A nonlinear generalized subdivision scheme of arbitrary degree with a tension parameter

In this paper, a nonlinear generalized subdivision scheme of arbitrary degree with a tension parameter is presented, which refines 2D point-normal pairs. The construction of the scheme is built upon the stationary linear generalized subdivision scheme of arbitrary degree with a tension parameter, by replacing the weighted binary arithmetic average in the linear scheme with the circle average. For such a nonlinear scheme, we investigate its smoothness and get that it can reach C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C^{1}$\end{document} with suitable choices of the tension parameter when degree m≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m\geq 3$\end{document}. Besides, the nonlinear scheme can reconstruct the circle and the selection of parameters and initial normal vectors can effectively control the shape of the limit curves.

In addition, the shape of the limit curve can be effectively controlled with the selection of parameters and initial normal vectors.
The outline of this paper is organized as follows. Section 2 provides a short survey of required background, including a summary on the stationary linear generalized subdivision scheme of arbitrary degree with a tension parameter introduced in [10] and a brief review on the circle average. In Sect. 3, we modify the stationary linear generalized subdivision scheme and provide a nonlinear subdivision scheme. The convergence and smoothness are analyzed in Sect. 4. In Sect. 5, some examples are given to illustrate the performance of the nonlinear scheme. Finally, we conclude the paper with a short summary and further research work in Sect. 6.

Preliminaries
In this section, we recall the definition of the circle average and present the algorithm of the stationary linear generalized scheme of arbitrary degree with a tension parameter (LGP scheme) introduced in [10].

The circle average
Given a real weight t ∈ [0, 1] and in a 2D space two point-normal pairs (PNPs) P 0 = (p 0 , n 0 ), P 1 = (p 1 , n 1 ), each consisting of a point and a normal unit vector, the circle average produces a new pair P t = P 0 t P 1 = (p t , n t ). The point p t is on an auxiliary arcp 0 p 1 , at arc distance tθ from p 0 , where θ is the angle between n 0 and n 1 , and 0 ≤ θ ≤ π . The selection of the auxiliary arc depends upon whether the normal vectors n 0 and n 1 are in different half-planes (relative to p 0 p 1 ). The normal n t is the geodesic average of n 0 and n 1 , and the length of the line segment [p 0 , p 1 ] is denoted by |p 0 p 1 |.
For the selection criterion of the auxiliary arcp 0 p 1 and for specific details on the circle average, refer to [7].

The stationary linear generalized scheme of arbitrary degree with a tension parameter
In this subsection, we recall the stationary linear generalized subdivision scheme of arbitrary degree with a tension parameter (LGP scheme) introduced in [10]. Let f = {p 1 , p 2 , . . . , p n } be the initial control polygon and u be the tension parameter. The LGP scheme of degree 1 is as follows: ). The algorithm of the LGP scheme is presented in Algorithm 1, which can be seen as a generalization of the Lane-Riesenfeld algorithm.

The nonlinear generalized subdivision scheme of arbitrary degree with a tension parameter
In this section, we give the rule of the nonlinear generalized subdivision scheme of arbitrary degree with a tension parameter (NGP scheme). The NGP scheme refining 2D PNPs is obtained from the LGP scheme. We substitute the weighted binary arithmetic average Algorithm 1 The refinement step of the LGP algorithm Input: The tension parameter u(u ∈ R + ) and the data to be refined f = {p i } i∈Z . The degree m(m ∈ N 0 ) of the LGP scheme. Output: The refinement data S(f).
end for(i) 8: for k = 1, 2, . . . , m -1 do 9: for i ∈ Z do 10: end for(i) 12: end for(k) end for(i) 16: end for(j) Algorithm 2 The refinement step of the NGP algorithm Input: The tension parameter u(u ∈ R + ) and the data to be refined P = {P i = (p i , n i )} i∈Z . The degree m(m ∈ N 0 ) of the NGP subdivision scheme. Output: The refinement data T(P). 1: P 0 2i ← P i 2: P 0 2i+1 ← P i 3: for j ∈ N 0 do 4: for i ∈ Z do 5: end for(i) 8: for k = 1, 2, . . . , m -1 do 9: for i ∈ Z do 10: end for(i) 12: end for(k) in the LGP scheme by the circle average and obtain the NGP scheme, which is presented in Algorithm 2.
Remark 1 When u = 1, the NGP scheme reduces to the modified Lane-Riesenfeld scheme introduced in [7], which is C 1 for m ≥ 2. In particular, the NGP scheme of degree 3 becomes the nonlinear 3-point approximating scheme with a tension parameter introduced in [9], when u = 1 2w -1.
Remark 2 As the weighted circle averages of the two point-normal pairs are located on a circle, the NGP scheme can reconstruct the circle when the initial data is sampled from a circle (see Lemma 2.1 of [7]).

The convergence and smoothness analysis
In this section, we discuss the convergence and smoothness of the NGP scheme. If p is a sequence of points, we use the symbol p for the sequence of differences:

Convergence analysis
In this subsection, we analyze the convergence of the proposed nonlinear scheme. The convergence analysis is based on the proximity condition and the displacement-safe condition, i.e., if a nonlinear scheme is displacement-safe for manifold data with a contractivity factor η ∈ (0, 1), then the nonlinear scheme is convergent for any input manifold(see Theorem 3.6 of [11]). First, we introduce some notation related to the NGP algorithm. For k = 0, 1, . . . , m -1 and j ∈ N 0 , define P j,k For k = 0, 1, . . . , m -2, and j ∈ N 0 , define μ j,k = 1 2 cos θ j,k 4 . We also define, for j ∈ N 0 , Lemma 1 (Contractivity) There exists J ∈ N 0 such that the NGP algorithm is contractive in the refinement levels above J, equivalently e j+1 ≤ ηe j with η ∈ (0, 1) for j ≥ J.
for j ≥ j * . By the subdivision of the normals, we have Thus Furthermore, based on (4) and (5), we obtain is monotone decreasing with j. Let j * * be the minimal j for ( 1 , then η j ≤ η J < 1 for j ≥ J. Define η = η J , we get e j+1 ≤ ηe j with η ∈ (0, 1) for j ≥ J, which completes the proof.

Lemma 2 (Safe displacement)
The NGP scheme is displacement safe in refinement levels above j * described in Lemma 1, namely satisfies |p Since Set c = B2 m-1 , we easily get |p j+1 2ip j i | ≤ ce j (c > 0) from (6), which completes the proof.
By Lemma 1 and Lemma 2, we conclude the convergence of the points. It remains to prove the convergence of the normals. The convergence of the normals is an immediate consequence of [12], which is a special case of Corollary 3.3 in [12] with α 1 = u and α 2 = α 2 = · · · = α m = 1.
With the above results, we can now obtain the following theorem.

Theorem 3
The NGP scheme of degree m (m ≥ 1) is convergent.

Smoothness analysis
In this subsection, we study the smoothness of the NGP scheme. To conveniently study the smoothness, we introduce some additional notation and rewrite Algorithm 2 as Algorithm 3, which includes one nonlinear 3-point approximating refinement step and m -3 smoothing steps. . We also define, for j ∈ N 0 , For given two PNPs, P i = (p i , n i ), i = 0, 1, we denote by e the length of the segment [p 0 , p 1 ], θ the angle between n 0 and n 1 , and t = |p t q t |, where p t is the point of P 0 t P 1 and q t denotes the linear average q t = (1t)p 0 + tp 1  In the triangle p t p 0 q t , we have the following result by applying the cosine theorem.

Algorithm 3
The refinement step of the NGP algorithm Input: The parameter u(u ∈ R + ) and the data to be refined P = {P i = (p i , n i )} i∈Z . The degree m(m ∈ N 0 ) of the NGP subdivision scheme. Output: The refinement data T(P). 1: P 0 2i ← P i 2: P 0 2i+1 ← P i 3: for j ∈ N 0 do 4: for i ∈ Z do 5: end for(i) 10: for k = 1, 2, . . . , m -3 do 11: for i ∈ Z do 12 end for(i)

18: end for(j)
In the rest of this subsection, we prove several lemmas about the NGP scheme, which show that an alternative proximity condition (16) holds.
To study the convergence of the nonlinear scheme, we recall the following result (see Lemma 3.3 in [8]).
Lemma 7 In the above notation, for k ≥ 1, we have With the above lemmas, we obtain the following results.

Lemma 8 Let S be the linear scheme corresponding to the NGP scheme, and j = j,m-3 .
Then, for j ≥ J, Lemma 9 Let S be the linear scheme corresponding to the NGP scheme. Then Remark 3 Condition (16) in Lemma 9 is an alternative proximity condition.
With the above lemmas in this subsection, we can obtain the following theorem.
which is essential for the proof of C 1 smoothness. When u ∈ ( √ 2 -1, √ 2 + 1), it follows that 2B 2 1 < 1, which is required in the proof of the theorem. Similar to the proof of Theorem 3.8 and Theorem 3.13 in [8], we complete the proof according to (17) and (19).
Remark 4 The NGP scheme of arbitrary degree m reduces to the nonlinear one introduced in [7] for u = 1, which is C 1 for m ≥ 2.

Examples
In this section, we present some examples to illustrate the performance of the nonlinear subdivision scheme presented in this paper. Figure 3 and Fig. 4 illustrate limit curves generated by the LGP scheme of degree 3 and the NGP scheme of degree 3 with different tension parameters, which have the same control polygon. The resulting curves generated by the LGP scheme of degree 3 progressively tend to shrink towards the control polygon with the increasing of the tension parameter u, but cannot reconstruct the circle in Fig. 3, while the limit curves generated by the NGP scheme of degree 3 with different tension parameters can reproduce the circle in Fig. 4.
The limit curves generated by the LGP scheme of degree 3 and the NGP scheme of degree 3 with the parameter u = 1 2 and four different initial normals at P are shown in Fig. 5. It shows that the selection of the initial normals effectively controls the shape of the limit curves generated by the NGP scheme. Figure 6 shows that some examples illustrating Limit curves generated by the NGP scheme of degree 3 with u = 1 10 , 1 2 , 1, 2, 10 after six iterations, respectively(from left to right)

Figure 5
Limit curves generated by the LGP scheme of degree 3(dash-dot) and the NGP scheme of degree 3(solid) with u = 1 2 , starting from the same polygon with four different initial normals at P Figure 6 Limit curves generated by the NGP scheme of degree 2, 3, 4, 5 with the parameter u = 1 2 (from left to right), respectively the shape of the resulting curve are produced by the same control polygon, but with the tension parameter u = 1 2 and different degrees 2, 3, 4, 5 of the NGP scheme. It shows that the resulting curve becomes more and more smooth with increase in the degree of the NGP scheme.

Conclusion
In this paper, by suitably using the circle average, we have presented a nonlinear generalized subdivision scheme of arbitrary degree m with a tension parameter based on the stationary linear generalized subdivision scheme of arbitrary degree with a tension parameter. The scheme can be seen as the generalization of the nonlinear modified Lane-Riesenfeld algorithm presented in [7] and the nonlinear 3-point approximating scheme with a parameter introduced in [9], and can reach C 1 for m ≥ 3. The nonlinear scheme can reconstruct the circle by the circle average, and suitable choices of parameters and initial normal vectors can effectively control the shape of the limit curve. It can be seen that the limit curves become smoother with increase in the degree of the NGP scheme in Fig. 6. So we may focus on proving higher order of smoothness of the NGP scheme in the future.