AbstractIn this paper a combined approximating and interpolating subdivision scheme is presented. The relationship between approximating subdivision and interpolating subdivision is derived by directly performing operations on geometric rules. The behavior of the limit curve produced by our combined subdivision scheme is analyzed by the Laurent polynomial and attains C2 degree of smoothness. Furthermore, a non-uniform combined subdivision with shape control parameters is introduced, which allows a different tension value for every edge of the original control polygon

Lin, Shujin

Luo, Xiaonan

Pan, Jun

Elsevier - Publisher Connector 

A combined approximating and interpolating subdivision scheme with C2 continuity 

Applied Mathematics Letters 25 (2012) 2140–2146
Contents lists available at SciVerse ScienceDirect
Applied Mathematics Letters
journal homepage: www.elsevier.com/locate/aml
A combined approximating and interpolating subdivision scheme with
C2 continuity
Jun Pan a,b,∗, Shujin Lin a,c, Xiaonan Luo a,∗∗
a National Engineering Research Center of Digital Life, State-Province Joint Laboratory of Digital Home Interactive Applications, School of Information Science and
Technology, Sun Yat-sen University, Guangzhou, 510006, China
b College of Computer and Information Engineering, Henan University of Economics and Law, Zhengzhou, 450000, China
c School of Communication and Design, Sun Yat-sen University, Guangzhou, 510006, China
a r t i c l e i n f o
Article history:
Received 8 November 2010
Received in revised form 20 May 2012
Accepted 21 May 2012
Keywords:
Combined subdivision
Approximation
Interpolation
Shape control parameters
a b s t r a c t
In this paper a combined approximating and interpolating subdivision scheme is presented.
The relationship between approximating subdivision and interpolating subdivision is
derived by directly performing operations on geometric rules. The behavior of the
limit curve produced by our combined subdivision scheme is analyzed by the Laurent
polynomial and attains C2 degree of smoothness. Furthermore, a non-uniform combined
subdivision with shape control parameters is introduced, which allows a different tension
value for every edge of the original control polygon.
© 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Subdivision is a very popular geometric modeling tool. In general, subdivision can be distinguished into two categories:
interpolating schemes and approximating schemes. Although approximating schemes yield smoother curves with higher
order continuity [1–3], interpolating schemes are more useful for engineering applications, especially the schemes with the
shape control [4,5]. The approximating schemes and interpolating schemes are related by a deep connection. Maillot et al.
[6] and Zhang [7] introduced a push-back method to progressively interpolate the control vertices. To obtain interpolating
refinement rules from approximating ones, Rossignac [8] used aweak operation and Li et al. [9] used a sequence of weighted
averaging operations. Lin et al. [10] proceededwith relationship between the cubic B-spline and four-point curve subdivision
scheme. Beccari et al. [11] employed generating functions to extend a unified univariate subdivision. All the considered
methods only produced C1-continuous limit function.
In this paper,we focus on constructing combined subdivision schemewith higher order continuity by directly performing
some simple operations on geometric rules. We also offer a method of intuitively modifying the approximating or
interpolating tension of the limit shape. The new combined approximating and interpolating four-point C2 subdivision
scheme is defined as follows:
Pk+13i = a0Pki−1 + a1Pki + a0Pki+1,
Pk+13i+1 = b0Pki−1 + b1Pki + b2Pki+1 + b3Pki+2,
Pk+13i+2 = b3Pki−1 + b2Pki + b1Pki+1 + b0Pki+2,
(1)
∗ Corresponding author at: College of Computer and Information Engineering, Henan University of Economics and Law, Zhengzhou, 450000, China.∗∗ Corresponding author.
E-mail addresses: panjun@mail2.sysu.edu.cn (J. Pan), lnslxn@mail.sysu.edu.cn (X. Luo).
0893-9659/$ – see front matter© 2012 Elsevier Ltd. All rights reserved.
doi:10.1016/j.aml.2012.05.012
J. Pan et al. / Applied Mathematics Letters 25 (2012) 2140–2146 2141
where
a0 = 427α,
a1 = 1− 827α,
b0 = 127 −
1
27
(1− α)(ω + 1),
b1 = 1627 +
1
27
(1− α)(2+ 2ω − ν),
b2 = 1027 +
1
27
(1− α)(−1− ω + 2ν),
b3 = − 127 (1− α)ν,
with ω = 32 (1 + µ), ν = 32 (1 − µ), µ is free parameter. If α = 0, (1) generates the interpolating subdivision; if α = 1,
(1) generates approximating subdivision; and if 0 < α < 1, (1) generates the subdivision that produces the limit curves
intervening between approximating subdivision and interpolating subdivision. It is also proved that the proposed combined
subdivision generates curves with C2 continuity when 1/5 < µ < 1/3 and 0 ≤ α ≤ 1.
2. Preliminaries
In this section some fundamental definitions and results are recalled as the basis of the theory developed in the
remainder of this paper. For their derivation the reader is referred to [12,13]. Start from a set of initial control vertices
P0 = {p0i ∈ R, i ∈ Z}, the set of control vertices Pk+1 = {pk+1i , i ∈ Z} at the (k + 1)th level generated by a ternary
subdivision scheme is defined by
Pk+1i =

j∈Z
ai−3jPkj , i ∈ Z. (2)
The set of coefficients a = {ai ∈ R, i ∈ Z} determines the subdivision rule. Let S be a convergent subdivision scheme with
a mask a. Then
i∈Z
a3i+j = 1, j = 0, 1, 2. (3)
The generating function associated with the set of coefficients can defined as the following Laurent polynomial:
a(z) =

−∞<i<∞
aiz i. (4)
A necessary Cm-continuity condition for the subdivision scheme is
a(z) = 1
3m

1− z3
1− z
m+1
am+1(z), (5)
with am+1(z) denoting the (m + 1)th divided difference of a(z). In particular, if av+1(z) satisfies relation (3) for all v ∈
Z ∩ [0,m] and there exists an integer L > 0 such that ∥SL
av+1∥∞ < 1, then S∞P0 ∈ Cm.
Let Sk be k-th refinement of the subdivision and13Sk+1
∞ = 13 max

i∈Z
|ak+13i |,

i∈Z
|ak+13i+1|,

i∈Z
|ak+13i+2|

, (6)
where
ak+1m+1(z) =
k
j=0
am+1(z3
j
). (7)
If ∥ 13Sk+1∥∞ < 1 for all given initial control vertices P0, then S∞P0 ∈ Ck.
3. Combined approximating and interpolating four-point ternary subdivision scheme
In this section, we deduce our combined four-point ternary subdivision scheme by directly performing operations on
geometric rules. Given a set of initial control vertices {P0i ∈ R, i ∈ Z}, as shown in Fig. 1(a). First, insert two vertices P
1
2
3i+1
2142 J. Pan et al. / Applied Mathematics Letters 25 (2012) 2140–2146
Fig. 1. Relationship between ternary approximating and interpolating subdivision.
and P
1
2
3i+2 into the control polygon respectively at 1/3 and 2/3 parametric positions between P
0
i and P
0
i+1, as shown in Fig. 1(b).
Let
∆0i = −
1
27
P0i−1 +
2
27
P0i −
1
27
P0i+1, (8)
then move P
1
2
3i+1 and P
1
2
3i+2 to the new position P
1
3i+1 and P
1
3i+2 with displacement of∆
0
i and∆
0
i+1, while move P
0
3i to the new
position P13i with displacement of 4∆
0
i , as shown in Fig. 1(c). After k-th refinement iterations, P
k+1 can be deduced by the
ternary approximating scheme as follows:
Pk+13i = Pki − 4∆ki ,
Pk+13i+1 =

2
3
Pki +
1
3
Pki+1

−∆ki ,
Pk+13i+2 =

1
3
Pki +
2
3
Pki+1

−∆ki+1.
(9)
Based on (9), the corresponding interpolating subdivision curves can be deduced in this way: keep all the vertices P0i
fixed and move the two inserted vertices P
1
2
3i+1 and P
1
2
3i+2 to a new position P
′1
3i+1 and P
′1
3i+2 with ∆
′0
3i+1 and ∆
′0
3i+2, as shown
in Fig. 1(d).∆
′0
3i+1 and∆
′0
3i+2 are defined by
∆
′0
3i+d = ω∆0i+d−1 + ν∆0i+2−d, d = 1, 2, (10)
where ω = 32 (1+µ), ν = 32 (1−µ), µ is free parameter. Thus the new interpolating four-point ternary subdivision can be
written in the form
P ′k+13i = Pki ,
P ′k+13i+1 =

2
3
Pki +
1
3
Pki+1

+ ω∆ki + ν∆ki+1,
P ′k+13i+2 =

1
3
Pki +
2
3
Pki+1

+ ν∆ki + ω∆ki+1.
(11)
By adding a parameter α, we can formalize the connection between (9) and (11) as
Pk+13i = Pki − 4α∆ki ,
Pk+13i+1 =

2
3
Pki +
1
3
Pki+1

− α∆ki + (1− α)(ω∆ki + ν∆ki+1),
Pk+13i+2 =

1
3
Pki +
2
3
Pki+1

− α∆ki+1 + (1− α)(ν∆ki + ω∆ki+1).
(12)
(12) is called the combined approximating and interpolating ternary subdivision scheme. It is obvious that both the
approximating subdivision defined by (9) and the interpolating subdivision defined by (11) are the special cases of the
combined subdivision scheme defined by (12). If α = 1, (12) generates the approximating subdivision and if α = 0, (12)
generates the interpolating subdivision scheme. From (8) and (12), we can get the combined ternary approximating and
interpolating subdivision in the formof (1). Fig. 2 illustrates a family of curves produced by our combined ternary subdivision
schemewith different α. In Section 4, wewill prove that our combined approximating and interpolating subdivision scheme
yields C2 limit curves when 1/5 < µ < 1/3 and 0 ≤ α ≤ 1.
J. Pan et al. / Applied Mathematics Letters 25 (2012) 2140–2146 2143
Fig. 2. A family of limit curves generated by our combined subdivision scheme from approximation to interpolation with α = 1.0, 0.8, 0.6, 0.4, 0.2, 0
from inside to outside, respectively.
4. Smoothness analysis
Theorem 1. The scheme defined by (1) converges and has smoothness C2 when 0 ≤ α ≤ 1 and 1/5 < µ < 1/3.
Proof. The generating function corresponding to the proposed combined ternary subdivision defined by (1) has the
following sequence of coefficients:
a = (ai) = (. . . , b3, b0, a0, b2, b1, a1, b1, b2, a0, b0, b3, . . .). (13)
The Laurent polynomial a(z) here can be written as:
a(z) = b3z−5 + b0z−4 + a0z−3 + b2z−2 + b1z−1 + a1z0 + b1z1 + b2z2 + a0z3 + b0z4 + b3z5. (14)
In order to prove the smoothness of this scheme to be C2 according to (5), let
b(m)(z) = 1
3
am(z), m ∈ Z+, (15)
where
am(z) =

3z
1+ z

am−1(z) =

3z
1+ z
m
a(z). (16)
From Laurent polynomial, form = 1, we have
b(1)(z) = 1
3
a1(z) = 13

ξiz i, i = −3,−2, . . . , 5, (17)
where
ξ−3 = ξ5 = −16 (−1+ α)(−1+ µ),
ξ−2 = ξ4 = 19 (α − 3µ+ 3αµ),
ξ−1 = ξ3 = 16 (1+ α + µ− αµ),
ξ0 = ξ2 = 13 (3+ α(−1+ µ)− µ),
ξ1 = 19 (9+ 6µ− 2α(1+ 3µ)).
(18)
When 1/5 < µ < 1/3 and 0 ≤ α ≤ 1, the norm of subdivision 13S1 is13S1
 = max

β
|b(1)γ+3β | : γ = 0, 1, 2

= max

13
27
,
19
45
,
19
45

= 13
27
< 1. (19)
From Laurent polynomial, form = 2, we have
b(2)(z) = 1
3
a2(z) = 118

ξ ′i z
i, i = −1, 0, . . . , 5, (20)
2144 J. Pan et al. / Applied Mathematics Letters 25 (2012) 2140–2146
where
ξ ′−1 = ξ ′5 = 3(1− α)(−1+ µ),
ξ ′0 = ξ ′4 = 3− α − 9(1− α)µ,
ξ ′1 = ξ ′3 = 3+ α + 9(1− α)µ,
ξ ′2 = 12− 6α − 6(1− α)µ.
(21)
When 1/5 < µ < 1/3 and 0 ≤ α ≤ 1, the norm of subdivision 13S2 is13S2
 = max

β
|b(2)γ+3β | : γ = 0, 1, 2

= max

13
15
,
1
3
,
1
3

< 1. (22)
From Laurent polynomial, form = 3, we have
b(3)(z) = 1
3
a2(z) = 13

ξ ′′i z
i, i = 1, 2, 3, 4, 5, (23)
where
ξ ′′1 = ξ ′′5 =
3
2
(1− α)(−1+ µ),
ξ ′′2 = ξ ′′4 = 3− 6µ+ α(−2+ 6µ),
ξ ′′3 = α + 9µ− 9αµ.
(24)
When 1/5 < µ < 1/3 and 0 ≤ α ≤ 1, the norm of subdivision 13S3 is13S3
 = max

β
|b(3)γ+3β | : γ = 0, 1, 2

= max

11
15
,
1
3
,
1
3

< 1. (25)
The calculation of b(4) does not give us a Laurent polynomial, i.e., the Laurent polynomial is disconverge everywhere ex-
cept at the origin. So, for 1/5 < µ < 1/3, we have ∥ 13S1∥∞, ∥ 13S2∥∞, ∥ 13S3∥∞ < 1. It is easy to verify that a(0), a(1), a(2), a(3)
all satisfy

i∈Z a3i+j = 1. Hence, the limit curve produced by our ternary combined subdivision is C2. 
5. Non-uniform combined subdivision with shape control parameters
Let αi and βi be a substitution of α in (12), the non-uniform combined subdivision scheme can be defined by
Pk+13i = a0,i−1Pki−1 + a1,iPki + a0,i+1Pki+1,
Pk+13i+1 = b0,i−1Pki−1 + b1,iPki + b2,i+1Pki+1 + b3,i+2Pki+2,
Pk+13i+2 = b3,i−1Pki−1 + b2,iPki + b1,i+1Pki+1 + b0,i+2Pki+2,
(26)
where
a0,i = 427αi,
a1,i = 1− 827αi,
b0,i = 127 −
1
27
(1− βi)(ω + 1),
b1,i = 1627 +
1
27
(1− βi)(2+ 2ω − ν),
b2,i = 1027 +
1
27
(1− βi)(−1− ω + 2ν),
b3,i = − 127 (1− βi)ν,
with ω = 32 (1+ µ), ν = 32 (1− µ), µ is free parameter.
The parameter αi controls the interpolating property of the subdivision curve and the parameter βi controls the
approximating property of the subdivision curve. The two parameters can intuitively modify the approximating or
interpolating tension of the limit shape. If 0 ≤ αi ≡ βi ≤ 1 and 1/5 < µ < 1/3 for the adjacent vertices, it keeps the
J. Pan et al. / Applied Mathematics Letters 25 (2012) 2140–2146 2145
(a) αi = 0, βi = 0. (b) αi = 0, βi = 0.25. (c) αi = 0, βi = 0.5. (d) αi = 0, βi = 1. (e) αi = 0, βi = 2. (f) αi = 0, βi = 3.
(g) αi = 0, βi = 1. (h) αi = 0.25, βi = 1. (i) αi = 0.5, βi = 1. (j) αi = 1, βi = 1. (k) αi = 2, βi = 1. (l) αi = 3, βi = 1.
Fig. 3. µ = 310 . Top row: limit curves obtained by the combined subdivision scheme with αi ≡ 0 and βi = 0, 0.25, 0.5, 1, 2, 3, respectively. Bottom row:
limit curves obtained by the combined subdivision scheme with βi ≡ 1 and αi = 0, 0.25, 0.5, 1, 2, 3, respectively.
Fig. 4. The effect of local interpolation by our combined subdivision with µ = 310 and different αi , βi .
limit curve near these vertices is C2-continuous. The parameter µ is the free parameter. It is obvious that µ only has the
influence to the interpolation situation, because if the scheme is the approximation mode with βi ≡ 1, it means µ would
have no contribution to the shape of curves. In the interpolation mode, if 1/5 < µ < 1/3, the limit subdivision curve is
convergent; if µ ≤ 1/5, the curvature of curve becomes very large and if µ ≥ 1/3 the curvature becomes discontinuous
[12]. Fig. 3 shows some subdivision curves of fitting a square produced by (26) with different control parameters αi and βi.
Fig. 4 shows that one can easily interpolate the control vertices in a local manner and change tension value for every edge
of the original control polygon.
6. Conclusion
This paper presents a new combined approximating and interpolating subdivision scheme for designing curves. Our
combined subdivision scheme can produce limit curves intervening between approximating subdivision and interpolating
subdivision. Our combined subdivision scheme attains C2 degree of smoothness. Furthermore, a non-uniform combined
subdivision with shape control parameters is introduced, which allows a different approximating or interpolating tension
value for every edge of the original control polygon.
Acknowledgments
This research is supported by NSFC-Guangdong Joint Fund (U0935004, U1135003), the National Key Technology R&D
Program (2011BAH27B01, 2011BHA16B08), the Major Science and Technology Projects of Guangdong (2011A080401007),
2146 J. Pan et al. / Applied Mathematics Letters 25 (2012) 2140–2146
the Industry-academy-research Project of Guangdong (2009A090100028), the NSFC (61103162) and the Fundamental
Research Funds for the Central Universities of China (No. 1109021170001137105).
References
[1] M.F. Hassan, N.A. Dodgson, Ternary and three-point univariate subdivision schemes, in: A. Cohen, J.-L. Merrien, L.L. Schumaker (Eds.), Curve and
Surface Fitting: Saint-Malo 2002, Nashboro Press, Brentwood, 2003, pp. 199–208.
[2] L. Zhang, Z. Luo, W. Liu, An approximating C3 stationary subdivision scheme, Journal of Information and Computational Science 1 (1) (2004) 157–161.
[3] S.S. Siddiqi, N. Ahmad, A C6 approximating subdivision scheme, Applied Mathematics Letters 21 (2008) 722–728.
[4] C.V. Beccari, G. Casciola, L. Romani, An interpolating 4-point C2 ternary non-stationary subdivision scheme with tension control, Computer Aided
Geometric Design 24 (4) (2007) 210–219.
[5] C.V. Beccari, G. Casciola, L. Romani, Shape controlled interpolatory ternary subdivision, Applied Mathematics and Computation 215 (3) (2009)
916–927.
[6] J. Maillost, J. Stam, A unified subdivision scheme for polygonal modeling, Computer Graphics Forum 20 (3) (2001) 471–479.
[7] H. Zhang, G. Wang, Semi-stationary push-back subdivision schemes, Journal of Software 13 (4) (2002) 1–10.
[8] J. Rossignac, Education-driven research in CAD, Computer Aided Design 36 (3) (2004) 1461–1469.
[9] G. Li, W. Ma, A method for constructing interpolatory subdivision schemes from approximating subdivision schemes, Computer Graphics Forum 26
(2) (2007) 185–201.
[10] S. Lin, X. Luo, F. You, Z. Li, Deducing interpolating subdivision schemes from approximating subdivision schemes, ACM Transaction on Graphics 27 (5)
(2008) (article 146).
[11] C.V. Beccari, G. Casciola, L. Romani, A unified framework for interpolating and approximating univariate subdivision, Applied Mathematics and
Computation 216 (4) (2010).
[12] M.F. Hassan, I. Ivrissimitzis, N. Dodgson, M. Sabin, An interpolating 4-point C2 ternary stationary subdivision scheme, Computer Aided Geometric
Design 19 (2002) 1–18.
[13] N. Dyn, Subdivision schemes in computer-aided geometric design, in: W. Light (Ed.), Advances in Numerical Analysis, Vol. 2, Clarendon Press, 1992,
pp. 36–104.


file:///data/remote/core/dit/data/elsevier/pdf/5e6/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDg5Mzk2NTkxMjAwMjY1MA==.pdf

A combined approximating and interpolating subdivision scheme with C2 continuity

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Elsevier - Publisher Connector