The Quaternary Interpolating Scheme for Geometric Design

A novel 4-point interpolating subdivision scheme is presented, which generates the family of C2 limiting curves and its limiting function has support on [−7/3, 7/3]. It behaves better than classical 4-point binary and ternary schemes with the same approximation order in many aspects that it has smaller support size, higher smoothness, and is computationally more efficient. The proposed nonstationary scheme can reproduce the functions of linear spaces spanned by {1, sin(αx), cos(αx)} for 0 < α < π/3. Moreover, some examples are illustrated to show that the proposed scheme can also reproduce asteroids, cardioids, and conic sections as well.


Introduction
Subdivision schemes have become one of the most important, significant, and emerging modelling tools in computer applications, medical image processing, scientific visualization, reverse engineering, computer aided geometric designing, and so forth, because of their simple, elegant, and efficient ways to create smooth curves from initial control polygon by subdividing them according to some refining rules, recursively. These refining rules take the initial polygon to produce a sequence of finer polygons converging to a smooth limiting curve.
In the field of nonstationary subdivision schemes, binary and ternary schemes are present in the literature [1][2][3][4][5][6]. But the quaternary schemes are not discussed as others. The aim of this paper is to introduce a quaternary nonstationary scheme which reproduces the linear functions spanned by {1, sin( ), cos( )}. Moreover, the proposed scheme can also generate asteroids, cardioids, and in particular conic sections like circles, ellipses, and so forth. Some properties of the proposed scheme (like support size and symmetry of basic limit function, approximation order, convergence, and smoothness) have also been investigated. Besides the convergence and approximation order, the two important and interesting properties of basic limit function of a subdivision scheme are the support size and the smoothness. They are commonly manipulated with each other in the sense that a higher degree of smoothness generally requires a larger support, thus leading to a more global influence of each initial data value on the limit function. Raising the arity of the subdivision scheme provides a way to overcome this problem to some extent. The proposed scheme has smaller support and higher smoothness than the classical binary 4-point scheme, and all other schemes reproduce cubic polynomials by construction whether they are stationary or nonstationary [1-4, 6, 7].
As interpolating subdivision schemes retain the points of stage as the subset of the points of next stage + 1, thus the structure of a quaternary 4-point interpolating scheme in general form can be written as respectively, for a large value of .

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This paper is organized as follows. In Section 2 the basic notion and definitions of the quaternary subdivision scheme are considered. The construction of the proposed scheme has been shown in Section 3. In Section 4, the convergence and smoothness 2 of the scheme are being calculated using the theory of asymptotic equivalence. The basic limit function and its properties are being discussed in Section 5. The conclusion is drawn in Section 6.

Preliminaries
A quaternary subdivision scheme, to generate smooth curves or surfaces, is defined as the limit of refined control polygons or meshes, according to some refinement rules, recursively, in which the set of control points = { ∈ | ∈ } of polygon at th level is mapped to a refined polygon to generate the new set of control points +1 = { +1 ∈ | ∈ } at the ( + 1)st level by applying the following subdivision rule: with the sequence of finite sets of real coefficients { ( ) : ≥ 0}, constitute the so-called mask of the subdivision scheme. If the mask of a scheme is independent of (namely, if ( ) = for all ≥ 0), then it is called stationary ; otherwise it is called nonstationary ( ) . The convergent subdivision scheme with the corresponding mask { , ∈ } necessarily satisfies ∑ ∈ 4 + = 1, = 0, 1, 2, 3.
For this, we introduce a symbol called the Laurent polynomial of a mask { , ∈ } with finite support. In view of Dyn [10], the sufficient and necessary conditions for a uniform convergent scheme are defined as follows.
A quaternary subdivision scheme is uniformly convergent if and only if there is an integer ≥ 1, such that and subdivision 1 with symbol 1 ( ) is related to with symbol ( ), where 1 ( ) = (4 3 /(1 + + 2 + 3 )) ( ) satisfying the property where = 0 = { : ∈ } and = {( ) = 4 ( +1 − ) : ∈ }. The norm ‖ ‖ ∞ of a subdivision scheme with a mask , ∈ , is defined by ‖ ‖ ∞ and where The notation of convergence for quaternary subdivision scheme, similar to that given by Dyn and Levin in [11] for a binary scheme, is given in the following.
and is not identically zero for some initial data 0 .

Definition 2. Two quaternary subdivision schemes
where then nonstationary scheme { } is also .

The Nonstationary Subdivision Scheme
In this paper, an interpolating 4-point nonstationary scheme is presented which is associated with Lagrange-like cubic trigonometric interpolants. For this, consider a space := span{1, sin( ), cos( )} for some such that 0 < < , for the data set, = {( −1 , ( −1 )), ( 0 , ( 0 )), ( 1 , ( 1 )), ( 2 , ( 2 ))}, and also consider the function, ISRN Computer Graphics is a unique function in which interpolates the data set with having minimum amplitude among other interpolants from [12], where the function ( ) is called the trigonometric Lagrange-like interpolant of the previous data. Since the function ( ) is defined for and similarly, it can be obtained for the following.

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The proof of the previous boundedness can be followed from Lemma 4.
Proof. Since where From Lemmas 6 and 7, it can be written as  (36) So, it can be written as Thus, the scheme corresponding to symbol ( ) ( ) is 1 and hence the scheme (21) is 2 . Figure 1 shows the reproduction of the unit circle and ellipse by the proposed scheme (21), taking regular hexagon as control points for circle.
In Figure 2, twelve points have been taken as control points on the parametric curves of asteroid: ( ) = 3 cos + cos 3 , ( ) = 3 sin − sin 3 and cardioid: ( ) = (1 − cos ) sin , ( ) = (1 − cos ) cos to reconstruct the asteroid and cardioid, respectively, by the proposed scheme. Figure 3 shows the behavior of basic limit function for different values of .

Properties of the Scheme
In this section some properties (approximation order, reproduction property, support size, and symmetry of basic limit function) are being discussed. The results show that the proposed scheme with the same approximation order has the higher smoothness and smaller support than the existing 4point interpolating subdivision schemes (see also Table 1).
Another advantage (see also [14]) of the scheme is the computational cost. In one subdivision step, for a sequence of points the proposed scheme requires 12 multiplications and 9 additions. On the other hand, computing the same number of 4 refined points with the binary scheme requires two subdivision steps with a total cost of 18 multiplications and 9 additions, so the quaternary scheme is more efficient.
The basic limit function of the scheme is symmetric about -axis for the data Theorem 9 (see [3]). The basic limit function is symmetric about the -axis.
Proof. Let denote the basic limit function and define := { /4 | ∈ } such that restriction of to satisfies ( /4 ) = for all ∈ . The symmetry of basic function is proved using induction on .
In the following the support of basic limit function of the proposed scheme has been calculated generated by the control point 0  Theorem 11 (see [15]). A convergent subdivision scheme that reproduces polynomial has an approximation order + 1.
So in the light of the previous theorem, the proposed scheme has approximation order 4 because the construction of mask is formulated through the cubic trigonometric polynomials.
The polynomial reproduction property has its own importance, as the reproduction property of the polynomials up to certain degree implies that the scheme has + 1 approximation order. For this, polynomial reproduction can be made from initial data which has been sampled from some polynomial functions. So, in view of the previous theorem, scheme (32) has approximation order 4.

Conclusion
A 4-point novel quaternary interpolating nonstationary subdivision scheme has been developed which generates the family of 2 limiting curves and its limiting function has support on [−7/3, 7/3]. The nonstationary scheme is associated with trigonometric polynomials which reproduces the functions of linear spaces spanned by {1, sin( ), cos( )} for 0 < < /3. The comparison shows that, with existing stationary and nonstationary interpolating schemes, the proposed scheme is better. Moreover, it also can reconstruct or reproduce the conic sections, asteroids, and cardioids.