Generalized 5-point Approximating Subdivision Scheme of Varying Arity

: The Subdivision Schemes (SSs) have been the heart of Computer Aided Geometric Design (CAGD) almost from its origin, and various analyses of SSs have been conducted. SSs are commonly used in CAGD and several methods have been invented to design curves/surfaces produced by SSs to applied geometry. In this article, we consider an algorithm that generates the 5-point approximating subdivision scheme with varying arity. By applying the algorithm, we further discuss several properties: continuity, Hölder regularity, limit stencils, error bound, and shape of limit curves. The efﬁciency of the scheme is also depicted with assuming different values of shape parameter along with its application.


Introduction
Computer Aided Geometric Design (CAGD) deals with studies of curves and surfaces used in computer graphics, data structure, and computational algebra. In CAGD, geometric shapes are related to the mathematical representations that satisfy approximation and interpolation properties of curves and surfaces. Surface modeling is one of the important studies in the fields of CAGD and computer graphics. It links mathematical sciences with computer science and engineering such as the animation industry, automotive and industrial design, aerospace, mechanical engineering, and numerical computing. Subdivision is an interesting subject and one of the common tools in CAGD, which provides an elegant way for the description of curves and surfaces modeling. Initially, Rham [1] worked on subdivision schemes and made a scheme which generates a function with the Theorem 1. [30] S a scheme converges iff the scheme S c is contractive. S c is contractive if c l < 1 for some l > 0 with c l = max ∑ j |c l k−2 l j | : 0 ≤ k < 2 l , where c l j are the coefficients of the scheme S l c with Laurent polynomial c l (x) = c(x)c(x 2 )...c(x 2 l −1 ). Theorem 2. [30] If S b converges, then the limit curves of the scheme S a with Laurent polynomial a(x) = ( 1+x 2 ) m b(x) are C m continuous, where S b is the scheme for the mth divided differences. An a-ary scheme is said to be linear if it generates level k+1 from level k with linear combination of control points, that is for all 'k' and 'j', there exists sets of real numbers known as masks a k = {a k } such that λ k+1 j = ∑ i∈Z a k j−ni λ k i .
If the mask of the scheme is independent of k, then the scheme has finite support. Similarly, if the mask is independent of 'j', that is each refinement rule operates in the same way at all locations, then the scheme is known as uniform.
A general formula for the mask of the proposed scheme is defined as and 4 ∑ i=0 a 8 where ω j = ω 3−j and j = 0, 1, 2, where w is called the shape controlling parameter and it is used to control the shape of the control of the polygon.

The 5-Point Approximating Schemes
This section consists of different 5-point approximating schemes together with the properties: convergence criteria, continuity, Hölder regularity, and limit stencils.

5-point Binary Approximating Scheme
By substituting a = 2 into Equations (1) and (2), we can get the scheme in the form λ j+1 2i called 5-point binary approximating scheme.

Continuity
To find continuity of the scheme, Equation (6) gives the Laurent polynomial of the form where Mathematics 2020, 8, 474

of 25
C 0 continuity of the scheme S a analogous to a(x), b 1 (x) should be convergent, where b 1 (x) should satisfy Theorem 1 with the given condition 1 2 S b 1 ∞ < 1. From Theorem 1, for −9 < ω < 72/5, we extract The given condition satisfies Theorem 1, thus it must satisfy Theorem 2. This shows that the 5-point scheme is C 0 continuous.
For C 1 continuity, Equation (6) takes the form which satisfies Theorem 1 with the given condition 1 2 S b 2 ∞ < 1. From Theorem 1, for −9 < ω < 72/5, we extract which shows that the scheme is C 1 continuous. For C 2 continuity, Equation (6) may be written as which satisfies Theorem 1 with the given condition 1 2 S b 3 ∞ < 1. From Theorem 1, for −6 < ω < 10, we can get Hence, the scheme is C 2 continuous. For C 3 continuity, Equation (6) takes the form which satisfies Theorem 1 with the given condition 1 2 S b 4 ∞ < 1. From Theorem 1, for −6 < ω < 10, we extract which shows that the scheme is C 3 continuous. For C 4 continuity, Equation (6) may be written as with the given condition 1 2 S b 5 ∞ < 1. Thus, from Theorem 1, for −2 < ω < 6, we can get This satisfies Theorem 1, thus it must satisfy Theorem 2. Thus, the scheme is C 4 continuous. For C 5 continuity, Equation (6) may be written as with the given condition 1 2 S b 6 ∞ < 1. From Theorem 1, for −2 < ω < 6, we extract which shows that the scheme is C 5 continuous. For C 6 continuity, Equation (6) may be written as with the given condition 1 2 S b 7 ∞ < 1. If we extract for 0 < ω < 2 The given condition satisfies Theorem 1, thus it must satisfy Theorem 2, which shows C 6 continuity of the scheme. Similarly, for C 7 continuity we substitute ω = 1 in Equation (6) and get From Theorem 2, we have Therefore, the scheme is C 7 continuous.

Hölder Regularity
We use Theorem 3 to find Hölder regularity of the scheme. The Laurent polynomial of the binary scheme using Equation (3) may be written as or where If ω = 1, we can get b = 2. Using Theorem 3 with m = 7 and b = 2, we have r ≥ 7 − log 2 (2) = 6.

Limit Stencils
The matrix form of the scheme using Equation (3) for w = 1 has the form which shows that the size of the invariant neighborhood is 8. After simplification, matrix X has eigenvalues λ = 1, 1 Mathematics 2020, 8, 474 Thus, the decomposition of the local subdivision matrix X has the form Since the ∧ is diagonal matrix and also the power of a diagonal matrix is equal to the power of each diagonal element. Therefore,   After simplification, we can get which shows that the limit stencils are stable/constant.

The 5-point Ternary Approximating Scheme
Substituting a = 3 into Equations (1) and (2), the 5-point ternary approximating scheme may be written as  14) or

Continuity
To find continuity of the scheme, further simplification of Equation (15) gives the Laurent polynomial in the form To check C 0 continuity of the scheme S β analogous to β(x), c 1 (x) should be convergent, where Since the given condition satisfies Theorem 1, it must satisfy Theorem 2. This means that the scheme is C 2 continuous.
For C 3 continuity, Equation (15) may be written as To check C 3 continuity, the scheme S β analogous to β(x), c 4 (x) should be convergent, where c 4 (x) should satisfy Theorem 1 with the given condition 1 This shows that the scheme is C 3 continuous. For C 4 continuity, Equation (15) may be written as To check C 4 continuity, the scheme S β analogous to β(x), c 5 (x) should be convergent, where c 5 (x) should satisfy Theorem 1 with the given condition 1 3 (S c 5 ) ∞ < 1. From Theorem 1, for −4 < ω < 12, we extract which satisfies Theorem 1. Thus, the scheme is C 4 continuous.
To check C 5 continuity we substitute ω = 4 3 in Equation (15) and get which implies that Since from Theorem 2, we have Therefore, the scheme is C 5 continuous.

Limit Stencils
The matrix form of the scheme using Equation (12) has the form and the local subdivision matrix After simplification on the same manner as presented in Section 3.1.4, we can get which shows that the limit stencils are stable/constant.

The 5-point Quaternary Approximating Scheme
By substituting a = 4 into Equations (1) and (2), we can get the scheme in the form The mask of the quaternary scheme using Equation ( (4 + 5ω), ω} .
Since the given condition satisfies Theorem 1, it must satisfy Theorem 2. This means that the scheme is C 1 continuous.
For C 2 continuity, Equation (21) may be written as To check C 2 continuity, the scheme S γ analogous to γ(x), e 3 (x) should be convergent, where e 3 (x) should satisfies Theorem 1 with the given condition 1 4 (S e 3 ) ∞ < 1. From Theorem 1, for −26 < ω < 38, we extract which satisfies Theorem 1. This shows that the scheme is C 2 continuous. For C 3 continuity, Equation (21) may be written as To check C 3 continuity, the scheme S γ analogous to γ(x), e 4 (x) should be convergent, where e 4 (x) should satisfy Theorem 1 with the given condition 1 4 (S e 4 ) ∞ < 1. From Theorem 1, for −24 < ω < 36, we extract Since the given condition satisfies Theorem 1, it must satisfy Theorem 2. Thus, the scheme is C 3 continuous.
For C 4 continuity, Equation (21) may be written as or To check C 4 continuity, the scheme S γ analogous to γ(x), e 5 (x) should be convergent, where e 5 (x) should satisfy Theorem 1 with the given condition 1 4 (S e 5 ) ∞ < 1. From Theorem 1, for −4 < ω < 8, we extract which satisfies Theorem 1. Since the given condition satisfies Theorem 1, it must satisfy Theorem 2. Hence, the given scheme is C 4 continuous. Now, for C 5 continuity, Equation (21) may be written as Hence, Since, from Theorem 2, we have the scheme is C 5 continuous.

Limit Stencil
The matrix form of the scheme using Equation (18) has the form After simplification on the same manner as presented in Section 3.1.4, we can get which shows that the limit stencils are stable/constant.

Results and Discussion
This section consists of three major parametric effects of the schemes presented by Equations (3), (12), and (18).

Error Bound
This section presents the error between control polygon and limit curve after kth subdivision level of 5-point binary, ternary, and quaternary subdivision schemes using different values mentioned in Tables 1-3 by applying the approach of Hashmi [31]. The error is minimum over the interval ω ∈ [0, 8], 15], and ω ∈ [0, 25] for binary, ternary, and quaternary, respectively, and increases on both sides of the intervals. In Tables 1-3, it is observed that increases in the arity of the schemes decrease the error of the proposed schemes. Figures 1-3 illustrate graphical representation of error. Moreover, the proposed computational cost decreases by increasing the arity of subdivision schemes. Therefore, our experiments show that higher arity scheme are better than the lower arity schemes in the sense of computational cost and error bounds. sides of the intervals. In Tables 1-3, it is observed that increases in the arity of the schemes decrease the error of the proposed schemes. Figures 1-3 illustrate graphical representation of error. Moreover, the proposed computational cost decreases by increasing the arity of subdivision schemes. Therefore, our experiments show that higher arity scheme are better than the lower arity schemes in the sense of computational cost and error bounds.     (3)). the proposed computational cost decreases by increasing the arity of subdivision schemes. Therefore, our experiments show that higher arity scheme are better than the lower arity schemes in the sense of computational cost and error bounds.    (12)). Figure 2. Error bounds of ternary scheme (Equation (12)).   (18)).

Continuity
This section describes the effects of parameters for the schemes in Equations (3), (12), and (18). The order of continuity and effects of parameters ω of the schemes are shown in Tables 4-6, respectively. This can easily be found over the parametric intervals using the approach of Hassan [8].

Continuity
This section describes the effects of parameters for the schemes in Equations (3), (12), and (18). The order of continuity and effects of parameters ω of the schemes are shown in Tables 4-6, respectively. This can easily be found over the parametric intervals using the approach of Hassan [8]. Table 4. Continuity order of binary scheme (Equation (3)).

Shapes of Limit Curves
The parametric effect and continuity of the limit curve of the schemes are shown in Figures 4-6, respectively. These figures illustrate the role of free parameters when 5-point binary, ternary ,and quaternary approximating schemes are applied on discrete data point. One can see the looseness/tightness of the limit curves in

Conclusion
In this work, we introduce the family of 5-point schemes which depict the representation of a wide variety of shapes with high smoothness (continuity) and less computational cost (processing time). These properties are useful in computer aided geometric design and geometric modeling. We apply Laurent polynomial to analyze our schemes. The shape parameter ω makes it able to provide different results along with its applications.

Conclusions
In this work, we introduce the family of 5-point schemes which depict the representation of a wide variety of shapes with high smoothness (continuity) and less computational cost (processing time). These properties are useful in computer aided geometric design and geometric modeling. We apply Laurent polynomial to analyze our schemes. The shape parameter ω makes it able to provide different results along with its applications.