+1-x,where θ(min{xk+1,x},max{xk+1,x}) and c4 is a positive constant independent of k. Now we show that k=0xk+1-x< so that the scheme {Sek}k0 satisfies approximate sum rules of order 3.

In fact, similar to the sequence {vk}k0, for L=1/22, we still have D1ψxL for x(-2,) and that (32)xk+1-xc5Lk+1x0-x,where c5 is a positive constant independent of k. Thus, k=0xk+1-xk=0c5Lk+1x0-x< and the scheme {Sek}k0 satisfies approximate sum rules of order 3. Then, by Theorem 3, the scheme {Sek}k0 is C2-convergent, since it is asymptotic similar to the C2-convergent cubic B-spline scheme.

Figures 4 and 5 show some curves generated by the scheme {Sek}k0 with different values of x0. From Figures 4 and 5, we see that, with x0(-2,), the scheme {Sek}k0 can indeed generate a wide variety of curves, including some interesting ones.

Curves generated the scheme {Sek}k0 with x0=-1.25,-1.2,-1 (left to right: a qudarifomun, a simple ‘China knot’ and an astroid).

Curves generated the scheme {Sek}k0 with x0=-0.6,-0.1,1.1,2.5 (left to right).

6. Conclusion

In this paper, a generalized cubic exponential B-spline scheme is presented, which can generate different kinds of curves, including the conics. The key ingredients of the construction and analysis is the iteration (15) coming from the generation of exponential polynomials and a suitable function h(·) with two parameters μ and ν. By adjusting the values of μ and ν, we can change the sensitivity of the shape of the obtained curve to the initial control value so as to generate various kinds of curves, including the conics. For this new scheme, we show that, with any admissible choice of μ and ν, it keeps the same smoothness order and support as the cubic exponential B-spline scheme. Some hints on how to choose μ and ν to generate the curves we want are also given. Besides, based on a different iteration and another suitable function, we also obtained a different nonstationary scheme, which can also generate a wide variety of curves with the change of the initial control value. This shows the role of fixed point iterations and suitably chosen functions in the construction and analysis of such schemes.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This is paper is supported by the National Science Foundation for Young Scientists of China (Grant No. 61801389).

Badoual A. Novara P. Romani L. Schmitter D. Unser M. A non-stationary subdivision scheme for the construction of deformable models with sphere-like topology Graphical Models 2017 94 38 51 10.1016/j.gmod.2017.10.001 MR3732544 2-s2.0-85033593080 Vonesch C. Blu T. Unser M. Generalized Daubechies wavelet families IEEE Transactions on Signal Processing 2007 55 9 4415 4429 10.1109/TSP.2007.896255 MR2464454 2-s2.0-34548211453 Conti C. Romani L. Algebraic conditions on non-stationary subdivision symbols for exponential polynomial reproduction Journal of Computational and Applied Mathematics 2011 236 4 543 556 2-s2.0-80051924743 10.1016/j.cam.2011.03.031 Zbl1232.65035 Siddiqi S. S. Salam W. u. Rehan K. Hyperbolic forms of ternary non-stationary subdivision schemes originated from hyperbolic B-splines Journal of Computational and Applied Mathematics 2016 301 16 27 10.1016/j.cam.2016.01.001 MR3464988 Zbl1382.65039 Zheng H. Zhang B. A non-stationary combined subdivision scheme generating exponential polynomials Applied Mathematics and Computation 2017 313 209 221 10.1016/j.amc.2017.05.066 MR3669696 2-s2.0-85020721083 Asghar M. Mustafa G. Family of a-Ary univariate subdivision schemes generated by Laurent polynomials Mathematical Problems in Engineering 2018 2018 11 7824279 Jeong B. Lee Y. J. Yoon J. A family of non-stationary subdivision schemes reproducing exponential polynomials Journal of Mathematical Analysis and Applications 2013 402 1 207 219 2-s2.0-84875376701 10.1016/j.jmaa.2013.01.026 Zbl1267.65017 Fang M.-e. Ma W. Wang G. A generalized curve subdivision scheme of arbitrary order with a tension parameter Computer Aided Geometric Design 2010 27 9 720 733 10.1016/j.cagd.2010.09.001 MR2737589 Zbl1216.65026 2-s2.0-77957991013 Jena M. K. Shunmugaraj P. Das P. C. A subdivision algorithm for trigonometric spline curves Computer Aided Geometric Design 2002 19 1 71 88 10.1016/S0167-8396(01)00090-5 MR1879681 Zbl0984.68165 2-s2.0-0036144133 Beccari C. Casciola G. Romani L. An interpolating 4-point C2 ternary non-stationary subdivision scheme with tension control Computer Aided Geometric Design 2007 24 4 210 219 10.1016/j.cagd.2007.02.001 MR2320435 Tan J. Sun J. Tong G. A non-stationary binary three-point approximating subdivision scheme Applied Mathematics and Computation 2016 276 37 43 10.1016/j.amc.2015.12.002 MR3451996 Zbl07039566 2-s2.0-84951732202 Morin G. Warren J. Weimer H. A subdivision scheme for surfaces of revolution Computer Aided Geometric Design 2001 18 5 483 502 10.1016/S0167-8396(01)00043-7 MR1841462 Zbl0970.68177 2-s2.0-0035369739 Conti C. Dyn N. Manni C. Mazure M.-L. Convergence of univariate non-stationary subdivision schemes via asymptotic similarity Computer Aided Geometric Design 2015 37 1 8 10.1016/j.cagd.2015.06.004 MR3370381 2-s2.0-84935029980 Zbl06993475 Charina M. Conti C. Guglielmi N. Protasov V. Regularity of non-stationary subdivision: a matrix approach Numerische Mathematik 2017 135 3 639 678 10.1007/s00211-016-0809-y MR3606458 2-s2.0-84966709671 Faires J. D. Burden R. L. Numerical Analysis 2011 9th Brooks/Cole Publishing. 10.2514/1.J052430