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Shape preservation of 4-point interpolating non-stationary subdivision scheme
Journal of Computational and Applied Mathematics 319 (2017) 480–492
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Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
Shape preservation of 4-point interpolating non-stationary subdivision scheme Ghazala Akram a,∗ , Khalida Bibi a , Kashif Rehan b , Shahid S. Siddiqi c a
Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan
b
Department of Mathematics, University of Engineering and Technology, KSK Campus, Lahore, Pakistan
c
Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan
article
info
Article history: Received 30 June 2016 Received in revised form 18 January 2017 Keywords: Shape preservation Subdivision schemes Binary Interpolating Non-stationary
abstract In this paper, the shape preserving properties of the binary four-point interpolating nonstationary scheme (Beccari et al., 2007) are analyzed, which we obtain for the hyperbolic case of the scheme, when ν0 > 1. Sufficient conditions on the original control points are developed that allow to generate positivity, monotonicity and convexity preserving curves after a finite number of subdivision steps. Moreover, the results are generalized to derive conditions for the shape preservation of the limit curves. Also the limit curves with specific shape preserving properties are depicted by significant application of derived conditions on the initial data. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Subdivision schemes are an essential tool for the construction of smooth curves and surfaces in the fields of Computer Aided Geometric Designing (CAGD), Computer graphics and reverse engineering. The edges formed by the original set of control points can be refined to generate a sequence of gradually more refined shapes with the aid of subdivision schemes. The most desirable property in the execution of a subdivision rule is shape preservation to create a curve in a predictable manner for exhibiting substantial variation of geometric designs. A reasonable amount of research on the topic of shape preservation has been publicized in the past years. Subdivision techniques were established for the first time by G. de Rham [1] in 1956. Later in 1974, Chaikin [2] reformulated a binary approximating scheme achieving C 1 -continuity. In 1987, Dyn et al. [3] introduced the first famous binary fourpoint interpolating scheme generating limit curves of C 1 -continuity. In 1994, Cai [4] proposed the convexity preserving algorithm for the scheme [3]. Kuijt and Damme [5,6] constructed shape preserving interpolating subdivision schemes while dealing with non-uniform data. In 1999, Dyn et al. [7] analyzed the convexity preservation of the scheme [3] by developing conditions on the initial data. In 2009, Cai [8] deduced conditions on the parameter of a ternary four-point scheme [9] generating limit curves of C 2 -continuity. In 2013, Amat et al. [10] established a novel approach to identify the convexity preserving conditions on reconstruction operators of univariate interpolating subdivision schemes. Later in 2014, Pitolli [11] analyzed the shape preserving properties of ternary subdivision schemes associating with the bell shaped masks. In 2014, Tan et al. [12] presented a binary five-point approximating scheme and investigated its convergence and convexity preserving properties.
∗
Corresponding author. E-mail addresses: [email protected] (G. Akram), [email protected] (K. Bibi), [email protected] (K. Rehan), [email protected] (S.S. Siddiqi). http://dx.doi.org/10.1016/j.cam.2017.01.026 0377-0427/© 2017 Elsevier B.V. All rights reserved.
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Despite the fact that stationary subdivision schemes have interesting features, reconstruction of special types of curves including polynomial functions, conic sections and spiral curves, could not be accomplished without the non-stationary subdivision schemes, e.g., [13–15]. In 2007, Beccari et al. [13] introduced a binary four-point interpolating non-stationary scheme generating limit curves of C 1 -continuity. The aim of this paper is to fully investigate the specific shape preserving properties of non-stationary scheme [13] reproducing the hyperbolic functions. In this regard, three important shape preserving properties, namely positivity (see Section 3), monotonicity (see Section 4) and convexity (see Section 5) are examined and conditions depending on the original data are derived for some finite number of subdivision steps. Although this might be interesting in many practical applications, where the finite number of subdivision steps are sufficient for generating smooth curves. Furthermore, results are generalized to obtain shape preserving conditions in the limiting case. 2. Non-stationary subdivision scheme The definition of interpolating non-stationary subdivision scheme [13] is given below: Given a set of initial control points {(x0i , fi0 ) ∈ Rd }i∈Z and denoting the set of control points at kth subdivision step by k {(xi , fik )}i∈Z (k ≥ 0, k ∈ Z), then the control points at (k + 1)th subdivision step are defined by the rule
k+1 k f2i = fi , 1 (2νk+1 + 1)2 1 k k k f2ik+ (fik + fi+ (fi− 1) − 1 + fi+2 ), +1 = 8νk+1 (1 + νk+1 ) 8νk+1 (1 + νk+1 )
(1)
where
νk+1 =
t 1 k+ − t e 2 1 + e 2k+1 . 2
(2)
Moreover, the parameter νk+1 satisfies the recurrence relation
νk+1 =
1 + νk 2
·
(3)
The subdivision scheme (1) generates limit curves of C 1 -continuity. Observe that for ν0 defined in Eq. (2), that is ν0 = 1 (et + e−t ), it holds respectively: 2
• if −1 < ν0 < 1, then ν0 = cos(t ) for some t ∈ (0, π ) and the scheme reproduces the trigonometric functions • if ν0 = 1, then the scheme reproduces the cubic polynomials • if ν0 > 1, then ν0 = cosh(t ) for some t ∈ R+ and the scheme reproduces the hyperbolic functions. In the following sections, a method is illustrated that allows to formulate sufficient conditions required for shape preserving limit curves and identified whenever scheme (1) is exact for the hyperbolic functions (ν0 > 1). Note that, νk+1 ∈]1, ∞[ at each subdivision step for this choice of the initial tension parameter ν0 . 3. Positivity preservation In the following manner, the positivity preserving property of the subdivision scheme (1) can be obtained. Taking, pki =
fik+1 fik
and P k = maxi {pki ,
1 pki
}, k ≥ 0, k ∈ Z.
To derive the positivity preserving condition for a finite (but arbitrary) number of subdivision steps, Lemma 3.1 is given below. Lemma 3.1. For any n ∈ Z+ , assume that the initial control points {(x0i , fi0 ) : i ∈ Z} are positive, i.e., fi0 > 0, i ∈ Z, such that P 0 < 4νn+1 (1 + νn+1 ) − 1 = αn ,
(4)
then fi > 0, P < αn , i ∈ Z, n ∈ Z+ , i.e., the control points generated by the subdivision scheme (1) at (n)th subdivision step are still positive. n
n
Proof. As νk+1 ∈]1, ∞[, ∀k ∈ Z+ , so it conveniently gives
αk = 4νk+1 (1 + νk+1 ) − 1 > 1. Since ν0 > 1 and it is easy to get from the recurrence relation given in Eq. (3) that νk+1 < νk , therefore immediately leads to νn+1 ≤ νk+1 for all k ≤ n. Simultaneously, it follows that αn ≤ αk for all k ≤ n. Alternatively, αn is the least finite value obtained after (n)th subdivision step, thus the initial control points should satisfy the condition involving αn . The proof of Lemma 3.1 proceeds by induction on n.
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1. By hypothesis, the assertion holds for n = 0, i.e., fi0 > 0, P 0 < αn , i ∈ Z. 2. Now, suppose by induction hypothesis fin > 0 and P n < αn , i ∈ Z for some n ∈ Z+ , then it will be proved that fin+1 > 0 and P n+1 < αn . Obviously, α1 < pni < αn and α1 < p1n < αn . n
n
i
By the definition of scheme (1), f2in+1 > 0,
(5)
and +1 f2in+ 1 =
= > > = = = =
1 (2νn+1 + 1)2 n n n (fin + fi+ (fi− 1) − 1 + fi+2 ) 8νn+1 (1 + νn+1 ) 8νn+1 (1 + νn+1 ) 1 fi n 2 n n n (2νn+1 + 1) (1 + pi ) − + pi+1 pi 8νn+1 (1 + νn+1 ) pni−1 fi n (2νn+1 + 1)2 + [(2νn+1 + 1)2 − αn ]pni − αn 8νn+1 (1 + νn+1 ) fi n 1 (2νn+1 + 1)2 + [(2νn+1 + 1)2 − αn ] − αn 8νn+1 (1 + νn+1 ) αn fi n (2νn+1 + 1)2 αn + [(2νn+1 + 1)2 − αn ] − αn2 8αn νn+1 (1 + νn+1 ) fi n (2νn+1 + 1)2 + [(2νn+1 + 1)2 − 1]αn − αn2 8αn νn+1 (1 + νn+1 ) fi n 8αn νn+1 (1 + νn+1 )
[8νn+1 (1 + νn+1 )]
fi n
αn > 0.
(6)
> 0. Combining Eqs. (5) and (6), we have fi In order to prove P n+1 < αn , it is shown that pni +1 < αn and n +1
Since, n +1 p2i =
=
< αn .
+1 f2in+ 1
f2in+1 fin
=
1 n+1
pi
8νn+1 (1+νn+1 )
(2νn+1 + 1)2 (1 + pni ) − pn1 + pni+1 pni i−1
fi n
1 8νn+1 (1 + νn+1 )
(2νn+1 + 1)2 (1 + pni ) −
1 pni−1
+ pni+1 pni
,
thus pn2i+1
− αn = < < = = = <
1 8νn+1 (1 + νn+1 )
1 2 n n n (2νn+1 + 1) (1 + pi ) − + pi+1 pi − 8αn νn+1 (1 + νn+1 ) n pi−1
1 1 1 2 2 n (2νn+1 + 1) + (2νn+1 + 1) − pi − − 8αn νn+1 (1 + νn+1 ) 8νn+1 (1 + νn+1 ) αn αn 1 1 1 (2νn+1 + 1)2 + (2νn+1 + 1)2 − αn − − 8αn νn+1 (1 + νn+1 ) 8νn+1 (1 + νn+1 ) αn αn 1 {1 − 4νn+1 (1 + νn+1 )}αn2 + 4αn νn+1 (1 + νn+1 ) − 1 8αn νn+1 (1 + νn+1 ) 2 (αn − 1) − (4νn+1 + 4νn+1 − 1)αn − 1 8αn νn+1 (1 + νn+1 ) (αn − 1)2 (αn + 1) − 8αn νn+1 (1 + νn+1 ) 0,
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which has been obtained by using the fact that αn > 1 and νn+1 ∈]1, ∞[· Hence, n +1 p2i < αn .
(7)
Since, n +1 p2i +1 =
+1 f2in+ 2 +1 f2in+ 1 n fi+ 1
=
fin+1
8νn+1 (1+νn+1 )
(2νn+1 + 1)2 1 +
1 pni
−
1
pni−1 pni
+ pni+1
8νn+1 (1 + νn+1 ) , = (2νn+1 + 1)2 1 + p1n − pn 1 pn + pni+1 i
i−1 i
thus
n+1 p2i +1 − αn =
8νn+1 (1 + νn+1 ) − αn (2νn+1 + 1)2 1 +
(2νn+1 + 1)2 1 +
1 pni
−
1
pni−1 pni
+ pni+1
− pn 1 pn + pni+1 i−1 i 1 2 8νn+1 (1 + νn+1 ) − αn (2νn+1 + 1) + pn − (2νn+1 + 1)2 αpnn + αn pni+1 i i−1 = (2νn+1 + 1)2 1 + p1n − pn 1 pn + pni+1 1 pni
i
=
N D
i−1 i
·
(8)
n+1 8νn+1 (1+νn+1 )f2i+1
> 0. Moreover, N of Eq. (8) satisfies 1 αn 2 N = 8νn+1 (1 + νn+1 ) − αn (2νn+1 + 1)2 + − ( 2 ν + 1 ) + αn pni+1 n +1 n n
Using Eq. (6), as D =
fin+1
pi−1
< 8νn+1 (1 + νn+1 ) − αn (2νn+1 + 1)2 + αn − (2νn+1 + 1)
2 αn pni
pi
+ αn2
< 8νn+1 (1 + νn+1 ) − αn (2νn+1 + 1)2 + αn − (2νn+1 + 1)2 + αn2 = αn2 − 4αn νn+1 (1 + νn+1 ) + (4νn2+1 + 4νn+1 − 1) = (αn − 1)[αn − {4νn+1 (1 + νn+1 ) − 1}] = 0, this follows n+1 p2i +1 < αn .
(9)
Combining Eqs. (7) and (9), gives pni +1 < αn . In the same manner, it can be verified that Since, P
n +1
=
maxi pni +1
{
,
1 n+1 pi
}, therefore P
By induction, fi > 0, P < αn . n
n
1 n+1
pi
n +1
< αn by showing that < αn .
1 n+1 p2i
< αn ,
1 n+1 p2i+1
< αn .
Lemma 3.1 discusses the positivity preservation of the subdivision scheme (1) for the finite number of n subdivision steps, but it does not hold for the limit curve. Hence, Theorem 3.2 is given to develop the positivity preserving condition in the limiting case, as n → ∞. It can be observed that the parameter νn+1 given in Eq. (2) satisfies lim νn+1 = 1.
n→∞
Consequently, limn→∞ αn = 7 in Theorem 3.2 and note that the proof can be followed from Lemma 3.1. Theorem 3.2. Assume that the initial control points {(x0i , fi0 ) : i ∈ Z} are positive, such that P 0 < 7, then the limit curves generated by the subdivision scheme (1) are positive.
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4. Monotonicity preservation k k Define the first order divided difference by Dki = 2k (fi+ 1 − fi ).
Taking, qki =
Dki+1 Dki
and Q k = maxi {qki ,
1 qki
}, k ≥ 0, k ∈ Z.
The following lemma in this same spirit is given to establish the monotonicity preserving condition for the finite number of n subdivision steps. Lemma 4.1. For n ∈ Z+ , assume that the initial control points {(x0i , fi0 ) : i ∈ Z} are strictly monotonic increasing, i.e., D0i > 0, i ∈ Z, such that
Q 0 ≤ 2νn+1 (1 + νn+1 ) − 1 + 2 νn4+1 + 2νn3+1 − νn+1 = βn ,
(10)
then Dni > 0, Q n ≤ βn , i ∈ Z, n ∈ Z+ , i.e., the control points generated by the subdivision scheme (1) at (n)th subdivision step are still strictly monotonic increasing. Proof. First order divided difference for the scheme (1) can be obtained as, k+1 D2i = Dki + k+1 k D2i +1 = Di +
1 4νk+1 (1 + νk+1 ) 1 4νk+1 (1 + νk+1 )
(Dki−1 − Dki+1 ),
(11)
(Dki+1 − Dki−1 ).
(12)
As νk+1 ∈]1, ∞[, ∀k ∈ Z+ , so it automatically gives
βk = 2νk+1 (1 + νk+1 ) − 1 + 2 νk4+1 + 2νk3+1 − νk+1 = 2νk+1 (1 + νk+1 ) − 1 + 2 νk+1 (1 + νk+1 )[νk+1 (1 + νk+1 ) − 1] > 1. Since νk+1 < νk , ∀k ∈ Z+ , therefore follows that {νk }k∈Z+ is strictly decreasing sequence. Correspondingly, βn ≤ βk for all k ≤ n. Which implies that βn is the least finite value obtained after (n)th subdivision step, thus the initial control points should satisfy the condition involving βn . To prove Lemma 4.1, we proceed by mathematical induction on n. 1. By hypothesis, the assertion holds for n = 0, i.e., D0i > 0, Q 0 ≤ βn , i ∈ Z. 2. Now, suppose by induction hypothesis Dni > 0 and Q n ≤ βn , i ∈ Z, for some n ∈ Z+ , then it will be shown that Dni +1 > 0 and Q n+1 ≤ βn . It is easy to obtain β1 ≤ qni ≤ βn and β1 ≤ q1n ≤ βn . n
n
i
Using Eq. (11),
4νn+1 (1 + νn+1 )
≥ Dni 1 + = = = =
1
n +1 D2i = Dni 1 +
4νn+1 (1 + νn+1 ) Dni
4βn νn+1 (1 + νn+1 ) Dni 4βn νn+1 (1 + νn+1 )
Dni
1+
βn
1−
1
− qni
− βn
βn
(13)
4βn νn+1 (1 + νn+1 ) + 1 − βn2
βn νn+1 (1 + νn+1 )
qni−1
1
Dni
1
4νn+1 (1 + νn+1 ) + 4 ν
4 n +1
+ 2ν
3 n +1
− νn+1
νn+1 (1 + νn+1 ) + νn+1 (1 + νn+1 ) νn+1 (1 + νn+1 ) − 1
1
νn+1 (1 + νn+1 )
> 0,
(14)
since βn > 1 and νn+1 ∈]1, ∞[, so Eq. (14) satisfies automatically. From Eq. (12), Dn2i++11
=
Dni
1+
1 4νn+1 (1 + νn+1 )
qni
−
1 qni−1
,
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485
similar reasoning as in Eqs. (13)–(14) leads to n+1 D2i +1 > 0 .
(15)
Combining Eqs. (14) and (15), gives Din+1 > 0. Further to prove Q n+1 ≤ βn , it is shown that qni +1 ≤ βn and
1 n+1
qi
≤ βn .
Since, qn2i+1 =
=
=
Dn2i++11 Dn2i+1 Dni + 4ν (11+ν ) (Dni+1 − Dni−1 ) n+1 n+1 Dni + 4ν (11+ν ) (Dni−1 − Dni+1 ) n+1 n+1 Dni
Dni
4νn+1 (1+νn+1 ) 4νn+1 (1+νn+1 )
4νn+1 (1 + νn+1 ) + qni − 4νn+1 (1 + νn+1 ) +
4νn+1 (1 + νn+1 ) + qni −
= 4νn+1 (1 + νn+1 ) +
1
qni−1
1
qni−1
− qni
1 qni−1
1 qni−1
− qni
,
thus
− βn 4νn+1 (1 + νn+1 ) + qn1 − qni i−1 qn2i+1 − βn = 4νn+1 (1 + νn+1 ) + qn1 − qni i−1 n 4νn+1 (1 + νn+1 )(1 − βn ) + (1 + βn )qi − (1 + βn ) qn1 i−1 = 1 n 4νn+1 (1 + νn+1 ) + qn − qi 4νn+1 (1 + νn+1 ) + qni −
1
qni−1
i−1
=
N D
Using Eq. (14), as D =
·
(16)
n+1 4νn+1 (1+νn+1 )D2i
Dni
> 0. Moreover, N of Eq. (16) satisfies
N = 4νn+1 (1 + νn+1 )(1 − βn ) + (1 + βn )qni − (1 + βn )
≤ 4νn+1 (1 + νn+1 )(1 − βn ) + (1 + βn )βn − (1 + βn ) =
1
βn
1 qni−1 1
βn
[βn3 + (1 − 4νn+1 − 4νn2+1 )βn2 + (−1 + 4νn+1 + 4νn2+1 )βn − 1]
1
(βn − 1) βn2 + (2 − 4νn+1 − 4νn2+1 )βn + 1 1 = (βn − 1) βn − 2νn+1 (1 + νn+1 ) − 1 + 2 νn4+1 + 2νn3+1 − νn+1 βn × βn − 2νn+1 (1 + νn+1 ) − 1 − 2 νn4+1 + 2νn3+1 − νn+1
=
βn
= 0. Hence, qn2i+1 ≤ βn .
(17)
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Since, qn2i++11 =
n +1 D2i +2
Dn2i++11 n
Di+1 4νn+1 (1+νn+1 )
=
4νn+1 (1 + νn+1 ) +
n
Di+1 4νn+1 (1+νn+1 )
4νn+1 (1+νn+1 ) qni
4νn+1 (1 + νn+1 ) +
=
4νn+1 (1+νn+1 ) qni
1 qni
+ 1−
1 qni
+ 1−
− qni+1
1 qni−1 qni
− qni+1 ,
1 qni−1 qni
thus
4ν (1+ν ) − βn n+1 qn n+1 + 1 − qn 1 qn i i−1 i qn2i++11 − βn = 4νn+1 (1+νn+1 ) 1 + 1 − qn qn qni i−1 i β 4νn+1 (1 + νn+1 ) − βn − qni+1 + 1 − 4βn νn+1 (1 + νn+1 ) + qn n q1n i i−1 = 4νn+1 (1+νn+1 ) 1 + 1 − qn qn qn
4νn+1 (1 + νn+1 ) +
1 qni
− qni+1
i−1 i
i
=
N D
·
(18)
n+1 4νn+1 (1+νn+1 )D2i+1
> 0. Moreover, N of Eq. (18) satisfies βn 1 N = 4νn+1 (1 + νn+1 ) − βn − qni+1 + 1 − 4βn νn+1 (1 + νn+1 ) + n n
Using Eq. (15), as D =
Dni+1
qi−1
≤ 4νn+1 (1 + νn+1 ) − βn − ≤ 4νn+1 (1 + νn+1 ) − βn − =
1
βn 1
βn
qi
1 2
+ 1 − 4βn νn+1 (1 + νn+1 ) + βn
qni
1 + 1 − 4βn νn+1 (1 + νn+1 ) + βn2 βn
1 4βn νn+1 (1 + νn+1 ) − βn2 − 1 + 1 − 4βn νn+1 (1 + νn+1 ) + βn2
βn = 0.
Hence, qn2i++11 ≤ βn .
(19)
Combining Eqs. (17) and (19), leads to ≤ βn . Using the same arguments, it can be proved that qni +1
Since, Q
n +1
=
maxi qni +1
{
By induction, it follows,
,
Dni
1 n+1
qi
}, thus Q
n +1
> 0, Q ≤ βn . n
≤ βn .
1 n+1
qi
≤ βn by verifying
1 n+1 q2i
≤ βn ,
1 n+1 q2i+1
≤ βn .
Similarly, Lemma 4.1 is followed to generalize the result in the limit n → ∞ taking into account that limn→∞ βn = √ 3 + 2 2. Theorem 4.2. Assume that the initial control points {(x0i , fi0 ) : i ∈ Z} are strictly monotonic increasing, such that
√
Q 0 ≤ 3 + 2 2, then the limit curves generated by the subdivision scheme (1) are strictly monotonic increasing. Along the same lines Lemma 4.1 and Theorem 4.2 can be verified in the case of monotonically decreasing data. 5. Convexity preservation k k k Define the second order divided difference by dki = 22k−1 (fi− 1 − 2fi + fi+1 ).
Taking, rik =
dki+1 dki
, and Rk = maxi {rik ,
1 rik
}, k ≥ 0, k ∈ Z.
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487
Finally, the convexity preserving condition for the finite number of n subdivision steps is developed in the same fashion. Lemma 5.1. For n ∈ Z+ , assume that the initial control points {(x0i , fi0 ) : i ∈ Z} are strictly convex, i.e., d0i > 0, i ∈ Z, such that R0 ≤ νn+1 (1 + νn+1 ) − 1 +
νn4+1 + 2νn3+1 − νn2+1 − 2νn+1 = γn ,
(20)
then dni > 0, Rn ≤ γn , i ∈ Z, n ∈ Z+ , i.e., the points generated by the subdivision scheme (1) at (n)th subdivision step are still strictly convex. Proof. Second order divided difference for the scheme (1) can be obtained as, dk2i+1 = k+1 d2i +1 =
2−
1
νk+1 (1 + νk+1 ) 1
dki −
1 2νk+1 (1 + νk+1 )
(dki−1 + dki+1 ),
(21)
(dki + dki+1 ).
νk+1 (1 + νk+1 )
(22)
As νk+1 ∈]1, ∞[, ∀k ∈ Z+ , so it conveniently gives
νk4+1 + 2νk3+1 − νk2+1 − 2νk+1 = νk+1 (1 + νk+1 ) − 1 + νk+1 (1 + νk+1 )[νk+1 (1 + νk+1 ) − 2] > 1.
γk = νk+1 (1 + νk+1 ) − 1 +
Since νk+1 < νk , ∀k ∈ Z+ , therefore follows that {νk }k∈Z+ is strictly decreasing sequence. Correspondingly, γn ≤ γk for all k ≤ n. Which implies that γn is the least finite value obtained after (n)th subdivision step, thus the initial control points should satisfy the condition involving γn . Lemma 5.1 is proved by mathematical induction on n. 1. By hypothesis, the assertion holds for n = 0, i.e., d0i > 0, R0 ≤ γn , i ∈ Z. 2. Now, suppose by induction hypothesis dni > 0 and Rn ≤ γn , i ∈ Z, for some n ∈ Z+ , then it will be shown that dni +1 > 0 and Rn+1 ≤ γn . It is easy to obtain γ1 ≤ rin ≤ γn and γ1 ≤ r1n ≤ γn . n
n
i
Using Eq. (21), n+1 = d2i
≥ = =
dni
4νn+1 (1 + νn+1 ) − 2 −
2νn+1 (1 + νn+1 ) dni 2νn+1 (1 + νn+1 ) dni
νn+1 (1 + νn+1 ) dni
νn+1 (1 + νn+1 )
=
dni
1−
1−
rin−1
+ rin
[2νn+1 (1 + νn+1 ) − 1 − γn ]
νn+1 (1 + νn+1 ) dni
1
[4νn+1 (1 + νn+1 ) − 2 − 2γn ]
=
νn+1 (1 + νn+1 ) −
νn+1 (1 + νn+1 ) −
2
ν
4 n +1
+ 2ν
3 n +1
−ν
2 n +1
− 2νn+1
νn+1 (1 + νn+1 ) νn+1 (1 + νn+1 ) − 2
νn+1 (1 + νn+1 )
> 0.
(23)
As νn+1 ∈]1, ∞[, so Eq. (23) satisfies easily. It is obvious from Eq. (22), n+1 d2i +1 > 0 .
(24)
Combining Eqs. (23) and (24), gives dni +1 > 0. To prove Rn+1 ≤ γn , it suffices to show that rin+1 ≤ γn and
1 n+1
ri
≤ γn .
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Since, n+1 r2i =
n+1 d2i +1 n +1 d2i dni
νn+1 (1+νn+1 )
=
dni
(1 + rin )
4νn+1 (1 + νn+1 ) − 2 − 2νn+1 (1+νn+1 )
=
2(1 + rin ) 4νn+1 (1 + νn+1 ) − 2 −
1 rin−1
+ rin
1
rin−1
+ rin
,
thus
n+1 r2i − γn =
2(1 + rin ) − γn 4νn+1 (1 + νn+1 ) − 2 −
4νn+1 (1 + νn+1 ) − 2 −
1
rin−1
1
rin−1
+ rin
+ rin
2(1 + γn ) − 4γn νn+1 (1 + νn+1 ) + (2 + γn )rin +
= 4νn+1 (1 + νn+1 ) − 2 −
=
N D
Using Eq. (23), as D =
1
rin−1
+ rin
γn
rin−1
·
(25)
n+1 2νn+1 (1+νn+1 )d2i
dni
> 0. Moreover, N of Eq. (25) satisfies
N = 2(1 + γn ) − 4γn νn+1 (1 + νn+1 ) + (2 + γn )rin +
γn rin−1
≤ 2(1 + γn ) − 4γn νn+1 (1 + νn+1 ) + (2 + γn )γn + γn2 = 2γn2 + (4 − 4νn+1 − 4νn2+1 )γn + 2 = 2 γn − νn+1 (1 + νn+1 ) − 1 + νn4+1 + 2νn3+1 − νn2+1 − 2νn+1 × γn − νn+1 (1 + νn+1 ) − 1 − νn4+1 + 2νn3+1 − νn2+1 − 2νn+1 = 0. Hence, n+1 r2i ≤ γn .
(26)
Since, n+1 r2i +1 =
n+1 d2i +2 n+1 d2i +1 n
=
di+1 2νn+1 (1+νn+1 )
4νn+1 (1 + νn+1 ) − 2 − dni+1
νn+1 (1+νn+1 )
4νn+1 (1 + νn+1 ) − 2 −
= 2
1 rin
1 rin
1 rin
1 rin
+ rin+1
+1
+ rin+1
+1
,
thus n+1 r2i +1 − γn =
4νn+1 (1 + νn+1 ) − 2 −
1 rin
2
+ rin+1 − 2γn r1n + 1 i +1
1 rin
G. Akram et al. / Journal of Computational and Applied Mathematics 319 (2017) 480–492
=
4νn+1 (1 + νn+1 ) − 2 − 2γn − (1 + 2γn ) r1n − rin+1 i
2
=
489
N D
Using Eq. (24), as D =
1 rin
+1
·
(27) n+1 2νn+1 (1+νn+1 )d2i+1
dni+1
> 0. Moreover, N of Eq. (27) satisfies
N = 4νn+1 (1 + νn+1 ) − 2 − 2γn − (1 + 2γn )
≤ 4νn+1 (1 + νn+1 ) − 2 − 2γn − (1 + 2γn ) 2
1 rin 1
γn
− rin+1 −
1
γn
γn2 + (2 − 2νn+1 − 2νn2+1 )γn + 1 2 =− γn − νn+1 (1 + νn+1 ) − 1 + νn4+1 + 2νn3+1 − νn2+1 − 2νn+1 γn γn − νn+1 (1 + νn+1 ) − 1 − νn4+1 + 2νn3+1 − νn2+1 − 2νn+1
=−
γn
= 0, it follows, n+1 r2i +1 ≤ γn .
(28)
Combining Eqs. (26) and (28), gives rin+1 ≤ γn . In the same manner, it can be verified that leads to
1 n+1
ri
≤ γn .
n +1
Since, R
= maxi {rin+1 ,
By induction, it follows,
1 n+1
ri dni
1 n+1 r2i
≤ γn and
1 n+1 r2i+1
≤ γn , so finally
}, therefore Rn+1 ≤ γn .
> 0, Rn ≤ γn .
Lemma 5.1 derives the convexity preserving condition of the scheme (1) for finitely many subdivision steps and not for the limit curve. Consequently, Lemma 5.1 is followed to satisfy the convexity preserving property in the limiting case taking into account that limn→∞ γn = 1. Theorem 5.2. Assume that the initial control points {(x0i , fi0 ) : i ∈ Z} are strictly convex, such that R0 = 1, then the limit curves generated by the subdivision scheme (1) are strictly convex. 6. Examples To analyze and verify the efficiency of shape preserving conditions that are proposed for the scheme (1), the following numerical examples are performed. The significant feature of the last series of results computed for the shape preservation is that it restricts our search for a particular set of the initial control points. Examples 1–3 reveal that if the distribution of the initial data is not restricted to the derived conditions, the scheme (1) may not own specific shape preserving properties in the generated curves. In Examples 4–6, the initial control polygons are indicated by red broken line segments and limit curves are identified by blue solid lines, such that the initial control points satisfy specific shape preserving conditions. Example 1. The subdivision curves generated by the scheme (1) using two different sets of control points are obtained that demonstrate the effect of sufficient condition derived for positivity. Considering the set of control points {(−2, 0.01), (−1, 0.15), (0, 0.01), (1, 0.01), (2, 0.15), (3, 0.01)}, which do not satisfy the additional condition of positivity, curves are drawn in Fig. 1(a). In the other case, control points are considered as {(−2, 0.01), (−1, 0.07), (0, 0.01), (1, 0.01), (2, 0.07), (3, 0.01)} that fulfill the derived condition of positivity and curves are drawn in Fig. 1(b). In Fig. 1(a) and (b), the results are depicted after five subdivision steps by mauve and green solid lines for ν0 = 1.05 and ν0 = 2, respectively. It can be noticed that although the control points in both of cases are positive, the new points generated using first set of control points are not positive as shown in Fig. 1(a). It is necessary to mention that when an additional condition of positivity is imposed on the initial data, then the subdivision scheme (1) is capable of producing positive curves as shown in Fig. 1(b).
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G. Akram et al. / Journal of Computational and Applied Mathematics 319 (2017) 480–492
Fig. 1. The curves after five subdivision steps generated by the scheme (1) using positive initial data.
Fig. 2. The curves after five subdivision steps generated by the scheme (1) using monotonic increasing initial data.
Example 2. In this example, two different sets of control points are considered so as to reveal the importance of sufficient condition of monotonicity in generation of monotonicity preserving curve. In Fig. 2(a) and (b), curves generated by the scheme (1) are indicated by mauve and green solid lines for ν0 = 1.05 and ν0 = 15, respectively. Firstly, curves are drawn in Fig. 2(a), using the set of control points {(−2, 0), (−1, 1), (0, 21), (1, 22), (2, 42), (3, 43)} that are monotonic increasing but do not satisfy the additional condition of monotonicity. Secondly, curves are drawn in Fig. 2(b), using the set of control points {(−2, 0), (−1, 1), (0, 6), (1, 7), (2, 12), (3, 13)} that are not only monotonic increasing but also satisfy the additional condition of monotonicity. As a consequence, the vertices generated after five steps of refinement are monotonically increasing for the second set of control points. Example 3. Fig. 3(a) displays curves generated by the scheme (1) using the set of control points {(−2, 14.5), (−1, 7), (0, 0.5), (1, 0), (2, 0.5), (3, 7), (4, 14.5)}. Notice that the initial set of control points are strictly convex but do not satisfy the sufficient condition of convexity for ν0 = 1.05 and ν0 = 15, which give rise to inflection points in the curves generated after five subdivision steps, as shown by mauve and green solid lines in Fig. 3(a). Fig. 3(b) displays curves generated by the scheme (1) for the same choices of tension parameter using the set of control points {(−2, 4.5), (−1, 2), (0, 0.5), (1, 0), (2, 0.5), (3, 2), (4, 4.5)}. This set of control points are strictly convex and fulfill the required condition of convexity, consequently, curves constructed in Fig. 3(b) preserve convexity. Example 4. The graphs of positivity preserving limit curves generated by the scheme (1) with positive set of control points {(−3, 0.01), (−2, 0.06), (−1, 0.06), (0, 0.01), (1, 0.01), (2, 0.06), (3, 0.06), (4, 0.01)} are depicted in Fig. 4(a) and (b) for ν0 = 1.05 and ν0 = 2, respectively. Example 5. The graphs of monotonicity preserving limit curves generated by the scheme (1) with strictly monotonic increasing set of control points {(−3, 0), (−2, 1), (−1, 5), (0, 6), (1, 11), (2, 12), (3, 17), (4, 18)} are depicted in Fig. 5(a) and (b) for ν0 = 1.05 and ν0 = 15, respectively.
G. Akram et al. / Journal of Computational and Applied Mathematics 319 (2017) 480–492
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Fig. 3. The curves after five subdivision steps generated by the scheme (1) using strictly convex initial data.
Fig. 4. The positivity preserving limit curves generated by the scheme (1) with control polygon.
Fig. 5. The monotonicity preserving limit curves generated by the scheme (1) with control polygon.
Example 6. The graphs of convexity preserving limit curves generated by the scheme (1) with strictly convex set of control points {(−3, 8), (−2, 4.5), (−1, 2), (0, 0.5), (1, 0), (2, 0.5), (3, 2), (4, 4.5), (5, 8)} are depicted in Fig. 6(a) and (b) for ν0 = 1.05 and ν0 = 15, respectively. 7. Conclusion In this paper, the specific shape preserving properties of the non-stationary interpolating scheme (1) generating C 1 continuous limit curves have been discussed. The sufficient conditions on original data are derived for the preservation of positivity, monotonicity and convexity in the limit curves, when the initial parameter ν0 ranges over the interval ]1, ∞[. As
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G. Akram et al. / Journal of Computational and Applied Mathematics 319 (2017) 480–492
Fig. 6. The convexity preserving limit curves generated by the scheme (1) with control polygon.
the subdivision scheme is non-stationary, so the conditions are also derived for some finite number of subdivision steps and exhibits extensive variations of curves in the context of shape preservation. Experimental results demonstrate that positivity, monotonicity or convexity of the initial data does not always guarantee the positivity, monotonicity or convexity of the resulting limit curves, so sufficient conditions obtained for the shape preservation have its importance. The full significance of the derived conditions should become apparent with the given examples. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
G. de Rham, Sur une courbe plane, J. Math. Appl. 35 (1956) 25–42. G.M. Chaikin, An algorithm for high speed curve generation, Comput. Graph. Image Process. 3 (4) (1974) 346–349. N. Dyn, J.A. Gregory, D. Levin, A four-point interpolatory subdivision scheme for curve design, Comput. Aided Geom. Design 4 (4) (1987) 257–268. Z. Cai, Four point scheme and convexity preserving algorithm, Comput. Aided Des. Comput. Graph. 6 (1) (1994) 33–36. F. Kuijt, Convexity preserving interpolation-stationary nonlinear subdivision and splines (Ph.D. thesis), University of Twente, 1998. F. Kuijt, R.V. Damme, Shape preserving interpolatory subdivision schemes for nonuniform data, J. Approx. Theory 114 (1) (2002) 1–32. N. Dyn, F. Kuijt, D. Levin, R.V. Damme, Convexity preservation of the four-point interpolatory subdivision scheme, Comput. Aided Geom. Design 16 (8) (1999) 789–792. Z. Cai, Convexity preservation of the interpolating four- point c 2 ternary stationary subdivision scheme, Comput. Aided Geom. Design 26 (5) (2009) 560–565. M.F. Hassan, I.P. Ivrissimitzis, N.A. Dodgson, M.A. Sabin, An interpolating 4-point c 2 ternary stationary subdivision scheme, Comput. Aided Geom. Design 19 (1) (2002) 1–18. S. Amat, R. Donat, J.C. Trillo, Proving convexity preserving properties of interpolatory subdivision schemes through reconstruction operators, Appl. Math. Comput. 219 (14) (2013) 7413–7421. F. Pitolli, Ternary shape preserving subdivision subdivision schemes, Math. Comput. Simulation 106 (2014) 185–194. J. Tan, Y. Yao, H. Cao, L. Zhang, Convexity preservation of five-point binary subdivision scheme with a parameter, Appl. Math. Comput. 245 (2014) 279–288. C. Beccari, G. Casciola, L. Romani, A non-stationary uniform tension controlled interpolating 4-point scheme reproducing conics, Comput. Aided Geom. Design 24 (1) (2007) 1–9. M.K. Jena, P. Shunmugaraj, P.C. Das, A non-stationary subdivision scheme for curve interpolation, ANZIAM J. 44 (E) (2003) E216–E235. L. Romani, From approximating subdivision schemes for exponential splines to high-performance interpolating algorithms, J. Comput. Appl. Math. 224 (1) (2009) 383–396.
Contents lists available at ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
Shape preservation of 4-point interpolating non-stationary subdivision scheme Ghazala Akram a,∗ , Khalida Bibi a , Kashif Rehan b , Shahid S. Siddiqi c a
Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan
b
Department of Mathematics, University of Engineering and Technology, KSK Campus, Lahore, Pakistan
c
Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan
article
info
Article history: Received 30 June 2016 Received in revised form 18 January 2017 Keywords: Shape preservation Subdivision schemes Binary Interpolating Non-stationary
abstract In this paper, the shape preserving properties of the binary four-point interpolating nonstationary scheme (Beccari et al., 2007) are analyzed, which we obtain for the hyperbolic case of the scheme, when ν0 > 1. Sufficient conditions on the original control points are developed that allow to generate positivity, monotonicity and convexity preserving curves after a finite number of subdivision steps. Moreover, the results are generalized to derive conditions for the shape preservation of the limit curves. Also the limit curves with specific shape preserving properties are depicted by significant application of derived conditions on the initial data. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Subdivision schemes are an essential tool for the construction of smooth curves and surfaces in the fields of Computer Aided Geometric Designing (CAGD), Computer graphics and reverse engineering. The edges formed by the original set of control points can be refined to generate a sequence of gradually more refined shapes with the aid of subdivision schemes. The most desirable property in the execution of a subdivision rule is shape preservation to create a curve in a predictable manner for exhibiting substantial variation of geometric designs. A reasonable amount of research on the topic of shape preservation has been publicized in the past years. Subdivision techniques were established for the first time by G. de Rham [1] in 1956. Later in 1974, Chaikin [2] reformulated a binary approximating scheme achieving C 1 -continuity. In 1987, Dyn et al. [3] introduced the first famous binary fourpoint interpolating scheme generating limit curves of C 1 -continuity. In 1994, Cai [4] proposed the convexity preserving algorithm for the scheme [3]. Kuijt and Damme [5,6] constructed shape preserving interpolating subdivision schemes while dealing with non-uniform data. In 1999, Dyn et al. [7] analyzed the convexity preservation of the scheme [3] by developing conditions on the initial data. In 2009, Cai [8] deduced conditions on the parameter of a ternary four-point scheme [9] generating limit curves of C 2 -continuity. In 2013, Amat et al. [10] established a novel approach to identify the convexity preserving conditions on reconstruction operators of univariate interpolating subdivision schemes. Later in 2014, Pitolli [11] analyzed the shape preserving properties of ternary subdivision schemes associating with the bell shaped masks. In 2014, Tan et al. [12] presented a binary five-point approximating scheme and investigated its convergence and convexity preserving properties.
∗
Corresponding author. E-mail addresses: [email protected] (G. Akram), [email protected] (K. Bibi), [email protected] (K. Rehan), [email protected] (S.S. Siddiqi). http://dx.doi.org/10.1016/j.cam.2017.01.026 0377-0427/© 2017 Elsevier B.V. All rights reserved.
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Despite the fact that stationary subdivision schemes have interesting features, reconstruction of special types of curves including polynomial functions, conic sections and spiral curves, could not be accomplished without the non-stationary subdivision schemes, e.g., [13–15]. In 2007, Beccari et al. [13] introduced a binary four-point interpolating non-stationary scheme generating limit curves of C 1 -continuity. The aim of this paper is to fully investigate the specific shape preserving properties of non-stationary scheme [13] reproducing the hyperbolic functions. In this regard, three important shape preserving properties, namely positivity (see Section 3), monotonicity (see Section 4) and convexity (see Section 5) are examined and conditions depending on the original data are derived for some finite number of subdivision steps. Although this might be interesting in many practical applications, where the finite number of subdivision steps are sufficient for generating smooth curves. Furthermore, results are generalized to obtain shape preserving conditions in the limiting case. 2. Non-stationary subdivision scheme The definition of interpolating non-stationary subdivision scheme [13] is given below: Given a set of initial control points {(x0i , fi0 ) ∈ Rd }i∈Z and denoting the set of control points at kth subdivision step by k {(xi , fik )}i∈Z (k ≥ 0, k ∈ Z), then the control points at (k + 1)th subdivision step are defined by the rule
k+1 k f2i = fi , 1 (2νk+1 + 1)2 1 k k k f2ik+ (fik + fi+ (fi− 1) − 1 + fi+2 ), +1 = 8νk+1 (1 + νk+1 ) 8νk+1 (1 + νk+1 )
(1)
where
νk+1 =
t 1 k+ − t e 2 1 + e 2k+1 . 2
(2)
Moreover, the parameter νk+1 satisfies the recurrence relation
νk+1 =
1 + νk 2
·
(3)
The subdivision scheme (1) generates limit curves of C 1 -continuity. Observe that for ν0 defined in Eq. (2), that is ν0 = 1 (et + e−t ), it holds respectively: 2
• if −1 < ν0 < 1, then ν0 = cos(t ) for some t ∈ (0, π ) and the scheme reproduces the trigonometric functions • if ν0 = 1, then the scheme reproduces the cubic polynomials • if ν0 > 1, then ν0 = cosh(t ) for some t ∈ R+ and the scheme reproduces the hyperbolic functions. In the following sections, a method is illustrated that allows to formulate sufficient conditions required for shape preserving limit curves and identified whenever scheme (1) is exact for the hyperbolic functions (ν0 > 1). Note that, νk+1 ∈]1, ∞[ at each subdivision step for this choice of the initial tension parameter ν0 . 3. Positivity preservation In the following manner, the positivity preserving property of the subdivision scheme (1) can be obtained. Taking, pki =
fik+1 fik
and P k = maxi {pki ,
1 pki
}, k ≥ 0, k ∈ Z.
To derive the positivity preserving condition for a finite (but arbitrary) number of subdivision steps, Lemma 3.1 is given below. Lemma 3.1. For any n ∈ Z+ , assume that the initial control points {(x0i , fi0 ) : i ∈ Z} are positive, i.e., fi0 > 0, i ∈ Z, such that P 0 < 4νn+1 (1 + νn+1 ) − 1 = αn ,
(4)
then fi > 0, P < αn , i ∈ Z, n ∈ Z+ , i.e., the control points generated by the subdivision scheme (1) at (n)th subdivision step are still positive. n
n
Proof. As νk+1 ∈]1, ∞[, ∀k ∈ Z+ , so it conveniently gives
αk = 4νk+1 (1 + νk+1 ) − 1 > 1. Since ν0 > 1 and it is easy to get from the recurrence relation given in Eq. (3) that νk+1 < νk , therefore immediately leads to νn+1 ≤ νk+1 for all k ≤ n. Simultaneously, it follows that αn ≤ αk for all k ≤ n. Alternatively, αn is the least finite value obtained after (n)th subdivision step, thus the initial control points should satisfy the condition involving αn . The proof of Lemma 3.1 proceeds by induction on n.
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1. By hypothesis, the assertion holds for n = 0, i.e., fi0 > 0, P 0 < αn , i ∈ Z. 2. Now, suppose by induction hypothesis fin > 0 and P n < αn , i ∈ Z for some n ∈ Z+ , then it will be proved that fin+1 > 0 and P n+1 < αn . Obviously, α1 < pni < αn and α1 < p1n < αn . n
n
i
By the definition of scheme (1), f2in+1 > 0,
(5)
and +1 f2in+ 1 =
= > > = = = =
1 (2νn+1 + 1)2 n n n (fin + fi+ (fi− 1) − 1 + fi+2 ) 8νn+1 (1 + νn+1 ) 8νn+1 (1 + νn+1 ) 1 fi n 2 n n n (2νn+1 + 1) (1 + pi ) − + pi+1 pi 8νn+1 (1 + νn+1 ) pni−1 fi n (2νn+1 + 1)2 + [(2νn+1 + 1)2 − αn ]pni − αn 8νn+1 (1 + νn+1 ) fi n 1 (2νn+1 + 1)2 + [(2νn+1 + 1)2 − αn ] − αn 8νn+1 (1 + νn+1 ) αn fi n (2νn+1 + 1)2 αn + [(2νn+1 + 1)2 − αn ] − αn2 8αn νn+1 (1 + νn+1 ) fi n (2νn+1 + 1)2 + [(2νn+1 + 1)2 − 1]αn − αn2 8αn νn+1 (1 + νn+1 ) fi n 8αn νn+1 (1 + νn+1 )
[8νn+1 (1 + νn+1 )]
fi n
αn > 0.
(6)
> 0. Combining Eqs. (5) and (6), we have fi In order to prove P n+1 < αn , it is shown that pni +1 < αn and n +1
Since, n +1 p2i =
=
< αn .
+1 f2in+ 1
f2in+1 fin
=
1 n+1
pi
8νn+1 (1+νn+1 )
(2νn+1 + 1)2 (1 + pni ) − pn1 + pni+1 pni i−1
fi n
1 8νn+1 (1 + νn+1 )
(2νn+1 + 1)2 (1 + pni ) −
1 pni−1
+ pni+1 pni
,
thus pn2i+1
− αn = < < = = = <
1 8νn+1 (1 + νn+1 )
1 2 n n n (2νn+1 + 1) (1 + pi ) − + pi+1 pi − 8αn νn+1 (1 + νn+1 ) n pi−1
1 1 1 2 2 n (2νn+1 + 1) + (2νn+1 + 1) − pi − − 8αn νn+1 (1 + νn+1 ) 8νn+1 (1 + νn+1 ) αn αn 1 1 1 (2νn+1 + 1)2 + (2νn+1 + 1)2 − αn − − 8αn νn+1 (1 + νn+1 ) 8νn+1 (1 + νn+1 ) αn αn 1 {1 − 4νn+1 (1 + νn+1 )}αn2 + 4αn νn+1 (1 + νn+1 ) − 1 8αn νn+1 (1 + νn+1 ) 2 (αn − 1) − (4νn+1 + 4νn+1 − 1)αn − 1 8αn νn+1 (1 + νn+1 ) (αn − 1)2 (αn + 1) − 8αn νn+1 (1 + νn+1 ) 0,
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483
which has been obtained by using the fact that αn > 1 and νn+1 ∈]1, ∞[· Hence, n +1 p2i < αn .
(7)
Since, n +1 p2i +1 =
+1 f2in+ 2 +1 f2in+ 1 n fi+ 1
=
fin+1
8νn+1 (1+νn+1 )
(2νn+1 + 1)2 1 +
1 pni
−
1
pni−1 pni
+ pni+1
8νn+1 (1 + νn+1 ) , = (2νn+1 + 1)2 1 + p1n − pn 1 pn + pni+1 i
i−1 i
thus
n+1 p2i +1 − αn =
8νn+1 (1 + νn+1 ) − αn (2νn+1 + 1)2 1 +
(2νn+1 + 1)2 1 +
1 pni
−
1
pni−1 pni
+ pni+1
− pn 1 pn + pni+1 i−1 i 1 2 8νn+1 (1 + νn+1 ) − αn (2νn+1 + 1) + pn − (2νn+1 + 1)2 αpnn + αn pni+1 i i−1 = (2νn+1 + 1)2 1 + p1n − pn 1 pn + pni+1 1 pni
i
=
N D
i−1 i
·
(8)
n+1 8νn+1 (1+νn+1 )f2i+1
> 0. Moreover, N of Eq. (8) satisfies 1 αn 2 N = 8νn+1 (1 + νn+1 ) − αn (2νn+1 + 1)2 + − ( 2 ν + 1 ) + αn pni+1 n +1 n n
Using Eq. (6), as D =
fin+1
pi−1
< 8νn+1 (1 + νn+1 ) − αn (2νn+1 + 1)2 + αn − (2νn+1 + 1)
2 αn pni
pi
+ αn2
< 8νn+1 (1 + νn+1 ) − αn (2νn+1 + 1)2 + αn − (2νn+1 + 1)2 + αn2 = αn2 − 4αn νn+1 (1 + νn+1 ) + (4νn2+1 + 4νn+1 − 1) = (αn − 1)[αn − {4νn+1 (1 + νn+1 ) − 1}] = 0, this follows n+1 p2i +1 < αn .
(9)
Combining Eqs. (7) and (9), gives pni +1 < αn . In the same manner, it can be verified that Since, P
n +1
=
maxi pni +1
{
,
1 n+1 pi
}, therefore P
By induction, fi > 0, P < αn . n
n
1 n+1
pi
n +1
< αn by showing that < αn .
1 n+1 p2i
< αn ,
1 n+1 p2i+1
< αn .
Lemma 3.1 discusses the positivity preservation of the subdivision scheme (1) for the finite number of n subdivision steps, but it does not hold for the limit curve. Hence, Theorem 3.2 is given to develop the positivity preserving condition in the limiting case, as n → ∞. It can be observed that the parameter νn+1 given in Eq. (2) satisfies lim νn+1 = 1.
n→∞
Consequently, limn→∞ αn = 7 in Theorem 3.2 and note that the proof can be followed from Lemma 3.1. Theorem 3.2. Assume that the initial control points {(x0i , fi0 ) : i ∈ Z} are positive, such that P 0 < 7, then the limit curves generated by the subdivision scheme (1) are positive.
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4. Monotonicity preservation k k Define the first order divided difference by Dki = 2k (fi+ 1 − fi ).
Taking, qki =
Dki+1 Dki
and Q k = maxi {qki ,
1 qki
}, k ≥ 0, k ∈ Z.
The following lemma in this same spirit is given to establish the monotonicity preserving condition for the finite number of n subdivision steps. Lemma 4.1. For n ∈ Z+ , assume that the initial control points {(x0i , fi0 ) : i ∈ Z} are strictly monotonic increasing, i.e., D0i > 0, i ∈ Z, such that
Q 0 ≤ 2νn+1 (1 + νn+1 ) − 1 + 2 νn4+1 + 2νn3+1 − νn+1 = βn ,
(10)
then Dni > 0, Q n ≤ βn , i ∈ Z, n ∈ Z+ , i.e., the control points generated by the subdivision scheme (1) at (n)th subdivision step are still strictly monotonic increasing. Proof. First order divided difference for the scheme (1) can be obtained as, k+1 D2i = Dki + k+1 k D2i +1 = Di +
1 4νk+1 (1 + νk+1 ) 1 4νk+1 (1 + νk+1 )
(Dki−1 − Dki+1 ),
(11)
(Dki+1 − Dki−1 ).
(12)
As νk+1 ∈]1, ∞[, ∀k ∈ Z+ , so it automatically gives
βk = 2νk+1 (1 + νk+1 ) − 1 + 2 νk4+1 + 2νk3+1 − νk+1 = 2νk+1 (1 + νk+1 ) − 1 + 2 νk+1 (1 + νk+1 )[νk+1 (1 + νk+1 ) − 1] > 1. Since νk+1 < νk , ∀k ∈ Z+ , therefore follows that {νk }k∈Z+ is strictly decreasing sequence. Correspondingly, βn ≤ βk for all k ≤ n. Which implies that βn is the least finite value obtained after (n)th subdivision step, thus the initial control points should satisfy the condition involving βn . To prove Lemma 4.1, we proceed by mathematical induction on n. 1. By hypothesis, the assertion holds for n = 0, i.e., D0i > 0, Q 0 ≤ βn , i ∈ Z. 2. Now, suppose by induction hypothesis Dni > 0 and Q n ≤ βn , i ∈ Z, for some n ∈ Z+ , then it will be shown that Dni +1 > 0 and Q n+1 ≤ βn . It is easy to obtain β1 ≤ qni ≤ βn and β1 ≤ q1n ≤ βn . n
n
i
Using Eq. (11),
4νn+1 (1 + νn+1 )
≥ Dni 1 + = = = =
1
n +1 D2i = Dni 1 +
4νn+1 (1 + νn+1 ) Dni
4βn νn+1 (1 + νn+1 ) Dni 4βn νn+1 (1 + νn+1 )
Dni
1+
βn
1−
1
− qni
− βn
βn
(13)
4βn νn+1 (1 + νn+1 ) + 1 − βn2
βn νn+1 (1 + νn+1 )
qni−1
1
Dni
1
4νn+1 (1 + νn+1 ) + 4 ν
4 n +1
+ 2ν
3 n +1
− νn+1
νn+1 (1 + νn+1 ) + νn+1 (1 + νn+1 ) νn+1 (1 + νn+1 ) − 1
1
νn+1 (1 + νn+1 )
> 0,
(14)
since βn > 1 and νn+1 ∈]1, ∞[, so Eq. (14) satisfies automatically. From Eq. (12), Dn2i++11
=
Dni
1+
1 4νn+1 (1 + νn+1 )
qni
−
1 qni−1
,
G. Akram et al. / Journal of Computational and Applied Mathematics 319 (2017) 480–492
485
similar reasoning as in Eqs. (13)–(14) leads to n+1 D2i +1 > 0 .
(15)
Combining Eqs. (14) and (15), gives Din+1 > 0. Further to prove Q n+1 ≤ βn , it is shown that qni +1 ≤ βn and
1 n+1
qi
≤ βn .
Since, qn2i+1 =
=
=
Dn2i++11 Dn2i+1 Dni + 4ν (11+ν ) (Dni+1 − Dni−1 ) n+1 n+1 Dni + 4ν (11+ν ) (Dni−1 − Dni+1 ) n+1 n+1 Dni
Dni
4νn+1 (1+νn+1 ) 4νn+1 (1+νn+1 )
4νn+1 (1 + νn+1 ) + qni − 4νn+1 (1 + νn+1 ) +
4νn+1 (1 + νn+1 ) + qni −
= 4νn+1 (1 + νn+1 ) +
1
qni−1
1
qni−1
− qni
1 qni−1
1 qni−1
− qni
,
thus
− βn 4νn+1 (1 + νn+1 ) + qn1 − qni i−1 qn2i+1 − βn = 4νn+1 (1 + νn+1 ) + qn1 − qni i−1 n 4νn+1 (1 + νn+1 )(1 − βn ) + (1 + βn )qi − (1 + βn ) qn1 i−1 = 1 n 4νn+1 (1 + νn+1 ) + qn − qi 4νn+1 (1 + νn+1 ) + qni −
1
qni−1
i−1
=
N D
Using Eq. (14), as D =
·
(16)
n+1 4νn+1 (1+νn+1 )D2i
Dni
> 0. Moreover, N of Eq. (16) satisfies
N = 4νn+1 (1 + νn+1 )(1 − βn ) + (1 + βn )qni − (1 + βn )
≤ 4νn+1 (1 + νn+1 )(1 − βn ) + (1 + βn )βn − (1 + βn ) =
1
βn
1 qni−1 1
βn
[βn3 + (1 − 4νn+1 − 4νn2+1 )βn2 + (−1 + 4νn+1 + 4νn2+1 )βn − 1]
1
(βn − 1) βn2 + (2 − 4νn+1 − 4νn2+1 )βn + 1 1 = (βn − 1) βn − 2νn+1 (1 + νn+1 ) − 1 + 2 νn4+1 + 2νn3+1 − νn+1 βn × βn − 2νn+1 (1 + νn+1 ) − 1 − 2 νn4+1 + 2νn3+1 − νn+1
=
βn
= 0. Hence, qn2i+1 ≤ βn .
(17)
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G. Akram et al. / Journal of Computational and Applied Mathematics 319 (2017) 480–492
Since, qn2i++11 =
n +1 D2i +2
Dn2i++11 n
Di+1 4νn+1 (1+νn+1 )
=
4νn+1 (1 + νn+1 ) +
n
Di+1 4νn+1 (1+νn+1 )
4νn+1 (1+νn+1 ) qni
4νn+1 (1 + νn+1 ) +
=
4νn+1 (1+νn+1 ) qni
1 qni
+ 1−
1 qni
+ 1−
− qni+1
1 qni−1 qni
− qni+1 ,
1 qni−1 qni
thus
4ν (1+ν ) − βn n+1 qn n+1 + 1 − qn 1 qn i i−1 i qn2i++11 − βn = 4νn+1 (1+νn+1 ) 1 + 1 − qn qn qni i−1 i β 4νn+1 (1 + νn+1 ) − βn − qni+1 + 1 − 4βn νn+1 (1 + νn+1 ) + qn n q1n i i−1 = 4νn+1 (1+νn+1 ) 1 + 1 − qn qn qn
4νn+1 (1 + νn+1 ) +
1 qni
− qni+1
i−1 i
i
=
N D
·
(18)
n+1 4νn+1 (1+νn+1 )D2i+1
> 0. Moreover, N of Eq. (18) satisfies βn 1 N = 4νn+1 (1 + νn+1 ) − βn − qni+1 + 1 − 4βn νn+1 (1 + νn+1 ) + n n
Using Eq. (15), as D =
Dni+1
qi−1
≤ 4νn+1 (1 + νn+1 ) − βn − ≤ 4νn+1 (1 + νn+1 ) − βn − =
1
βn 1
βn
qi
1 2
+ 1 − 4βn νn+1 (1 + νn+1 ) + βn
qni
1 + 1 − 4βn νn+1 (1 + νn+1 ) + βn2 βn
1 4βn νn+1 (1 + νn+1 ) − βn2 − 1 + 1 − 4βn νn+1 (1 + νn+1 ) + βn2
βn = 0.
Hence, qn2i++11 ≤ βn .
(19)
Combining Eqs. (17) and (19), leads to ≤ βn . Using the same arguments, it can be proved that qni +1
Since, Q
n +1
=
maxi qni +1
{
By induction, it follows,
,
Dni
1 n+1
qi
}, thus Q
n +1
> 0, Q ≤ βn . n
≤ βn .
1 n+1
qi
≤ βn by verifying
1 n+1 q2i
≤ βn ,
1 n+1 q2i+1
≤ βn .
Similarly, Lemma 4.1 is followed to generalize the result in the limit n → ∞ taking into account that limn→∞ βn = √ 3 + 2 2. Theorem 4.2. Assume that the initial control points {(x0i , fi0 ) : i ∈ Z} are strictly monotonic increasing, such that
√
Q 0 ≤ 3 + 2 2, then the limit curves generated by the subdivision scheme (1) are strictly monotonic increasing. Along the same lines Lemma 4.1 and Theorem 4.2 can be verified in the case of monotonically decreasing data. 5. Convexity preservation k k k Define the second order divided difference by dki = 22k−1 (fi− 1 − 2fi + fi+1 ).
Taking, rik =
dki+1 dki
, and Rk = maxi {rik ,
1 rik
}, k ≥ 0, k ∈ Z.
G. Akram et al. / Journal of Computational and Applied Mathematics 319 (2017) 480–492
487
Finally, the convexity preserving condition for the finite number of n subdivision steps is developed in the same fashion. Lemma 5.1. For n ∈ Z+ , assume that the initial control points {(x0i , fi0 ) : i ∈ Z} are strictly convex, i.e., d0i > 0, i ∈ Z, such that R0 ≤ νn+1 (1 + νn+1 ) − 1 +
νn4+1 + 2νn3+1 − νn2+1 − 2νn+1 = γn ,
(20)
then dni > 0, Rn ≤ γn , i ∈ Z, n ∈ Z+ , i.e., the points generated by the subdivision scheme (1) at (n)th subdivision step are still strictly convex. Proof. Second order divided difference for the scheme (1) can be obtained as, dk2i+1 = k+1 d2i +1 =
2−
1
νk+1 (1 + νk+1 ) 1
dki −
1 2νk+1 (1 + νk+1 )
(dki−1 + dki+1 ),
(21)
(dki + dki+1 ).
νk+1 (1 + νk+1 )
(22)
As νk+1 ∈]1, ∞[, ∀k ∈ Z+ , so it conveniently gives
νk4+1 + 2νk3+1 − νk2+1 − 2νk+1 = νk+1 (1 + νk+1 ) − 1 + νk+1 (1 + νk+1 )[νk+1 (1 + νk+1 ) − 2] > 1.
γk = νk+1 (1 + νk+1 ) − 1 +
Since νk+1 < νk , ∀k ∈ Z+ , therefore follows that {νk }k∈Z+ is strictly decreasing sequence. Correspondingly, γn ≤ γk for all k ≤ n. Which implies that γn is the least finite value obtained after (n)th subdivision step, thus the initial control points should satisfy the condition involving γn . Lemma 5.1 is proved by mathematical induction on n. 1. By hypothesis, the assertion holds for n = 0, i.e., d0i > 0, R0 ≤ γn , i ∈ Z. 2. Now, suppose by induction hypothesis dni > 0 and Rn ≤ γn , i ∈ Z, for some n ∈ Z+ , then it will be shown that dni +1 > 0 and Rn+1 ≤ γn . It is easy to obtain γ1 ≤ rin ≤ γn and γ1 ≤ r1n ≤ γn . n
n
i
Using Eq. (21), n+1 = d2i
≥ = =
dni
4νn+1 (1 + νn+1 ) − 2 −
2νn+1 (1 + νn+1 ) dni 2νn+1 (1 + νn+1 ) dni
νn+1 (1 + νn+1 ) dni
νn+1 (1 + νn+1 )
=
dni
1−
1−
rin−1
+ rin
[2νn+1 (1 + νn+1 ) − 1 − γn ]
νn+1 (1 + νn+1 ) dni
1
[4νn+1 (1 + νn+1 ) − 2 − 2γn ]
=
νn+1 (1 + νn+1 ) −
νn+1 (1 + νn+1 ) −
2
ν
4 n +1
+ 2ν
3 n +1
−ν
2 n +1
− 2νn+1
νn+1 (1 + νn+1 ) νn+1 (1 + νn+1 ) − 2
νn+1 (1 + νn+1 )
> 0.
(23)
As νn+1 ∈]1, ∞[, so Eq. (23) satisfies easily. It is obvious from Eq. (22), n+1 d2i +1 > 0 .
(24)
Combining Eqs. (23) and (24), gives dni +1 > 0. To prove Rn+1 ≤ γn , it suffices to show that rin+1 ≤ γn and
1 n+1
ri
≤ γn .
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G. Akram et al. / Journal of Computational and Applied Mathematics 319 (2017) 480–492
Since, n+1 r2i =
n+1 d2i +1 n +1 d2i dni
νn+1 (1+νn+1 )
=
dni
(1 + rin )
4νn+1 (1 + νn+1 ) − 2 − 2νn+1 (1+νn+1 )
=
2(1 + rin ) 4νn+1 (1 + νn+1 ) − 2 −
1 rin−1
+ rin
1
rin−1
+ rin
,
thus
n+1 r2i − γn =
2(1 + rin ) − γn 4νn+1 (1 + νn+1 ) − 2 −
4νn+1 (1 + νn+1 ) − 2 −
1
rin−1
1
rin−1
+ rin
+ rin
2(1 + γn ) − 4γn νn+1 (1 + νn+1 ) + (2 + γn )rin +
= 4νn+1 (1 + νn+1 ) − 2 −
=
N D
Using Eq. (23), as D =
1
rin−1
+ rin
γn
rin−1
·
(25)
n+1 2νn+1 (1+νn+1 )d2i
dni
> 0. Moreover, N of Eq. (25) satisfies
N = 2(1 + γn ) − 4γn νn+1 (1 + νn+1 ) + (2 + γn )rin +
γn rin−1
≤ 2(1 + γn ) − 4γn νn+1 (1 + νn+1 ) + (2 + γn )γn + γn2 = 2γn2 + (4 − 4νn+1 − 4νn2+1 )γn + 2 = 2 γn − νn+1 (1 + νn+1 ) − 1 + νn4+1 + 2νn3+1 − νn2+1 − 2νn+1 × γn − νn+1 (1 + νn+1 ) − 1 − νn4+1 + 2νn3+1 − νn2+1 − 2νn+1 = 0. Hence, n+1 r2i ≤ γn .
(26)
Since, n+1 r2i +1 =
n+1 d2i +2 n+1 d2i +1 n
=
di+1 2νn+1 (1+νn+1 )
4νn+1 (1 + νn+1 ) − 2 − dni+1
νn+1 (1+νn+1 )
4νn+1 (1 + νn+1 ) − 2 −
= 2
1 rin
1 rin
1 rin
1 rin
+ rin+1
+1
+ rin+1
+1
,
thus n+1 r2i +1 − γn =
4νn+1 (1 + νn+1 ) − 2 −
1 rin
2
+ rin+1 − 2γn r1n + 1 i +1
1 rin
G. Akram et al. / Journal of Computational and Applied Mathematics 319 (2017) 480–492
=
4νn+1 (1 + νn+1 ) − 2 − 2γn − (1 + 2γn ) r1n − rin+1 i
2
=
489
N D
Using Eq. (24), as D =
1 rin
+1
·
(27) n+1 2νn+1 (1+νn+1 )d2i+1
dni+1
> 0. Moreover, N of Eq. (27) satisfies
N = 4νn+1 (1 + νn+1 ) − 2 − 2γn − (1 + 2γn )
≤ 4νn+1 (1 + νn+1 ) − 2 − 2γn − (1 + 2γn ) 2
1 rin 1
γn
− rin+1 −
1
γn
γn2 + (2 − 2νn+1 − 2νn2+1 )γn + 1 2 =− γn − νn+1 (1 + νn+1 ) − 1 + νn4+1 + 2νn3+1 − νn2+1 − 2νn+1 γn γn − νn+1 (1 + νn+1 ) − 1 − νn4+1 + 2νn3+1 − νn2+1 − 2νn+1
=−
γn
= 0, it follows, n+1 r2i +1 ≤ γn .
(28)
Combining Eqs. (26) and (28), gives rin+1 ≤ γn . In the same manner, it can be verified that leads to
1 n+1
ri
≤ γn .
n +1
Since, R
= maxi {rin+1 ,
By induction, it follows,
1 n+1
ri dni
1 n+1 r2i
≤ γn and
1 n+1 r2i+1
≤ γn , so finally
}, therefore Rn+1 ≤ γn .
> 0, Rn ≤ γn .
Lemma 5.1 derives the convexity preserving condition of the scheme (1) for finitely many subdivision steps and not for the limit curve. Consequently, Lemma 5.1 is followed to satisfy the convexity preserving property in the limiting case taking into account that limn→∞ γn = 1. Theorem 5.2. Assume that the initial control points {(x0i , fi0 ) : i ∈ Z} are strictly convex, such that R0 = 1, then the limit curves generated by the subdivision scheme (1) are strictly convex. 6. Examples To analyze and verify the efficiency of shape preserving conditions that are proposed for the scheme (1), the following numerical examples are performed. The significant feature of the last series of results computed for the shape preservation is that it restricts our search for a particular set of the initial control points. Examples 1–3 reveal that if the distribution of the initial data is not restricted to the derived conditions, the scheme (1) may not own specific shape preserving properties in the generated curves. In Examples 4–6, the initial control polygons are indicated by red broken line segments and limit curves are identified by blue solid lines, such that the initial control points satisfy specific shape preserving conditions. Example 1. The subdivision curves generated by the scheme (1) using two different sets of control points are obtained that demonstrate the effect of sufficient condition derived for positivity. Considering the set of control points {(−2, 0.01), (−1, 0.15), (0, 0.01), (1, 0.01), (2, 0.15), (3, 0.01)}, which do not satisfy the additional condition of positivity, curves are drawn in Fig. 1(a). In the other case, control points are considered as {(−2, 0.01), (−1, 0.07), (0, 0.01), (1, 0.01), (2, 0.07), (3, 0.01)} that fulfill the derived condition of positivity and curves are drawn in Fig. 1(b). In Fig. 1(a) and (b), the results are depicted after five subdivision steps by mauve and green solid lines for ν0 = 1.05 and ν0 = 2, respectively. It can be noticed that although the control points in both of cases are positive, the new points generated using first set of control points are not positive as shown in Fig. 1(a). It is necessary to mention that when an additional condition of positivity is imposed on the initial data, then the subdivision scheme (1) is capable of producing positive curves as shown in Fig. 1(b).
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G. Akram et al. / Journal of Computational and Applied Mathematics 319 (2017) 480–492
Fig. 1. The curves after five subdivision steps generated by the scheme (1) using positive initial data.
Fig. 2. The curves after five subdivision steps generated by the scheme (1) using monotonic increasing initial data.
Example 2. In this example, two different sets of control points are considered so as to reveal the importance of sufficient condition of monotonicity in generation of monotonicity preserving curve. In Fig. 2(a) and (b), curves generated by the scheme (1) are indicated by mauve and green solid lines for ν0 = 1.05 and ν0 = 15, respectively. Firstly, curves are drawn in Fig. 2(a), using the set of control points {(−2, 0), (−1, 1), (0, 21), (1, 22), (2, 42), (3, 43)} that are monotonic increasing but do not satisfy the additional condition of monotonicity. Secondly, curves are drawn in Fig. 2(b), using the set of control points {(−2, 0), (−1, 1), (0, 6), (1, 7), (2, 12), (3, 13)} that are not only monotonic increasing but also satisfy the additional condition of monotonicity. As a consequence, the vertices generated after five steps of refinement are monotonically increasing for the second set of control points. Example 3. Fig. 3(a) displays curves generated by the scheme (1) using the set of control points {(−2, 14.5), (−1, 7), (0, 0.5), (1, 0), (2, 0.5), (3, 7), (4, 14.5)}. Notice that the initial set of control points are strictly convex but do not satisfy the sufficient condition of convexity for ν0 = 1.05 and ν0 = 15, which give rise to inflection points in the curves generated after five subdivision steps, as shown by mauve and green solid lines in Fig. 3(a). Fig. 3(b) displays curves generated by the scheme (1) for the same choices of tension parameter using the set of control points {(−2, 4.5), (−1, 2), (0, 0.5), (1, 0), (2, 0.5), (3, 2), (4, 4.5)}. This set of control points are strictly convex and fulfill the required condition of convexity, consequently, curves constructed in Fig. 3(b) preserve convexity. Example 4. The graphs of positivity preserving limit curves generated by the scheme (1) with positive set of control points {(−3, 0.01), (−2, 0.06), (−1, 0.06), (0, 0.01), (1, 0.01), (2, 0.06), (3, 0.06), (4, 0.01)} are depicted in Fig. 4(a) and (b) for ν0 = 1.05 and ν0 = 2, respectively. Example 5. The graphs of monotonicity preserving limit curves generated by the scheme (1) with strictly monotonic increasing set of control points {(−3, 0), (−2, 1), (−1, 5), (0, 6), (1, 11), (2, 12), (3, 17), (4, 18)} are depicted in Fig. 5(a) and (b) for ν0 = 1.05 and ν0 = 15, respectively.
G. Akram et al. / Journal of Computational and Applied Mathematics 319 (2017) 480–492
491
Fig. 3. The curves after five subdivision steps generated by the scheme (1) using strictly convex initial data.
Fig. 4. The positivity preserving limit curves generated by the scheme (1) with control polygon.
Fig. 5. The monotonicity preserving limit curves generated by the scheme (1) with control polygon.
Example 6. The graphs of convexity preserving limit curves generated by the scheme (1) with strictly convex set of control points {(−3, 8), (−2, 4.5), (−1, 2), (0, 0.5), (1, 0), (2, 0.5), (3, 2), (4, 4.5), (5, 8)} are depicted in Fig. 6(a) and (b) for ν0 = 1.05 and ν0 = 15, respectively. 7. Conclusion In this paper, the specific shape preserving properties of the non-stationary interpolating scheme (1) generating C 1 continuous limit curves have been discussed. The sufficient conditions on original data are derived for the preservation of positivity, monotonicity and convexity in the limit curves, when the initial parameter ν0 ranges over the interval ]1, ∞[. As
492
G. Akram et al. / Journal of Computational and Applied Mathematics 319 (2017) 480–492
Fig. 6. The convexity preserving limit curves generated by the scheme (1) with control polygon.
the subdivision scheme is non-stationary, so the conditions are also derived for some finite number of subdivision steps and exhibits extensive variations of curves in the context of shape preservation. Experimental results demonstrate that positivity, monotonicity or convexity of the initial data does not always guarantee the positivity, monotonicity or convexity of the resulting limit curves, so sufficient conditions obtained for the shape preservation have its importance. The full significance of the derived conditions should become apparent with the given examples. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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