Curve Variations in Non-Stationary Three-Point Subdivision Schemes

Subdivision schemes are acknowledged as an important tool in computer aided geometric design. The new binary non-stationary three-point approximating subdivision schemes have been proposed that generate wide variations of C and C continuous curves using shape control parameter 0 ξ . The proposed schemes are the counterpart of stationary schemes introduced by Hormann and Sabin (2008) and Siddiqi and Ahmad (2007). Curve variations using the shape control parameter 0 ξ have been demonstrated by the several examples.


INTRODUCTION
For the generation of smooth curves and surfaces, subdivision schemes have proven itself the building bricks in Computer Aided Geometric Design.
Hassan and Dodgson ( 2 0 0 3 ) proposed a binary three-point approximating subdivision scheme that generates C 3 limiting curve.Siddiqi and Ahmad (2007) introduced another binary three-point approximating subdivision scheme that gives C 2 smooth limiting curve.Hormann and Sabin (2008) developed another binary three-point approximating subdivision scheme that generates the smooth limiting curve of C 1 continuity.The limiting curve generated by the scheme of Hormann and Sabin (2008) always lies outside the control polygon.The geometric behavior of Hassan and Dodgson (2003) scheme, Siddiqi and Ahmad (2007) scheme, Hormann and Sabin (2008) schemes have been illustrated in the Fig. 1.Dyn et al. (1987) developed a binary stationary four-point interpolating subdivision scheme that generates the smooth interpolating limiting curve of C 1 continuous.Beccari et al. (2007a) introduced a binary non-stationary four-point interpolating subdivision scheme that generates conics sections quite efficiently.Beccari et al. (2007b) proposed a ternary non-stationary four-point interpolating subdivision scheme that generates the smooth interpolating curve of C 2 continuous.Siddiqi et al. (2015) introduced a binary non-stationary three and four-point approximating subdivision schemes using hyperbolic function that generate the limiting Stationary and non-stationary subdivision schemes are defined as the set of coefficients k i ξ determines the subdivision rule at level k and is called the k-th level mask.If the mask k i ξ are independent of k, the subdivision scheme is called stationary subdivision scheme otherwise it is called nonstationary.In this study, we present the binary nonstationary three-point approximating subdivision schemes, providing the user with an initial control parameter k 0 ξ which can generate C 1 and C 2 continuous limit curves of different shapes.

BINARY STATIONARY THREE-POINT SUBDIVISION SCHEMES
The binary three-point approximating subdivision techniques to refine the discrete control polygon are: where, is the set of initial discrete control point at 0 level and the coefficients of the scheme satisfy the equation For the above refinement rules, a geometric designer can generate only one limiting curve with a small magnitude and no one can see significant variations (Fig. 1).

Binary non-stationary three-point subdivision schemes:
The binary non-stationary three-point approximating subdivision scheme, counterpart of stationary scheme of Hormann and Sabin (2008), is defined by the refinement techniques (1) and the mask of the scheme are given by: In this way, given an initial shape control parameter each different level k can be calculated using the above formulae.
Another non-stationary three-point approximating scheme, which is a counterpart of stationary scheme of Siddiqi and Ahmad (2007), is defined by the refinement techniques (1) and the coefficients of the scheme are: From this, using an initial parameter different level k can be calculated using the above formulae.
Starting from any 6 0 − ≥ ξ , this implies that ξ is always well defined.A wide range of shape control parameter allows user to get considerable variations of shapes in curves design (Fig. 2 and 3).
Remark 1: From the Fig. 2 and 3, as the initial shape control parameter 0 ξ increases in its domain, the shape of curve changes a lot at first and then tends to approximate the control polygon as .
0 +∞ → ξ Remark 2: From (3) and ( 5), we know: is stationary.Then the proposed scheme retrogrades to the stationary scheme.If Here convergence of the schemes (2) and (4) are analyzed using the results of Dyn and Levin (1995).Which describes the convergence of a non-stationary scheme to its asymptotically equivalent stationary counterpart.
Theorem 1: The non-stationary scheme (2) is asymptotically equivalent to the stationary scheme of Hormann and Sabin (2008)  Proof: To verify the smoothness C 1 of the proposed non-stationary scheme (2), its second divided difference mask are required.The coefficients of the scheme are: So, its first divided difference masks are: Now applying Remark 2 gives: . 32 Then the two schemes are asymptotically equivalent.And hence the scheme is C 1 (Dyn and Levin (1995)).Since: To prove (6), the convergence of the series is required to be prove:  2), the behavior of (7) as To verify the convergence of (7), the following three cases are: Case 1: For this, exploit the ratio test.Since Hence the convergence of ( 7) is verified.

Case 3.2:
which turns to case 2. Hence the convergence of ( 7) is verified.
Combining the three cases, the non-stationary subdivision scheme (2) is asymptotically equivalent to the stationary scheme of Hormann and Sabin (2008) with the coefficients And the scheme generates the smooth C 1 curves.
Corollary 1: A proposed binary non-stationary threepoint subdivision scheme (2) is not C t continuous for t > 1.
Theorem 2: The non-stationary scheme ( 4) is asymptotically equivalent to the stationary scheme of Siddiqi and Ahmad (2007)  Proof: To verify the smoothness C 2 of the proposed non-stationary scheme (4), its second divided difference mask are required.The coefficients of the scheme are: So, its first and second divided difference masks are: Now applying Remark 2 gives: . 32 Which are the coefficients of the second divided difference of the stationary scheme having mask in (1) with Then the two schemes are asymptotically equivalent.And hence the scheme is C 2 (Dyn and Levin (1995)).Since: To prove (8), the convergence of the series is required to be prove:  4), the behavior of (9) as To verify the convergence of (9), the following three cases are: Obviously, it gives: . 0 Here For this, exploit the ratio test.Since ).Here, ( ).  [ ] .
1 4 Two subcases are: Hence the convergence of ( 9) is verified.

Case
which turns to case 2. Hence the convergence of ( 9) is verified.
Combining the three cases, the non-stationary subdivision scheme (4) is asymptotically equivalent to the stationary scheme of Siddiqi and Ahmad (2007) with the coefficients Corollary 2: A proposed binary non-stationary threepoint subdivision scheme (4) is not C t continuous for t >2.  2) and (4) in Fig. 4 and 5.It is not difficult to observe that the limiting curves generated by the proposed non-stationary schemes outperform those by the stationary schemes.

CONCLUSION
The binary non-stationary three-point approximating subdivision schemes are introduced that generates a family of C 1 and C 2 limiting curves for the wider range of tension parameter.The proposed schemes give wide flexibility to construct C 1 and C 2 considerable variations of shapes in geometric designing.Important feature of the proposed schemes is that the schemes rectified the deficiency of its stationary counter parts (Hormann and Sabin, 2008;Siddiqi and Ahmad, 2007).The derivative continuity of the proposed schemes i s proved using the conclusion in Dyn and Levin (1995).

Fig. 1 :
Fig. 1: Comparison of binary stationary three-point approximating subdivision schemes.Continuous curve Harmann and Sabin (2008), dash curve Siddiqi and Ahmad (2007), dotted curve Hassan and Dodgson (2003) curves of C 1 and C 2 continuous respectively.Siddiqi and Rehan (2010) introduced a new ternary threepoint approximating subdivision scheme using quadratic B-spline basis functions that generates a limiting curve of C 2 continuous.Tan et al. (2014a) developed a new binary stationary four-point approximating subdivision scheme that generates the shape-preserving limiting curve of C 3 continuous.Tan et al. (2014b) also introduced a binary stationary five-point approximating subdivision scheme that preserve the convexity property with a parameter.Pan et al. (2012) introduced a new combined approximating and interpolating subdivision scheme which generates the limiting curve of C 2 continuous.Tan et al. (2016) proposed a new type of binary nonstationary three-point approximating subdivision
Which are the coefficients of the first divided difference of the stationary scheme having mask in (1 has C 2 smoothness.Now if: Hence the convergence of (9) is verified.

Case
the smooth C 2 curves.

Figure 1 ,Fig. 5 :
Figure1, comparison of the binary stationary three-point approximating subdivision schemesHassan and Dodgson (2003),Siddiqi and Ahmad (2007) and Harmann and Sabin (2008) have been illustrated using initial control points of the square.The schemes of Harmann and Sabin (2008),Siddiqi and Ahmad (2007) generate C 1 and C 2 -continuous a fix limiting curve respectively that restrict the user to design the variety of limiting curves.These restrictions have been removed using the binary non-stationary approximating subdivision schemes (2)