A New Approach to Increase the Flexibility of Curves and Regular Surfaces Produced by 4-Point Ternary Subdivision Scheme

Department of Mathematics, e Government Sadiq College Women University Bahawalpur, Bahawalpur, Pakistan Department of Mathematics, e Islamia University of Bahawalpur, Bahawalpur, Pakistan Department of Mathematics, Cankaya University, Ankara 06530, Turkey Institute of Space Sciences, Magurele-Bucharest 077125, Romania Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan Department of Mathematics, Faculty of Sciences, King Saud University, Riyadh 11451, Saudi Arabia Department of Mathematics, University of Sargodha, Sargodha, Pakistan Department of Mathematics, Huzhou University, Huzhou 313000, China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha 410114, China


Introduction
CAGD is considered as an emerging research field of computational mathematics, which has been fast growing in the last two decades due to a vast range of applications in a number of scientific fields and in real life. It has been extended into new directions owing to several generalizations and applications. e field is concerned with modeling and designing of different complex objects with the help of elegant mathematical algorithms. In CAGD, subdivision schemes have become one of the most important, efficient, and emerging modeling tools for designing and modeling of objects. It defines a smooth curve after applying a sequence of successive refinements. e subdivision schemes are main approaches used to create a curve from an initial control polygon or a surface from an initial control mesh by subdividing them according to the refining rules. ese refining rules take the initial control polygon or mesh to produce a sequence of finer polygons or meshes converging to a smooth limiting curve or surface.
Subdivision schemes are classified into interpolatory subdivision schemes and approximating subdivision schemes. Interpolatory subdivision schemes produce the limit curves that pass through all the initial points, whereas the approximating subdivision schemes generate the limit curves that do not pass through the initial control points. e combined subdivision schemes produce the limit curves that may or may not pass through the initial control points. So, their construction has become a new and important trend in CAGD. Different variants of a method to construct the ternary combined subdivision schemes from ternary approximating subdivision schemes have been discussed in [1][2][3]. Hameed and Mustafa [4] discussed a recursive process for constructing the family of combined binary subdivision schemes. Han and Jia [5] analyzed the approximation and smoothness properties of fundamental and refinable functions that arise from interpolatory subdivision schemes in multidimensional spaces. e push-back operators have been used by [6][7][8][9] for the construction of new subdivision schemes. In this article, we give a new method to construct a Modified Combined Ternary Subdivision Scheme (MCTSS). We start from two schemes, one of which is interpolatory with good approximation order and the other one is approximating with good continuity. Hence the MCTSS gives good approximation order and continuity. e motivation for the construction of a new combined subdivision scheme with shape parameters is explained in the following.

Motivation.
We construct a new subdivision scheme with shape parameters by using interpolatory and approximating subdivision schemes so that shape parameters allow the limit curves to move outside the interpolatory curve, inside the approximating curve, or in between the interpolatory and approximating curves.
is can be seen in Figure 1. In this figure, red bullets are the initial control points. Blue and green lines show the curves generated by schemes (10) and (11), respectively. ese schemes are also the special cases of the MCTSS. Black lines show the curves generated by the MCTSS for (α, β) � (0. 15 e remainder of this article is organized as follows. In Section 2, we present basic notations and results. In Section 3, the framework for the construction of MCTSS is presented. In Section 4, we study the properties of the MCTSS analytically. Comparison with existing schemes is given in Section 5. We give numerical examples of the MCTSS in Section 6. In Section 7, we extend the MCTSS into one of its bivariate versions. Conclusions are given in Section 8.

Basic Notations and Results
A univariate linear ternary subdivision scheme S a is based on repeated application of the refinement rules, which are used to map a polygon P k � P k i i∈Z ∈ l(Z) to a refined polygon P k+1 � P k+1 i i∈Z ∈ l(Z). e general compact form of these refinement rules is defined as where l(Z) denotes the space of scaler-valued sequences. e sequence a � a j j∈Z is called the refinement mask. e polynomial that uses this mask as coefficient is called the Laurent polynomial. erefore, the Laurent polynomial corresponding to subdivision scheme (1) is e necessary condition for the convergence of a ternary subdivision scheme is which is equivalent to the following relation: and (4) is also called the basic sum rule of the ternary subdivision scheme (1).
Definition 1 (see [2]). A combined ternary subdivision scheme is characterized by a parameter-dependent Laurent polynomial a(z), z ∈ C\ 0 { }, which satisfies the odd-symmetry property a(z) � a(z − 1 ) for all choices of the shape parameters and the interpolation property 2 j�0 a(e (2πi/3)j z) � 3 only for some special choices of the shape parameters.
Theorem 1 (see [10]). A convergent subdivision scheme S a corresponding to the Laurent polynomial, is C n -continuous if and only if the subdivision scheme S b corresponding to the Laurent polynomial b(z) is convergent.
Theorem 2 (see [10]). e scheme S a corresponding to the Laurent polynomial a(z) converges, if and only if the scheme S b corresponding to the Laurent polynomial b(z) is contractive, and the scheme Let ρ d be the space of polynomials of degree d and p ∈ ρ d . A subdivision operator S a is said to generate poly- Theorem 3 (see [11] where a (t) denotes the t-th derivative of a(z) with respect to z.
Theorem 4 (see [11] Convexity is an important shape property. e applications of convexity are in the following: (i) Designing of telecommunication system (ii) Nonlinear programming (iii) Engineering optimization theory (iv) Approximation theory, and many other fields In order to analyze this property for our subdivision scheme, we use the following notations and results.
Definition 2 (see [12]). e mask/coefficient of an n-th degree polynomial a(z) � n i�0 a i z i is said to be bell-shaped if it satisfies Supp(a) � [0, n], where a � a 0 , a 1 , . . . , a n is the set of masks/coefficients. A subdivision scheme is said to be monotonicity-preserving if it preserves the monotonicity of the starting sequence. at is, at any refinement/subdivision step k > 0, the difference sequence ΔP k is positive/negative whenever the difference sequence ΔP 0 is positive/negative, respectively. We use the following result of [12] to analyze this property for our subdivision scheme.
A subdivision scheme preserves convexity if it preserves the convexity of the starting sequence. at is, at any refinement step k > 0, the difference sequence Δ 2 P k is positive/ negative whenever the difference sequence Δ 2 P 0 is positive/ negative, respectively. We use the following result of [12] to analyze convexity-preserving property for our subdivision scheme.

Framework for the Construction of the MCTSS
We construct a new combined approximating and interpolating subdivision scheme with two shape parameters. e method that we adopt for the construction of the new subdivision scheme is described here. Here, we provide the subdivision rules of MCTSS in a vector approach. For this, firstly we take the ternary 4-point interpolating subdivision scheme presented in [13]: Now, we take the following ternary 4-point approximating B-spline scheme of degree 4:

Mathematical Problems in Engineering
Now differences between the points of (11) and the points of (10) give three displacement vectors, which are defined as follows: Now we obtain a new combined ternary subdivision scheme with two shape parameters by moving the points P k+1 3i , P k+1 3i+1 , and P k+1 3i+2 of (10) to the new position according to the displacement vectors αD k+1 3i+2 , respectively, where α and β are the shape parameters. e shape parameters are chosen in such a way that the interpolating and approximating behaviors of the new subdivision scheme depend on the shape parameter α, while β is used only for providing tension in the curves. Mathematically, Hence, we get the following combined subdivision scheme with two shape parameters: Hence, the mask of scheme (14) is and, equivalently, where e Laurent polynomial corresponding to scheme (14) is where c i,α,β : i � − 5, . . . , 5 are defined in (17).

Properties of the MCTSS
In this section, we analyze the behavior of the MCTSS. A detailed analysis of the scheme is presented here by discussing the important features of the scheme such as continuity, degree of polynomial generation, and degree of polynomial reproduction. We use the Laurent polynomial method to check the continuity of the MCTSS for different values of shape parameters. We also show that the MCTSS has a bell-shaped mask for the specific ranges of parameters. Moreover, we show that the MCTSS preserves monotonicity and convexity for the specific ranges of shape parameters.

Smoothness Analysis.
In this part of the paper, we discuss the level of continuity to which the MCTSS can produce smooth limiting curves or 2D models. It is well known that a continuous subdivision scheme must be convergent. So, we derive the following lemma.

Lemma 1.
e ternary subdivision scheme, which is defined in (14), satisfies the necessary conditions for the convergence.

(22)
Since (22) is proved. Hence, the necessary conditions for the convergence of scheme (14) are satisfied. is completes the proof. □ Theorem 7.

Proof 2.
e Laurent polynomial corresponding to the scheme S a 0,β is given by Now, we check the C 0 -continuity of the scheme S a 0,β . For this, we write (23) as which can also be written as where and, for C 0 -continuity of the scheme S a 0,β corresponding to Laurent polynomial a 0,β (z), we have to show that the scheme S b 0 is convergent. For this purpose, we develop a difference scheme S c 0 corresponding to the Laurent polynomial c 0 (z). Now, we have to show that the scheme S c 0 is contractive. For this, we use eorem 2 to calculate ‖c 0 ‖ ∞ ; that is, and, from above, we can easily calculate that ‖c 0 ‖ ∞ < 1 for − (16/5) < β < (5/2). It follows that scheme S c 0 is contractive, S b 0 is convergent, and S a 0,β is C 0 -continuous. So, the scheme S a 0,β is C 0 -continuous for − (16/5) < β < (5/2). Now we find C 1 -continuity of the scheme S a 0,β . From (23), we get where which can be written as where To find C 1 -continuity of the scheme S a 0,β , we have to show that the scheme corresponding to the Laurent polynomial (32) is contractive. To check the contractiveness of the scheme S c 1 , we use eorem 2 and calculate the following expression: It is easy to calculate that ‖c 1 ‖ ∞ < 1 for − (1/5) < β < (6/5) and − 2 < β < (6/5). e common interval for which ‖c 1 ‖ ∞ < 1 is − (1/5) < β < (6/5).
To increase the range of continuity for the shape parameter β, we apply eorem 2 for ℓ � 2 and derive the following result.

Remark 3.
e subdivision scheme S a 0,β is an interpolatory subdivision scheme. is scheme is a special case of scheme (14). In other words, MCTSS (14) is interpolatory for α � 0 and ∀β. e proofs of the rest of the theorems are similar to the proof of eorem 7.

Order of MCTSS for Generating and Reproducing
Polynomials. Polynomial generation and polynomial reproduction are the important properties of the subdivision schemes. In this section, we discuss the order/degree of polynomial generation and degree of polynomial reproduction of the scheme defined in (14). Generation is the highest degree of polynomials that are generated by the scheme. Any subdivision scheme that reproduces polynomials of degree d also generates polynomials of degree d. By [14], if a subdivision scheme reproduces polynomials of degree d, then it is said to have an approximation order d + 1. In the next part of paper, we check the capacity of the MCTSS (14) for generating and reproducing polynomials.

Theorem 11.
e scheme associated with the Laurent polynomial a 0,β (z) generally generates polynomials of degree 1 and for β � 0 it generates polynomials of degree 3.

Proof 3. We have
It is easy to see that the first four relations in (34) return zero when β � 0. is completes our proof by eorem 3. □ Theorem 12. e scheme associated with the Laurent polynomial a 0,β (z) generally reproduces polynomials of degree 1 and for specific value of β (i.e., β � 0) it reproduces polynomials of degree 3. Proof 4. By ( [11], Corollary 1), an interpolatory subdivision scheme that generates polynomials up to degree d also reproduces polynomials up to degree d. us, the result is proved. □ Remark 7. In a similar way, by using eorems 3 and 4, the following results can be proved: (i) e scheme S a α,0 generally generates polynomials of degree 1 and for α � 0, it generates polynomials of degree 3 (ii) e scheme S a α,0 generally reproduces polynomials of degree 1 and for α � 0, it reproduces polynomials of degree 3. (iii) e scheme associated with the Laurent polynomial a α,2α (z) generally generates polynomials of degree 3. (iv) e scheme associated with the Laurent polynomial a α,2α (z) generally reproduces polynomials of degree 1 and for α � 0, it reproduces polynomials of degree 3. (v) e scheme associated with the Laurent polynomial a (β/2),β (z) generally generates polynomials of degree 3.

Shape Preservation of MCTSS.
Here, we check the range of shape parameter for which MCTSS (14) preserves monotonicity and convexity. For this purpose, we use eorems 5 and 6.

Comparison with Existing Schemes
Here we give comparison of the MCTSS with the existing ternary schemes that produce limiting curves up to C 3 smoothness and summarize the results in Table 1. is table shows that MCTSS keeps detailed features better than existing schemes. In this table, G-D and R-D denote the degree of polynomials generation and the degree of polynomial reproduction of the ternary subdivision schemes, respectively. Moreover, scheme of [12] is a special case of the MCTSS for (α, β) � (− 1, − 2).

Numerical Experiments by MCTSS
Here, we show the behavior of the MCTSS (14) by presenting different models and show how it controls the shape of limiting curves. We develop the initial polygons by using functional and nonfunctional initial data. We also compare the results of the MCTSS (14) at different values of shape parameters. In the figures of these experiments, red solid circles and red lines represent initial points and initial polygons, respectively. Experiment 1. In this experiment, we draw the initial control model by using initial control points (12,10), (14,7), (14,5) (6,13), (6,9), (7,8), (7,5), (12,5), and (12,10).

Tensor Product Version of the MCTSS
In this section, we extend the MCTSS into its tensor product version. e tensor product scheme is designed for quadrilateral meshes. Since this scheme is the tensor product version of the MCTSS, it consists of nine refinement rules. One rule is for vertex, four rules are for edges, and four rules are for faces. Hence, at each subdivision step, the bivariate MCTSS splits each mesh into 9 meshes. e outer lawyer or boundary of the mesh which consists of different faces enclosed by edges constitutes the surface. Nine rules of the bivariate subdivision scheme are used for fitting the surface by initial quadrilateral mesh. e bivariate MCTSS, the construction of which is explained in Appendix A, is given by

P k+1
3i,3j � are defined in (A.1)-(A.15) of Appendix A, respectively. Figures 8-10 show the surfaces produced by the bivariate MCTSS at different values of the shape parameters α and β. In these figures, red bullets and red lines represent the initial points and initial meshes, respectively.

Conclusion
In this paper, we have introduced a new method to construct a combined ternary subdivision scheme with two shape parameters. We have analyzed the properties of the MCTSS for different ranges of shape parameters. We have also  showed that the MCTSS produces smooth 2D and 3D models at specific choices of shape parameters. Moreover, we have shown that the graphical results of the MCTSS verify the analytical results of the scheme. Furthermore, we have derived the bivariate subdivision scheme with nine refinement rules. is scheme is used to produce smooth surface when all the initial meshes are quadrilateral.