International Conference on Computer Aided Engineering (CAE-2013)Department of Mechanical Engineering, IIT Madras, India

A NEW CLASS OF RATIONAL FRACTAL FUNCTION FOR CURVE FITTING

A. K. B. Chand∗1, S. K. Katiyar1 and G. Saravana Kumar2

1Department of Mathematics,2Department of Engineering DesignIndian Institute of Technology Madras

Chennai, India - 600036

ABSTRACT

Fractal is a modern tool to model non-linear scientific and natural data effectively. Recent studies have developed the fractalinterpolation function (FIF) for modeling experimental and/or geometric data. The iterated function system (IFS) is used togenerate a FIF. The work deals with determination of IFS for approximating prescribed data using FIF. When the scalingfactors are restricted appropriately in an IFS, the resulting FIF is differentiable in nature. Shape preservation is significantaspect for geometric modeling of curves and surfaces from data points. The shape preserving capabilities of polynomialspline FIF are heavily limited. In the present paper, we introduce a new class of rational cubic FIF with shape preservingproperties. The developed rational IFS model uses cubic by cubic form with four shape parameters in each subinterval. Thisscheme offers an additional freedom over the classical rational cubic interpolants due to the presence of scaling factors andshape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants.The sufficiency conditions on the scaling factors and shape parameters are investigated and developed so that the resultingrational cubic FIF is positive when the data set is positive.This approach offers a single specification for a large classofpositive fractal interpolants. The developed rational cubic spline FIF can be used for curve fitting if the data shows thetrendof a positiveC1- smooth function, whereas its derivative is irregular. Thestudy also shows that for appropriate values of thescaling factors and shape parameters, the developed rational FIF converges uniformly to the data points coming from aC3

continuous curve at least as rapidly as the third power of themesh norm. A visual illustration of the shape preserving fractalcurves is provided to support our theoretical results.Keywords: Fractal Interpolation Function, Iterated Function System, Rational Cubic Fractal functions, Shape Preservation,Positivity, Convergence.

1. INTRODUCTIONSmooth curve representation of scientific data throughinterpolation curve/surface has great significance in ComputerAided Design, Computer Graphics, Information Sciences,Geometric Modeling, Data visualization etc. Howeverapplications of interpolation problem in CAGD, Cartography,Image Analysis, etc., demand the interpolant to mimicsome geometric properties hidden in the data. The problemof searching a sufficiently smooth function that preservesthe qualitative shape properties inherent in the data isgenerally referred as shape preserving interpolation. Theshape properties are mathematically expressed in termsof conditions such as positivity, monotony, and convexity.Positivity is the most common and fundamental shape aspect.For instance, positive data arise in resistance offered by anelectric circuit, progress of an irreversible process, monthlyrainfall amounts, level of gas discharge in certain chemicalreactions, volume, density etc. The negative graphicalrepresentations of these physical quantities have no meaning.The literature on classical shape preserving interpolation

∗chand@iitm.ac.in

where interpolants are polynomial, exponential and rationalspline is abundant,(see for instances [6], [8], [10], [12],[13]).Similar to shape preservation, shape modification whereinshape interpolant can be altered for a prescribed data is alsoa requirement in design environment. Rational splines withshape parameters are well suited for shape modification andshape preservation. Rational splines have vast applicationsin Reverse Engineering and Vectorization of object (CAD)[9]. These traditional shape preserving interpolants withCr-continuity is in fact infinitely differentiable except possiblyfor finite number of data sets in the interpolation interval.Therefore these methods are less satisfactory for preservingthe shape present in the data, where variables representingderivatives are to be visualized by using functions havingirregularity in a dense subset of the interpolation interval,for instance, the velocity of a particle undergoing Brownianmovement.It is not ideal to use (piecewise) smooth interpolant when thedata has very irregular structure. Fractal interpolation function(FIF) captures the irregularity of data very effectively incomparison with the classical interpolants/splines. The FIFprovides an effective tool for modeling data sampled from

real world signals, graphs which are usually difficult to modelby using classical functions. Fractal interpolation offersthe possibility of choosing either smooth or non-smoothapproximants, as per problem requirement. Barnsley [1]introduced the definition of Fractal Interpolation Function(FIF) based on the theory of Iterated function system (IFS),that consist of a set of mappings whose fixed point is thegraph of a continuous function that interpolates the data.FIFs are used to approximate naturally occurring functionswhich show some short of self similarity under magnification.By imposing appropriate conditions on the scaling factors,Barnsley et al. [2] observed that if the problem is ofdifferentiable type, then the elements of the IFS may besuitably chosen so that corresponding FIF is smooth. But itis difficult to get all types of boundary conditions for fractalsplines in this iterative construction. Fractal splines withgeneral boundary conditions have been studied in [3], [4] insimpler ways.We exploit the aforementioned flexibility and versatility offractal splines to overcome the stated limitations of existingshape preserving rational splines. To be precise we shallconstruct an IFS with component functions being rationalfunctions of the formsi(x) = pi(x)

qi(x); wherepi(x) and qi(x)

are cubic polynomials.pi(x) are to be determined throughthe Hermite interpolation conditions of the correspondingFIF and qi(x) are preassigned cubic polynomials with fourshape parameters. It is observed that when the scaling factortends to zero and shape parameters tends to∞, the FIFapproaches piecewise linear interpolant. This assures thatFIF preserves shape of data in the limiting configuration.Hence the problem is to find the extent to which scalingfactors can be deviated from zero and shape parameterscan be reduced from theoretical limit value∞ so that thecorresponding interpolant preserves the shape of given data.In this regard we have derived a set of sufficient conditions onIFS parameters so that corresponding FIF renders a positivecurve for given positive data. The convergence of the fractalinterpolant is studied by establishing the uniform error ofrational cubic FIF to the original data defining function. Testexamples are provided to illustrate our positivity preservingFIF scheme, tension effect and derivative analysis. It is feltthat the results are encouraging for the rational FIF treated inthis paper when compared to its classical counterpart.

2. FRACTAL INTERPOLATION FUNCTION

Let {(xi, fi) ∈ [x1, xn] ×K, i = 1, 2, . . . , n} be a given setof interpolation data, whereK is a compact set inR, −∞ <

x1 < x2 < · · · < xn < ∞. Let I = [x1, xn], Ii = [xi, xi+1],J = {1, 2, . . . , n− 1} andLi : I −→ Ii, i ∈ J be contractionhomeomorphisms such that

Li(x1) = xi, Li(xn) = xi+1. (1)

Construct continuous functionsFi : I ×K → D such that

Fi(x1, f1) = fi, Fi(xn, fn) = fi+1,

|Fi(x, y)− Fn(x, y∗)| ≤ |λi||y − y∗|,

}

(2)

wherex ∈ I, y, y∗ ∈ K, and−1 < λi < 1; i ∈ J. Define,wi(x, f) = (Li(x), Fi(x, f)) for all i ∈ J. The definition of aFIF originates from the following proposition:

Proposition 2.1. [1] The IFS {I × K;wi, i ∈ J} has aunique fixed pointG. G is the graph of a continuous functionh : I → K such thath(xi) = fi, i = 1, 2, . . . , n.

The functionh in Proposition 2.1 is called a FIF corre-sponding to the IFS{I × K;wi, i ∈ J}. Let G = {g : I →R | g is continuous, g(x1) = f1 andg(xn) = fn}. Define ametric onG by

ρ(g1, g2) = max{|g1(x)− g2(x)| : x ∈ I} g1, g2 ∈ G.

Then (G, ρ) is a complete metric space. Define the Read-Bajraktarevic operatorT on (G, ρ) as

Tg(x) = Fi(L−1i (x), g(L−1

i (x))), x ∈ Ii, i ∈ J . (3)

Since the functionsL−1i , g, Fi are all continuous, obviously

Tg is continuous onIi. Also Tg is continuous at each ofthe pointsx2, x3, . . . , xn−1, Tg(x1) = f1, andTg(xn) = fndue to Eqns.(1)-(2). ConsequentlyTg ∈ G. Moreover,T is acontraction mapping on the complete metric space(G, ρ), i.e.,ρ(Tf, T g) ≤ |λ|∞ρ(f, g), where|λ|∞ = max{|λi| : i ∈ J}.By Banach fixed point theorem,T possesses a unique fixedpointh (say) onG. This fixed pointh is the FIF correspondingto the{I ×K;wi, i ∈ J}. Thus it follows from Eqn. (3) thatthe FIF satisfies the following functional equation:

h(Li(x)) = Fi(x, h(x)), x ∈ I, i ∈ J. (4)

In practice, the popular FIFs are defined by the IFS:

Li(x) = aix+ bi, Fi(x, f) = λif + si(x), i ∈ J (5)

whereλi is called a vertical scaling factor ofwi and λ =(λ1, λ2, . . . , λn−1) is the scale vector of IFS. The scale vectorλ gives a degree of freedom to the FIF and allows us tomodify its properties. In our construction of the rational cubicFIF, it is assumed thatsi, i ∈ J are rational functionsinvolving four shape parameters. The following propositionguarantee the existence of a differentiable or spline FIF. Theproof of the following proposition follows through the suitablemodification of the arguments in Barnsley et al. [2].

Proposition 2.2. Let {(xi, fi)|i = 1, 2, . . . , n} be the interpo-lation data withx1 < x2 < · · · < xn. Let Li(x) = aix + bisatisfies Eqn. (1) andFi(x, f) = λif + si(x) verifies Eqn.(2) for i ∈ J . Let si(x) = pi(x)

qi(x), pi(x), qi(x) be suitable

chosen polynomials inx of degreeM,N respectively, andqi(x) 6= 0 ∀ x ∈ I. Suppose forr ∈ N, |λi| < ari ,

i ∈ J Let Fmi (x, f) =

λif+s(m)i

(x)

am

i

, f (m)1 =

s(m)1 (x1)

am

1 −λ1,

f(m)n =

s(m)n−1(xn)

am

n−1−λn−1, where s

(m)i (x) represents themth

derivative of si(x) with respect tox. If Fmi (xn, f

(m)n ) =

Fmi+1(x1, f

(m)1 ) for i = 1, 2, 3, . . . , n− 2 andm = 1, 2, . . . , r,

then {(Li(x), Fi(x, f))}n−1i=1 determines a rational FIFΦ ∈

Cr[x1, xn] and Φ(m) is the rational FIF determined by{(Li(x), F

mi (x, f))}n−1

i=1 for m = 1, 2, . . . , r.

3. RATIONAL CUBIC FIF WITH FOUR FAM-ILIES OF SHAPE PARAMETERSIn this section we construct the rational cubic FIFs with fourfamilies of shape parameters, which generalize a class ofclassical rational cubic interpolants described by Sarfraz etal. [11].

Theorem 3.1. Let {(xi, fi); i = 1, 2, . . . , n} be a given dataset, wherex1 < x2 < · · · < xn and di (i = 1, 2, . . . , n)be the derivative values at knots. Consider the IFSI∗ ≡{I × R;wi(x, f) = (Li(x), Fi(x, f)), i ∈ J} whereLi(x) =aix + bi satisfies Eqn. (1) and letFi(x, f) = λif + si(x),si(x) = pi(x)

qi(x), pi(x) are cubic polynomials andqi(x) 6= 0

are preassigned cubic polynomials with four shape parametersand |λi| < ai, i ∈ J . Let F 1

i (x, f) =λif+s′

i(x)

ai

, wheres′i(x)represents the derivative ofsi(x) with respect tox. If

Fi(x1, f1) = fi, Fi(xn, fn) = fi+1,

F 1i (x1, d1) = di, F

1i (xn, dn) = di+1,

(6)

then the fixed point of IFSI∗ is the graph of aC1-rationalcubic FIF.

Proof: Supposeχ = {ξ ∈ C1[x1, xn] : ξ(x1) =f1, ξ(xn) = fn, ξ

′(x1) = d1 andξ′(xn) = dn}. Then(χ, d∗)is a complete metric space, whered∗ is the metric inducedby theC1-norm onC1[x1, xn]. Define the Read-BajraktarevicoperatorT ∗ on χ as

T ∗ξ(x) = λiξ(L−1i (x)) + si(L

−1i (x)), x ∈ Ii, i ∈ J. (7)

Sinceai =xi+1−xi

xn−x1< 1, the condition|λi| < ai < 1 gives

that T ∗ is a contraction operator on(χ, d∗). From Eqn. (4)the fixed point ofψ of T ∗ is a fractal function that satisfiesthe functional equation

ψ(Li(x)) = λiψ(x) + si(x), x ∈ I, i ∈ J. (8)

The four real parameters in the rational cubic functionsi(x)are evaluated using Eqn. (6) in the following way:Substitutingx = x1 andx = xn in Eqn. (8), we get,

fi = λif1 + si(x1), fi+1 = λifn + si(xn), i ∈ J. (9)

Sinceψ ∈ C1[x1, xn]. ψ′ satisfies the functional equation:

ψ′(Li(x)) =λiψ

′(x)

ai+s′i(x)

ai, x ∈ I, i ∈ J. (10)

Since |λi|ai

< 1, ψ′ is a fractal function. Substitutingx = x1andx = xn in Eqn. (10), we have two equations involvingdianddi+1 respectively as

di =λid1

ai+s′i(x1)

ai, di+1 =

λidn

ai+s′i(xn)

ai, i ∈ J. (11)

The four real parameters ofpi(x) are determined from Eqn.(9) and Eqn. (11). By using similar arguments as in [1], it canbe shown that the IFSI∗ has a unique fixed point and it isthe graph of the rational cubic FIFψ ∈ C1[x1, xn].

4. EVALUATION OF C1 RATIONAL CUBICFIFDenote∆i = fi+1−fi

xi+1−xi

, i ∈ J . For i = 1, 2, . . . , n, assume thatpi(x) ≡ Pi(θ) = Ui(1−θ)

3+Vi(1−θ)2θ+Wi(1−θ)θ

2+Ziθ3,

qi(x) ≡ Qi(θ) = αi(1−θ)2+βi(1−θ)

2θ+γi(1−θ)θ2+δiθ

2,

whereθ = x−x1

xn−x1, x ∈ I, αi > 0, βi > 0, γi > 0 andδi > 0;

i ∈ J , are shape parameters, andUi, Vi,Wi, Zi i ∈ J are realparameters to be evaluated. By considering Theorem 3.1 withabovepi(x) and qi(x), we consider the following functionalequation:

ψ(Li(x)) = λiψ(x) +pi(x)

qi(x), x ∈ I, i ∈ J. (12)

To make the rational cubic fractal functionψ a C1-interpolant,one needs to imposeψ(xi) = fi, ψ(xi+1) = fi+1, ψ

′(xi) =di, ψ

′(xi+1) = di+1. Substitutingx = x1 in Eqn. (12),we obtainUi = αi(fi − λif1). Similarly, x = xn givesZi = δi(fi+1−λifn). Now usingψ′(xi) = di in Eqn. (12), wegetVi = (αi+βi)fi+αihidi−λi[αi(xn−x1)d1+(αi+βi)f1].Finally the conditionψ′(xi+1) = di+1 gives Wi = (γi +δi)fi+1 − δihidi+1 + λi[δi(xn − x1)dn − (γi + δi)fn].Substituting the values ofUi, Vi,Wi and Zi in (13), therational cubic FIF is obtained as

ψ(Li(x)) = λiψ(x) +pi(x)

qi(x), x ∈ I, i ∈ J, (13)

wherepi(x) = αi(fi−λif1)(1−θ)3+{(αi+βi)fi+αihidi−

λi[αi(xn−x1)d1+(αi+βi)f1]}(1− θ)2θ+ {(γi+ δi)fi+1−

δihidi+1+λi[δi(xn−x1)dn−(γi+δi)fn]}(1−θ)θ2+δi(fi+1−

λifn)θ3, qi(x) = αi(1−θ)

2+βi(1−θ)2θ+γi(1−θ)θ

2+δiθ2.

In the absence of derivative values at the grids, we computethem from the given data set. In this paper we use thearithmetic mean method (see for instance [7]) for derivativeapproximations for the shape preserving rational cubic FIFs:Arithmetic mean method:The three-point difference approx-imation at internal grids are given as fori = 2, 3, n − 1,di = hi∆i−1+hi−1∆i

hi+hi−1, and at the end points of interval, the

derivatives are approximated asd1 = ∆1 + (∆1−∆2)h1

h1+h2, and

dn = ∆n−1 +(∆n−1−∆n−2)hn−1

hn−1+hn−2.

Example 1: We construct the rational cubic FIFs (see Figs.1(a)-(d)) for the data{(0, 0.5), (2, 1.5), (3, 7), (9, 9), (11, 13)}by using the following rational IFS:

{I × R;wi(x, f) = (Li(x), Fi(x, f)), i ∈ J}, (14)

Li(x) = aix+ bi, ai, bi are evaluated by using Eqn. (1), andFi(x, f) = λif + Pi(θ)

Qi(θ)with the expressionPi(θ) andQi(θ)

as in Eqn. (13). In the construction of the rational cubic FIFs,the choice of the scaling vectors areλ = (0.1, 0.1, 0.1, 0.1)in Figs. 1(a)-(c) and λ = (0.001, 0.001, 0.001, 0.001) in

Fig.1(d), and αi = δi = 1 ∀ i = 1, 2, 3, 4 in all figures.At the knot points, the first order derivatives are approximatedby the arithmetic mean method. By analyzing Figs. 1(a)-(c),we conclude that when shape parameters increases the fractalcurve is tightened. In Fig. 1(d) whenλi is close to zero, thenthe fractal curves tends to a piecewise linear interpolant forlarge values of tension parametersβi andγi. Thus, it is verifiedgraphically if λi → 0+ and βi, γi → ∞ simultaneouslythen the rational cubic FIF modifies to the classical affineinterpolant, and hence it can be used for shape preservinginterpolants.Remark 1:(Tension property) Eqn. (13) can be expressed

0 2 4 6 8 10 120

2

4

6

8

10

12

14

(a) βi = γi = 5 .

0 2 4 6 8 10 120

2

4

6

8

10

12

14

(b) βi = γi = 500 .

0 2 4 6 8 10 120

2

4

6

8

10

12

14

(c) βi = γi = 5000.

0 2 4 6 8 10 120

2

4

6

8

10

12

14

(d) βi = γi = 5000.

Fig. 1. Illustrations of shape control analysis

asψ(Li(x)) = λiψ(x) + fi(1 − θ) + fi+1θ +R

Qi(θ), where

R = −λiαif1(1− θ)3 + ((αi + βi)fi + αihidi − λi[αi(xn −x1)d1 +(αi +βi)f1])(1− θ)2θ+((γi + δi)fi+1 − δihidi+1 +λi[δi(xn − x1)dn − (γi + δi)fn])(1− θ)θ2 − βifi(1− θ)3θ−γifi+1(1 − θ)θ3 − (βifi+1 + γifi)(1 − θ)2θ2 − λiδifnθ

3. Ifλi → 0+ and βi, γi → ∞ simultaneously, then the rationalcubic FIF modifies to the classical affine interpolant.Remark 2:If λi = 0, i ∈ J, the rational Cubic FIF becomesthe classical rational cubic interpolation functions(x) asψ(Li(x)) =

p(υ)q(υ) , wherep(υ) = αifi(1−υ)

3+[(αi+βi)fi+

αihidi](1 − υ)2υ + [(γi + δi)fi+1 − δihidi+1](1 − υ)υ2 +δifi+1υ

3, q(υ) = αi(1−υ)2+βi(1−υ)

2υ+γi(1−υ)υ2+δiυ

2

as described by Sarfraz and Hussain [11], (see the Fig. 2(c)).Remark 3:It is interesting to note that whenαi = δi = 1, βi =γi = 2 andλi = 0 for i ∈ J in Eqn. (12), we obtainψ(x) = (2υ3 − 3υ2 + 1)fi + (υ3 − 2υ2 + υ)dihi + (−2υ3 +3υ2)fi+1 + (υ3 − υ2)di+1hi, υ = x−xi

xi+1−xi

, x ∈ [xi, xi+1].Henceψ is theC1-Hermite interpolant overI.Example 2: By taking the same data as in Example 1, we wantto show the effect of the scaling factorλi and shape parametersin the rational cubic FIF. In Fig. 2(a), we takeλi = 0 andshape parameters asαi = δi = 1 andβi = γi = 2. we obtain

theC1-Hermite interpolant as mentioned in Remark 3. In Fig.2(b), we takeλi = (0.111, 0.0901, 0.541, 0.1810) and shapeparameters asαi = 100, δi = 80 and βi = γi = 2. In Fig.2(c), we takeλi = 0 and the shape parameters as in Fig.2(b), to obtain the classical rational cubic FIF as mentionedin Remark 2. In Fig. 2(d), we takeλi as in Fig. 2(b) and theshape parameters asαi = 100, δi = 1 and βi = γi = 2 toillustrate the effects ofδi. In comparison of Fig. 2(b) withrespect to Fig. 2(d), it is observed that for small values ofδi,we obtain a non-positive rational cubic FIF.

0 2 4 6 8 10 12−2

0

2

4

6

8

10

12

14

(a) αi = δi =1 andβi = γi = 2.

0 2 4 6 8 10 120

5

10

15

20

25

30

35

40

(b) αi = 100, δi =80 andβi = γi = 2.

0 2 4 6 8 10 12−4

−2

0

2

4

6

8

10

12

14

16

(c) αi = 100, δi =80 andβi = γi = 2 .

0 2 4 6 8 10 12−10

0

10

20

30

40

50

60

70

80

(d) αi = 100, δi =1 andβi = γi = 2.

Fig. 2. The rational cubic FIF with arbitrary parameter

5. CONVERGENCE PROPERTIES OF RA-TIONAL CUBIC FIFIn the following theorem, the uniform error bound for arational cubic FIFψ(x) is obtained. It can be proved usingthe definition of the Read-Bajraktarevic operators for whichψ is a fixed point and by applying the Mean Value Theorem.We provide a sketch of the proof.

Theorem 5.1. Let ψ and s be the rational cubic FIF andclassical rational cubic interpolant with respect to the data{(xi, fi), i =1,2,...,n} generated from an original functionf ∈C3[x1, xn]. Let di, i = 1, 2, ..., n be the bounded derivativesat the knots. Suppose|λi| < ai , |λ|∞ = max{|λi| : i ∈ J} ,h = max{|hi| : i ∈ J}, c = max{|ci| : i ∈ J}, and the shapeparametersαi ≥ Λ, βi ≥ Λ, γi ≥ Λ and δi ≥ Λ, i ∈ J, forsome0 < Λ < 1. Then‖ψ − f‖∞ ≤ |λ|∞

1−|λ|∞(‖s‖∞ +K) + 1

2‖f(3)‖∞h

3c

Proof: Suppose that the rational cubic polynomialri(λi, x) involved in the IFS that generate rational cubic FIFψ satisfies|∂ri(τi,x)

∂λi

| ≤ K for |τi| ∈ (0, kai) ,|λi| < k < 1,

x ∈ Ii = [xi, xi+1], i ∈ J . Then, from Lemma 4.1 of [3]‖ψ − s‖∞ ≤ |λi|∞

1−|λi|∞(‖s‖∞ +K).

Also it is known from Theorem 5 of [11], the uniform errorbound of the rational cubic function forf(x) ∈ C3[x1, xn],‖f − s‖∞ ≤ 1

2‖f(3)‖∞ max

1≤i≤n−1{h3i ci} for some suitable

constantci independent ofhi.Using both above results in‖ψ−f‖∞ ≤ ‖ψ−s‖∞+‖s−f‖∞,‖ψ − f‖∞ ≤ |λi|∞

1−|λi|∞(‖s‖∞ +K) + 1

2‖f(3)‖∞h

3c,

we have obtained the desired bound for‖ψ − f‖∞. Since|λ|∞ ≤ h

xn−x1, the above theorem establishes that‖ψ −

f‖∞ = O(h). That is, ψ converges tof uniformly asmesh norm approaches zero. Further if we electλi such that|λi| < a3i =

h3i

(xn−x1)∀ i ∈ J, then ψ converges tof

with O(h3) which is similar to the convergence result of itsclassical nonrecursive counterparts.

6. POSITIVITY PRESERVING FRACTAL IN-TERPOLANTFor an arbitrary selection of the scaling factors and shapeparameters, the rational cubic FIFψ described above may notbe positive, if the data set is positive. This is very similartothe ordinary spline schemes that do not provide the desiredshape features of data. We have to restrict the scaling factorand shape parameters for a positive preserving rational cubicFIF as described in the following theorem:

Theorem 6.1. Let {(xi, fi, di), i = 1, 2, . . . , n} be a givenpositive data. If(i) the scaling factorλi, is selected as

λi ∈ [0,min{ai,fi

f1,fi+1

fn}), i ∈ J, (15)

(ii) the shape parametersαi, βi, γi and δi, are restricted as

αi > 0, δi > 0, βi ≥ max{0,−αi[1 +hid

∗i

m∗i

]},

γi ≥ max{0,−δi[1−hid

∗i+1

n∗i

]}, i ∈ J,

(16)

wherem∗i = fi − λif1, n

∗i = fi+1 − λifn, d∗i = di −

λid1

ai

,

d∗i+1 = di+1 − λidn

ai

, then we obtain a positivity preservingC1-rational cubic FIFψ.

Proof: We haveψ(Li(x)) = λiψ(x)+Pi(θ)Qi(θ)

, θ = x−x1

xn−x1,

x ∈ I. If ψ(x) ≥ 0 it is easy to verify that fori ∈ J , whenλi ≥ 0, i ∈ J , the sufficient condition forψ(Li(x)) > 0 forall x ∈ I is Pi(θ)

Qi(θ)> 0 ∀ θ ∈ [0, 1].

So the initial condition on the scaling factors areλi ≥ 0, i ∈ J .SinceQi(θ) > 0 ∀ θ ∈ [0, 1], αi > 0, βi > 0, γi > 0

and δi > 0; i ∈ J , the positivity of rational functionPi(θ)Qi(θ)

depends on the positivity ofPi(θ). For our convenience, wewrite Pi(θ) asPi(θ) = Ui(1−θ)

3+Vi(1−θ)2θ+Wi(1−θ)θ

2+Ziθ3. Now

Pi(θ) > 0 if Ui > 0, Vi > 0,Wi > 0, Zi > 0 (cf. Section 4).Now Ui > 0, Zi > 0 if Eqn. (15) is true. AlsoVi > 0,Wi > 0when Eqn. (16) is valid. Sinceψ(x) is constructed iterativelyψ(Li(.)) ≥ 0 ∀ ψ(.) ≥ 0 impliesψ(x) ≥ 0; ∀ x ∈ I.

Consequence. Setting λi = 0 in Eqn. (17), we obtainαi > 0, δi > 0, βi ≥ max{0,−αi[1 + hidi

fi]}, γi ≥

max{0,−δi[1 − hidi+1

fi+1]}, that provide a set of sufficient

conditions for the positivity of rational cubic splines. Itis worthwhile to mention that these conditions on shapeparameters are weaker than those obtained by Sarfraz andHussain [11].

7. ILLUSTRATION OF POSITIVE PRESERV-ING RESULTS

Consider the positive data set{(0, 0.5), (2, 1.5), (3, 7),(9, 9), (11, 13)}. If the selection ofλi is not according toEqn. (15), then we may obtain a non-positive rational cubicFIF. For instance the selection ofλ1 is not according to Eqn.(15), and hence a part of the rational cubic FIF in Fig. 3(a)related to the first sub-interval is lying in the fourth quadrant.Using the rational IFS Eqn. (14), we have constructed all ourpositivity preserving rational cubic FIFs, where the derivativevalues are approximated by the arithmetic mean method asd1 = −2.8333, d2 = 3.8333, d3 = 4.7619, d4 = 0, andd5 =2.4167. In order to carry the positive nature of above positivedata by our rational cubic FIF, the IFS parameters are chosenwith respect to Eqns. (15)-(16), respectively. The restriction ofthe scaling factors prescribed by Eqn. (15) are:λ1 ∈ [0, 0.115),λ2 ∈ [0, 0.0909), λ3 ∈ [0, 0.545), λ4 ∈ [0, 0.181). A standardrational cubic positive FIF in Fig. 3(b) is constructed withscaling vector asλ = (0.111, 0.0901, 0.541, 0.1810)and shapeparameters asα = (0.5, 0.6, 0.7, 0.8), δ = (0.2, 0.3, 0.4, 0.5),β = (6, 0.2, 0.3, 0.4) andγ = (17, 0.1, 0.2, 0.3). The scalingfactors are the key parameters of our rational cubic FIF.Theoretically due to the implicit and recursive nature of thefractal interpolant, a change in a particularλi value maybe rippled through the entire configuration. To study this inpractice, we change a particular value ofλi and observeits effect on the shape of Figs. 3(c)-(f) are constructed bychanging the scaling factorλ1, λ2, λ3 andλ4 as 0.001 withrespect to other IFS parameter of Fig. 3(b), respectively. Byanalyzing Fig. 3(c) with respect to Fig. 3(b), we observe majorchanges in[0, 2] and minute changes in other intervals. Theeffects ofλ2 andλ4 (see Figs. 3(d),3(f)) are similar toλ1, andall these parameter can be taken as moderately local in nature.The effect ofλ3 is global in nature (see Fig. 3(e)). Thus theeffect of perturbation ofλi may be global or local in nature.he user can get the desired positive rational cubic FIF byplaying with the scaling factors (within the restricted interval)and shape parameters through some optimization technique.In Fig. 4, we have showed graphically that the derivative ofa rational cubic FIF are irregular but in the classical case,itis smooth except at knot points. The derivatives of rationalcubic FIFs as described in Figs. 3(b)-(e) are plotted in Figs.4(a)-(d). The fractality in the derivative of the rational cubicFIF can be controlled over any subinterval with proper choiceof the scaling factor.

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(a) Non-positive rationalcubic FIF.

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(b) Standard rationalcubic positive FIF .

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(c) Effects ofλ1 in Fig.3(b).

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(d) Effects ofλ2 in Fig.3(b).

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(f) Effects ofλ4 in Fig.3(b).

Fig. 3. Rational cubic FIF and its positivity aspects

8. CONCLUSION

We have developed aC1-rational cubic FIFs that contain fourfamilies of shape parameters. With a zero scaling vector,the developed rational cubic FIF reduces to the existingclassical rational cubic interpolant with four families ofshape parameters. A uniform error bound has been developedbetween the rational cubic FIF and an original function, fromwhich it is deduced that the rational cubic FIF convergesuniformly to the original function. Data-dependent sufficientconstraints have been developed on the scaling factors. Twoof the free parameters are used to preserve the positivity ofthe data, while the other two free parameters are used to refinethe shape of curve. The power of the proposed rational cubicFIF have been demonstrated through suitable examples. Thepresent interpolation scheme may find applications in areassuch as tomography, computer graphics, CAGD, animation,visual space simulation, VLSI and image processing.

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(a) Standard Fractalderivative.

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(b) Effects ofλ1 in Fig.4(a).

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(e) Effects ofλ2 in Fig.4(a).

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(f) Effects ofλ3 in Fig.4(a).

Fig. 4. Derivative of rational cubic FIF.

9. ACKNOWLEDGEMENTThe partial support of the Department of Science and Tech-nology of Govt. of India (SERC DST Project No. SR/S4/MS:694/10) is gratefully acknowledged.

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