Analysis of Geometric Properties of Ternary Four-Point Rational Interpolating Subdivision Scheme

Shape preservation has been the heart of subdivision schemes (SSs) almost from its origin, and several analyses of SSs have been established. Shape preservation properties are commonly used in SSs and various ways have been discovered to connect smooth curves/surfaces generated by SSs to applied geometry. With an eye on connecting the link between SSs and applied geometry, this paper analyzes the geometric properties of a ternary four-point rational interpolating subdivision scheme. These geometric properties include monotonicity-preservation, convexity-preservation, and curvature of the limit curve. Necessary conditions are derived on parameter and initial control points to ensure monotonicity and convexity preservation of the limit curve of the scheme. Furthermore, we analyze the curvature of the limit curve of the scheme for various choices of the parameter. To support our findings, we also present some examples and their graphical representation.


Introduction
Shape preservation has great practical importance in the designing of curves/surfaces tailored to industrial design (e.g., related to car, aeroplane or ship modelling where convexity is imposed by technical and physical conditions as well as by aesthetic requirements). These properties are used in the design of curves or surfaces to predict or control their 'shape' by the shape of the control points, that is, the vertices of a given polygonal arc or polyhedral surface. Efficient methods to j i } i∈Z to define polygon F j+1 = { f j+1 i } i∈Z by connecting the resultant subdivision rule and having set of control points at level k(k ≥ 0, k ∈ Z).
This motivated us to present the analysis of the geometric properties of the C 2 -continuous ternary scheme which is capable of producing fractal curves. In order to show the performance of the ternary scheme, we analyze the geometric properties such as monotonicity-preservation, convexity-preservation and curvature of the limit curve. Moreover, the limit curves with specific value of shape control parameter u are depicted by significant application of derived conditions on the initial data. The rest of the paper is organized as follows: In Section 2, we present geometric properties of the FPR-scheme. Numerical examples and conclusions are discussed in Section 3.

Geometric Properties of the FPR-Scheme
In this section, we present the geometric properties of the limit curves generated by the FPR-scheme. These geometric properties consist of monotonicity-preservation, convexity preservation and curvature of the limit curves of the geometric properties.

Monotonicity Preservation
Here we discuss the monotonicity preserving property of the FPR-scheme. For monotonicity preservation, we consider monotone control polygon since the limiting curve generated by the FPR-scheme preserves the monotonicity of initial data. The monotonicity preservation of the FPR-scheme can be obtained by applying the first order divided difference (DD) as D k i = f k i+1 − f k i . Thus the first order DD-scheme of (4) can be written as For convenience, we introduce a new parameter ω in terms of u. For this let ω = −108u+1 9(126u−1) , so the value of u in terms of ω, can be written as: By combining Equations (5) and (6) the first order DD-scheme of (4) takes the form Theorem 1. Given a set of initial control points { f 0 i } i∈Z which are monotonically increasing, such that D 0 i ≥ 0. Let Furthermore, the parameter ω satisfies −0.077 ≤ ω < 0.
Thus, the FPR-scheme preserves monotonicity for initial monotone data.
By assumption it is clear that (8) holds for k = 0. Suppose that (8) holds for k ≥ 1, next we verify that it also holds for k + 1.
In order to show that (8) is true for k + 1, we first prove that D k+1 For convenience put λ = − 2ω 1−2ω , we get Now consider Now consider By combining Equations (9)-(11), we have D k+1 By Equation (9), the denominator of the above equation is greater and equal to zero, and the numerator A satisfies Further we have By Equation (10), the denominator of the above equation is greater and equal to zero, and the numerator B satisfies By Equation (11), the denominator of the above equation is greater and equal to zero, and the numerator C satisfies Therefore r k+1 3i+2 ≤ λ. By following same steps, we can also prove 1 This completes the proof.

Convexity Preservation
Now we discuss the convexity preservation of the FPR-scheme. For convexity preservation, we consider convex control polygon so the limiting curve generated by the FPR-scheme preserves convexity of initial data. The convexity preservation of the FPR-scheme can be obtained by applying second order DD as P k For convenience, we introduce a new parameter σ instead of u. For this let σ = 1−135u 3(126u−1) and u = 1 126 , so the value of u in terms of σ can be expressed as By combining Equations (12) and (13), second order DD-scheme takes the form Theorem 2. Given a set of initial control points Thus, the FPR-scheme preserves convexity for initial convex data.
By assumption it is clear that (15) holds for k = 0. Assume that (15) also holds for k ≥ 1, next we verify that it also holds for k + 1.
In order to show that (15) is true for k + 1, we first show that P k+1 For convenience put µ = 2+3σ 1+3σ , σ = − 1 3 , we get Now consider Now consider So by combining Equations (16)-(18), we have P k+1 By Equation (16), the denominator of the above equation is greater and equal to zero, and the numerator D satisfies By Equation (17), the denominator of the above equation is greater than and equal to zero, and the numerator E satisfies By Equation (18), the denominator of the above equation is greater than and equal to zero, and the numerator F satisfies Therefore, q k+1 3i+2 ≤ µ. By following same steps we can also prove 1 This completes the proof.

Curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. Curvature is very important property not only in continuous geometry but also in network applications; see Reference [25].
The quality of SS can be measured quantitatively by finding curvature, as functions of cumulative chord length. We use the method described in Reference [17] to determine the curvature and numerical simulations are provided to illustrate the effects of various choices of shape parameter on curvature.
Since the FPR-scheme is parametric, it is obvious to see the results for various choices of the parameter u. Here we measure the curvature of limit curves of the FPR-scheme for various choices of u. In Figures 1 and 2, we consider a control polygon of a circle. The shape of the circle is obtained by applying an FPR-scheme five times on this initial control polygon for various choices of parameter u. These limit curves are shown in Figures 1a,b and 2a,b, while their corresponding curvatures are shown in Figures 1c,d and 2c,d. In Figures 3 and 4, we consider a star-shaped closed control polygon. The limit curves after applying the FPR-scheme five times on this initial control polygon for various choices of parameter u is shown in Figures 3a,b and 4a,b, while their corresponding curvatures are shown in Figures 1c,d and 2c,d. In Figures 5 and 6, we consider an open control polygon that presents the basic limit function of the scheme. The limit curves after applying the FPR-scheme five times on this initial control polygon for various choices of parameter u is shown in Figures 5a,b and 6a,b, while their corresponding curvatures are shown in Figures 5c,d and 6c,d.

Numerical Examples and Conclusions
In this section, we present some numerical examples to show monotonicity and convexity preserving behaviour of the FPR-scheme. At the end of the section, we discuss the conclusion of the work done so far.

Numerical Examples
We consider monotone data in Tables 1 and 2. Figure 7 explains monotonicity preserving property of the FPR-scheme. Figure 7a is generated by using monotone data as given in Table 1. In this figure, dotted lines show the initial set of values and the solid line represents the limit curve generated by the FPR-scheme which is clearly monotonically increasing curve. Figure 7b is generated by using monotone data as given in Table 2. In this figure, dotted lines show the initial set of values and the solid line represents the limit curve generated by the FPR-scheme, which is clearly monotonically increasing curve.  We consider convex data in Tables 3 and 4. Figure 8 explains convexity preserving property of the FPR-scheme. Figure 8a is generated by using monotone data as given in Table 3. In this figure, dotted lines show the initial set of values and the solid line represents the limit curve generated by the FPR-scheme which is clearly monotonically increasing curve. Figure 8b is generated by using monotone data as given in Table 4. In this figure, dotted lines show the initial set of values and the solid line represents the limit curve generated by the FPR-scheme which is clearly monotonically increasing curve.

Conclusions
In this paper, we have presented geometrical properties of the FPR-scheme, which improves on the scheme in various ways that meet different requirements. These geometric properties demonstrate that the shape-preservation of the limit curve is a useful mechanism for modifying the FPR-scheme. We have shown that by taking initial control data monotone and convex, the limit curves generated by the FPR-scheme are also monotone and convex. As observed, the FPR-scheme for various choice of shape control parameter can be considered more universal. Several examples are given which support our findings. Finally, the same idea could be applied to bivariate surfaces. A useful extension of this work is to analyze shape preserving behaviour of SSs when scattered data is considered.