6_556_hussain.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 2, 2010, 194-212 ISSN 1307-5543 – www.ejpam.com Scientific Data Visualization with Shape Preserving C1 Rational Cubic Interpolation Malik Zawwar Hussain1∗, Muhammad Sarfraz2, Maria Hussain3 1 Department of Mathematics, University of the Punjab, Lahore, Pakistan 2 Department of Information Sciences, Adailiya Campus, Kuwait University, Kuwait 3 Lahore College for Women University, Lahore, Pakistan Abstract. This paper deals with the shape preserving C1 rational cubic interpolation. The developed rational cubic interpolating function has only one free parameter. The approximation order of rational cubic function is investigated and range of optimal error constant is determined. Moreover, positive, constrained and monotone data preserving schemes are developed. 2000 Mathematics Subject Classifications: 68U05, 65D05, 65D07, 65D18. Key Words and Phrases: Rational cubic, Spline, Peano Kernel theorem, Interpolation, Visualization. 1. Introduction The development of interpolating and approximating techniques for shape control and shape preservation is a germane area of research in Computer Graphics and Data Visualization environment. The data are classified as positive, constrained, monotone and convex according to their shapes. Stability of radioactive substance and chemical reactions, solvability of solute in solvent, population statistics [3], resistance offered by an electric circuit, probability distri- bution [3] are a few examples of entities which are always positive. Monotonicity is applied in the specification of Digital to Analog Converters (DACs), Analog to Digital Converters (ADCs) and sensors. These devices are used in control system applications where non-monotonicity is unacceptable [9]. Erythrocyte sedimentation rate (E.S.R.) in cancer patients [9], uric acid level in patients suffering from gout [9], approximation of couples and quasi couples in statis- tics [2], data generated from stress of a material [9], rate of dissemination of drug in blood [2] are a few examples of entities which are always monotone. ∗Corresponding author. Email addresses: malikzawwar�math.pu.edu.pk (M. Z. Hussain), prof.m.sarfraz�gmail. om (M. Sarfraz), mariahussain_1�yahoo. om (M. Hussain) http://www.ejpam.com 194 c© 2010 EJPAM All rights reserved. M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 195 In the past quite a few authors have worked in the area of shape preserving interpolation. The positivity preserving scheme, by Butt and Brodlie [3], preserved the shape of data by interval subdivision technique. In the interval, where the piecewise cubic interpolant lost the positive shape of data, the authors inserted an additional knot (increase in number of data points) in such a way that original shape of the data was preserved. Fahr and Kallay [5] used a monotone rational B-spline of degree one to preserve the shape of monotone data. Goodman et al [7] developed rational cubic interpolating schemes to preserve the shape of data lying on the either side of straight line. The first scheme scaled weights by some scale factors and the second scheme adopted the method of insertion of a new interpolation point. Goodman [8] surveyed the shape preserving interpolating schemes for planar data. Hussain and Sarfraz used four parameter family of rational cubic function to preserve the shape of positive and monotone data in [10] and [11] respectively. Lamberti and Manni [12] used cubic Hermite in parametric form to preserve the shape of data. The step length was used as shape preserving parameters. The first order derivatives at the knots were estimated imposing second derivative continuity at the knots, which resulted in tridiagonal system of equations. Sarfraz et al [13] addressed the problems of positive and monotone curve data interpolation using rational cubic function with two parameters. Sarfraz et al [14] developed a GC1 cubic function with one free parameter to preserve the positive, monotone and convex data. Schmidt and Hess [15] developed sufficient conditions on derivatives at the knots to assure positivity of interpolating cubic polynomial. This paper presents shape preserving interpolating schemes for positive, constrained and monotone data. A one parameter family of rational cubic function has been developed to pre- serve the shape of the data. In different intervals, the free parameter adopts different value. That is, the interpolant has same structure but different functions in different intervals. The presented schemes are neither dependent on data nor on derivatives. Unlike the methodolo- gies in [3, 7,12], the scheme developed in this paper does not constrain interval length. The developed scheme is equally applicable whether the data is with derivatives or the derivative is estimated by derivative estimation techniques [10-11]. The schemes presented in this paper use one parameter family of rational function to preserve the shape of data thus computation- ally less expansive than [10-11, 13]. The order of continuity achieved is C1 , whereas, in [14] it was GC1. The remainder of the paper is organized as follows. In Section 2, the C1 rational cubic interpolant with one free parameter is developed and Section 2.1 discusses its approximation properties. The visualization problems of positive, constrained, and monotone data interpo- lation are discussed in Sections 3, 4 and 5 respectively. Section 6 concludes the paper. 2. Rational Cubic Function In this Section, a C1 rational cubic function with one parameter and linear denominator has been developed. Let {(x i, fi), i = 0,1,2, . . . , n} be a given set of data points defined over the interval [a, b], where a = x0 < x1 < x2 < · · · < xn = b. The C1 piecewise rational cubic function with one parameter αi is defined over each subinterval Ii = [x i, x i+1], i = M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 196 0,1,2, . . . , n− 1 as: S(x)≡ Si(x) = pi(θ) qi(θ) , (1) where pi(θ) = 3 ∑ i=0 (1− θ)3−iθAi , A0 = (αi + 1) fi, A1 = (2αi + 3) fi + hidi, A2 = (αi + 3) fi+1 − hidi+1, A3 = fi+1, qi(θ) = 1+αi(1− θ), hi = x i+1 − x i, θ = (x − x i) hi , S(x i) = fi , S(x i+1) = fi+1, S(1)(x i) = di, S(1)(x i+1) = di+1. (2) S(1)(x) denotes the derivative with respect to x and di denotes derivative values estimated or given. It is noted that when αi = 0, the piecewise rational cubic function (1) reduces to cubic Hermite spline. In this paper, the parameter αi can assume any positive real value. 2.1. Error Estimation of Interpolation In this Section, the error of interpolation is estimated when the function being interpolated is f (x) ∈ C2[x0, xn]. The interpolation scheme developed in Section 2 is local, which allows investigating the error in an arbitrary subinterval Ii = [x i, x i+1] without loss of generality. Using Peano Kernel Theorem [16] the error of interpolation in each subinterval Ii = [x i, x i+1] is: R[ f ] = f (x)− Si(x) = 1 2 ∫ xi+1 xi f (2)(τ)Rx[(x −τ)+]dτ. (3) It is assumed that the function being interpolated is f (x) ∈ C2[x0, xn]. The absolute error in Ii = [x i, x i+1] is: | f (x)− Si(x)| ≤ 1 2 ‖ f (2)(τ)‖ ∫ xi+1 xi |Rx[(x −τ)+]|dτ, (4) where Rx[(x −τ)+] = ¨ u(τ, x), x i < τ < x , v(τ, x), x < τ < x i+1. « , (5) The integral involved in (4) is expressed as: ∫ xi+1 xi |Rx[(x −τ)+]|dτ= ∫ x xi |u(τ, x)|dτ+ ∫ xi+1 x |v(τ, x)|dτ. (6) M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 197 For the C1 rational cubic function (1), u(τ, x) and v(τ, x) have the values u(τ, x) = (x −τ)− θ2 qi(θ) [(1− θ){(αi + 3)(x i+1−τ)− hi}+ θ(x i+1 −τ)], (7) v(τ, x) = − θ2 qi(θ) [(1− θ){(αi + 3)(x i+1−τ)− hi}+ θ(x i+1 −τ)]. (8) The roots of u(x , x) and v(x , x) in [0,1] are θ = 0 and θ = 1. The root of u(τ, x) = 0 is τ1 = x − hiθ 2(αi+2) (αi+1)+θ (αi+2) . The root of v(τ, x) = 0 is τ2 = x i+1 − hi(1−θ ) 1+(αi+2)(1−θ ) . The above discussion leads to the following manipulation: | f (x)− Si(x)| ≤ 1 2 ‖ f (2)(τ)‖h2 iω(αi,θ), ω(αi,θ) = ∫ x xi |u(τ, x)|dτ+ ∫ xi+1 x |v(τ, x)|dτ = ∫ τ1 xi u(τ, x)dτ− ∫ x τ1 u(τ, x)dτ− ∫ τ2 x v(τ, x)dτ+ ∫ xi+1 τ2 v(τ, x)dτ = θ2{−θ2(αi + 2)2qi(θ) + (αi + 1)2(1+ θ)2((αi + 3)− θ(αi + 2))} qi(θ){1+αi + θ(αi + 2)}2 + 2θ2(1− θ)2 qi(θ) − 2θ4(1− θ)(αi + 2) qi(θ){1+αi + θ(αi + 2)} + θ2(1− θ)2 qi(θ){1+ (αi + 2)(1− θ)} . The above discussion is summarized as follows: Theorem 1. The error of C1 rational cubic function (1), for f (x) ∈ C2[x0, xn], in each subin- terval Ii = [x i, x i+1] is | f (x)− Si(x)| ≤ 1 2 ‖ f (2)(τ)‖h2 i ci , (9) where ci is the maximum value of ω(αi,θ) for 0≤ θ ≤ 1 and αi ≥ 0. Theorem 2. For any given positive parameter αi, the error optimal constant ci in Theorem 1 are bounded with 0≤ ci ≤ 0.2685. 3. Positivity Preserving Interpolation This section provides sufficient conditions on parameters for positive interpolation of curve data. Let {(x i, fi), i = 0,1,2, . . . , n} be the positive data defined over the interval [a, b]. The necessary condition for the positivity of data is fi > 0, i = 0,1,2, . . . , n− 1. (10) M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 198 The piecewise rational cubic function (1) preserves positivity if Si(x)> 0, i = 0,1,2, . . . , n− 1. Now, Si(x)> 0 if pi(θ)> 0 and qi(θ). But, qi(θ) if αi > 0. Using the result developed by Schmidt and Hess in [15], the cubic polynomial pi(θ)> 0 if (p′i(0), p′i(1)) ∈ R1UR2, where R1 = � (a, b) : a > −3 fi hi , b < 3 fi+1 hi � , R2 = {(a, b) : 36 fi fi+1(a 2 + b2+ ab− 3∆i(a+ b) + 3∆2 i ) + 4hi(a 3 fi+1 − b3 fi) +3(a fi+1 − b fi)(2hiab− 3a fi+1 + 3b fi)− h2 i a2 b2 ≥ 0}. For the rational cubic function (1), we have p′i(0) = −αi fi + hidi hi and p′i(1) = −αi fi+1 + hidi+1 hi . (p′ i (0), p′ i (1)) ∈ R1 if −αi fi + hidi hi > −3 fi hi , (11) and −αi fi+1 + hidi+1 hi < 3 fi+1 hi . (12) The inequality (11) leads to the following relation: αi < hidi fi + 3. (13) The inequality (12) imposes the following restriction on the free parameter αi: αi > hidi+1 fi+1 − 3. (14) Further (p′i(0), p′i(1)) ∈ R2 if φ(αi) = 36 fi fi+1[φ 2 1(αi) +φ 2 2(αi) +φ1(αi)φ2(αi)− 3∆i(φ1(αi) +φ2(αi))+ 3∆2 i ] +3[ fi+1φ1(αi)− fiφ2(αi)][2hiφ1(αi)φ2(αi)− 3 fi+1φ1(αi) + 3 fiφ2(αi)] +4hi[ fi+1φ 3 1(αi)− fiφ 3 2(αi)]− h2 iφ 2 1(αi)φ 2 2(αi)> 0, (15) with φ1(αi) = p′i(0) and φ2(αi) = p′i (1). The C1 rational cubic function is positive if either (13)-(14) or (15) are satisfied, but (13)-(14) involves less computation hence more suitable for practical applications. This discussion can be summarized as: M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 199 Theorem 3. The piecewise C1 rational cubic interpolant S(x), defined over the interval [a, b], in (1), is positive if in each subinterval Ii = [x i, x i+1] the following sufficient conditions are satisfied: Con1 < αi < Con2, where Con1 = Max � 0, hidi+1 fi+1 − 3 � and Con2 = hidi fi + 3. The above constraints can be rearranged as: Con1 + ki = αi = Con2 − li , where ki > 0 and li > 0. Thus we have the following algorithm for the manipulation of positive curve design. Algorithm 1 Step 1. Enter the (n+ 1) positive data points {(x i, fi) : i = 0,1,2, . . . , n}. Step 2. Estimate the first order derivatives di, i = 0,1,2, . . . , n at knots. (Note: The step 2 is only applicable if data is not provided with derivatives). Step 3. Calculate the value of free parameter αi using Theorem 3. Step 4. Substitute the values of fi, di, i = 0,1,2, . . . , n and αi , i = 0,1,2, . . . , n− 1 in rational cubic function (1) to obtain positive curve through positive data. 3.1. Demonstration Example 1. A positive data set in Table 1 is taken from [6]. The Figure 1 is produced fromTable 1: A positive data taken from [6℄. x 0.5 0.6 0.7 0.8 0.9 1.0 1.1 f 0.4804 0.5669 0.7262 0.1 7985 0.8658 0.9281 the data set in Table 1 using the cubic Hermite spline interpolation technique which loses the positive shape of data. The positive curve in Figure 2 is produced by using positive curve data interpolation scheme developed in Section 3. Example 2. The positive data set in Table 2 is taken from [6]. The Figure 3 is produced from theTable 2: Another positive data taken from [6℄. x 0.0 0.05 0.10 0.15 0.20 f 0.302 0.278 0.10 0.268 0.298 M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 200 data set in Table 2 using the cubic Hermite spline interpolation technique which looses the positive shape of data. Positive curve is produced in Figure 4 using positive curve data interpolation scheme developed in Section 3. M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 201 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 −1000 0 1000 2000 3000 4000 5000 6000 7000 8000 x−axis y− ax is Figure 1: Cubi Hermite spline. 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0 1000 2000 3000 4000 5000 6000 7000 8000 x−axis y− ax is Figure 2: Positive C1 rational ubi fun tion. M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 202 0 0.05 0.1 0.15 0.2 0.25 −0.5 0 0.5 1 1.5 2 2.5 3 x−axis y− ax is Figure 3: Cubi Hermite spline. 0 0.05 0.1 0.15 0.2 0.25 0 0.5 1 1.5 2 2.5 3 x−axis y− ax is Figure 4: Positive C1 rational ubi fun tion. M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 203 4. Constrained Data Interpolation Let {(x i, fi), i = 0,1,2, . . . , n} be the given set of data points lying above the straight line y = mx + c i.e. fi > mx i + c, ∀ i = 0,1,2, . . . , n. (16) The curve will lie above the straight line if the C1 rational cubic function (1) satisfies the following condition: S(x)> mx + c, ∀ x ∈ [x0, xn]. For each subinterval Ii = [x i, x i+1], the above relation in parametric form is expressed as Si(x)> ai(1− θ) + biθ , (17) where θ = x−xi hi and ai(1−θ)+biθ is the parametric equation of straight line with ai = mx i+c and bi = mx i+1 + c. The rearrangement of (17) leads to the following relation: Ui(θ) = 3 ∑ i=0 (1− θ)3−iθ iBi , (18) where B0 = (αi + 1)( fi − ai), B1 = αi(2 fi − ai − bi) + 3 fi − 2ai − bi + hidi, B2 = (αi + 2)( fi+1 − bi) + fi+1 − hidi+1 − ai , B3 = fi+1 − bi. Now, Ui(θ) > 0 if Bi > 0, i = 0,1,2,3. It is straightforward to know that B0 > 0 and B3 > 0 are true from the necessary condition defined in (16) and αi > 0. B1 > 0 provides the follow- ing constraints on αi: If 2 fi−ai− bi > 0 then αi > − fi+bi−hi di 2 fi−ai−bi . Similarly, if 2 fi−ai− bi < 0 then αi < − fi+bi−hi di 2 fi−ai−bi . One can also note that B2 > 0 if αi > − fi+1+hi di+1+ai fi+1−bi . The above discussion can be summarized as: Theorem 4. The piecewise C1 rational cubic interpolant S(x), defined over the interval [a, b], in (1), preserves the shape of data that lies above the straight line if in each subinterval Ii = [x i, x i+1] the following sufficient conditions are satisfied: Con3 < αi < Con4, Con3 = Max � 0, − fi + bi − hidi 2 fi − ai − bi , − fi+1 + hidi+1 + ai fi+1 − bi � , Con4 = − fi + bi − hidi 2 fi − ai − bi . M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 204 The above constraints can be rearranged as: Con3 +mi = αi = Con4 − ni, where mi > 0 and ni > 0. Thus we have the following algorithm for the manipulation of curve design. Algorithm 2 Step 1. Enter the (n+ 1) positive data points {(x i, fi) : i = 0,1,2, . . . , n}. Step 2. Estimate the first order derivatives di, i = 0,1,2, . . . , n at knots. (Note: The step 2 is only applicable if data is not provided with derivatives). Step 3. Calculate the value of free parameter αiusing Theorem 4. Step 4. Substitute the values of fi , di, i = 0,1,2, . . . , n and αi, i = 0,1,2, . . . , n− 1 in rational cubic function (1) to obtain the curve lying above the straight line. 4.1. Demonstration Example 3. The data set in Table 3 is lying above the straight line y = x 2 + 1 and is taken from [1] with slight modification. The Figure 5 is produced from the data set in Table 3 using cubicTable 3: x 0 2 3 5 6 8 11 12 14 15 f 6.5 6.5 6.5 6.5 6.5 6.5 16 50 60 85 Hermite spline. It is clear from the Figure 5 that some part of the curve is lying below the line y = x 2 + 1. This flaw is recovered nicely in Figure 6 using constrained data interpolation scheme developed in Section 4. It is clear from Figure 6 shape of the data is recovered. Example 4. The data set [10] in Table 4 is lying above the straight line y = x 2 + 1. Figure 7Table 4: The data set lying above the straight line y = x 2 + 1. x 2 3 7 8 10 13 14 f 12 4.5 6.5 10 6.3 12 18 is produced from the data set in Table 4 using cubic Hermite spline. It is clear from Figure 7 that cubic Hermite spline loses the shape of data. Figure 8 is produced from the same data set using constrained curve data interpolation scheme developed in Section 4. It is clear from the Figure 8 that curve is lying above the straight line y = x 2 + 1. M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 205 0 2 4 6 8 10 12 14 16 0 10 20 30 40 50 60 70 80 90 x−axis y− ax is Figure 5: Cubi Hermite spline. 0 2 4 6 8 10 12 14 16 0 10 20 30 40 50 60 70 80 90 x−axis y− ax is Figure 6: Constrained C1rational ubi fun tion. M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 206 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 18 20 x−axis y− ax is Figure 7: Cubi Hermite spline. 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 18 20 x−axis y− ax is Figure 8: Constrained C1 rational ubi fun tion. M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 207 5. Monotonicity Preserving Interpolation Let {(x i, fi), i = 0,1,2, . . . , n} be the monotone data defined over the interval [a, b] such that ∆i = fi+1 − fi hi >, di > 0, i = 0,1,2, . . . , n− 1. (19) The piecewise rational cubic function (1) preserves monotonicity if S(1)(x)> 0, i = 0,1,2, . . . , n− 1, where S (1) i (x) = ∑3 i=0(1− θ) 3−iθ iCi (qi(θ)) 2 , (20) C0 = (αi + 1)di, C1 = (αi + 1){(2αi + 6)∆i − 2di+1 − di}, C2 = (4αi + 6)∆i − di+1 − 2di, C3 = di+1. From (20), S (1) i (x) if Ci > 0, i = 0,1,2,3. We know that C0 > 0 and C3 > 0 are always true from the necessary condition of monotonicity defined in (19) and αi > 0. One can see that C1 > 0 if αi > Max � 0, 2di+1+ di 2∆i � . Similarly, C2 > 0 if αi > Max � 0, di+1 + 2di 4∆i � . The above can be summarized as: Theorem 5. The piecewise C1 rational cubic interpolant S(x), defined over the interval [a, b], in (1), is monotone if in each subinterval Ii = [x i, x i+1] the following sufficient conditions are satisfied: αi > Max � 0, 2di+1+ di 2∆i , di+1 + 2di 4∆i � . The above constraints can be rearranged as: αi = mi +Max � 0, 2di+1+ di 2∆i , di+1 + 2di 4∆i � , mi > 0. Thus we have the following algorithm for the manipulation of curve design. M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 208 Algorithm 3 Step 1. Enter the (n+ 1) points {(x i, fi) : i = 0,1,2, . . . , n}. Step 2. Estimate the first order derivatives di , i = 0,1,2, . . . , n at knots. (Note: The step 2 is only applicable if data is not provided with derivatives). Step 3. Calculate the value of free parameter αi using Theorem 5. Step 4. If ∆i > 0 then di = 0 S(x)≡ Si(x) = fi else S(x)≡ Si(x) = pi(θ ) qi(θ ) end 5.1. Demonstration Table 5: A monotone data set. x 4.0 6.0 7.0 f 3.9 4.2 5.7 Example 5. A monotone data set is taken in Table 5. Non-monotone curve in Figure 9 is produced from the monotone data set taken in Table 5 using cubic Hermite spline. The monotone curve (from the same data set) is produced in Figure 10 using monotonicity preserving scheme developed in Section 5. Table 6: A monotone data set. x 1710 2650 2760 f 500 1360 2940 Example 6. A monotone data set is taken in Table 6. The Figure 11 is produced from the monotone data set taken in Table 6 using cubic Hermite spline which loses the monotone shape of data. Monotone curve in Figure 12 is produced using the monotonicity preserving scheme developed in Section 5. M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 209 4 4.5 5 5.5 6 6.5 7 7.5 3.5 4 4.5 5 5.5 6 x−axis y− ax is Figure 9: Cubi Hermite spline. 4 4.5 5 5.5 6 6.5 7 7.5 3.5 4 4.5 5 5.5 6 x−axis y− ax is Figure 10: Monotone C1 rational ubi fun tion. M. Z. Hussain, M. Sarfraz, M. Hussain / Eur. J. Pure Appl. Math, 3 (2010), 194-212 210 1600 1800 2000 2200 2400 2600 2800 −4000 −3000 −2000 −1000 0 1000 2000 3000 x−axis y− ax is Figure 11: Cubi Hermite spline. 1600 1800 2000 2200 2400 2600 2800 500 1000 1500 2000 2500 3000 x−axis y− ax is Figure 12: Monotone C1rational ubi fun tion. REFERENCES 211 6. Conclusion The work in this paper is concerned to the development of shape preserving interpolating schemes. The developed schemes have the following advantageous features over the existing schemes. These are equally worthy for data with and without derivatives, whereas, authors in [15] had to constrain derivatives at knots to preserve the shape of positive data. The developed schemers are based on automated selection of free parameters thus do not require the modification of data. But, the authors in [3, 7, 12] constrained the interval length to preserve the shape of data. In [13] and [10-11], rational functions with two and four family of free parameters were used to preserve the shape of data hence computationally expansive than shape preserving schemes developed in this paper. The order of continuity attained in [14] was GC1, whereas, in this paper it is C1. References [1] Akima, H., A new method of interpolation and smooth curve fitting based on local procedures, Journal of the Association for Computing Machinery, 17, (1970), 589-602. [2] Beliakov, G., Monotonicity preserving approximation of multivariate scattered data, BIT, 45(4), (2005), 653-677. [3] Butt, S. and Brodlie, K. W., Preserving positivity using piecewise cubic interpolation, Computers and Graphics, 17(1), (1993), 55-64. [4] Duan, Q., Zhang, H., Zhang, Y. and Twizell, E. 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Z. and Sarfraz, M., Positivity-preserving interpolation of positive data by rational cubics, Journal of Computational and Applied Mathematics, 218(2), (2008), 446-458. REFERENCES 212 [11] Hussain, M. Z. and Sarfraz, M., Monotone piecewise rational cubic interpolation, To be appeared in International Journal of Computer Mathematics, 86, (2009). [12] Lamberti, P. and Manni, C., Shape-preserving functional interpolation via parametric cubics, Numerical Algorithms, 28, (2001), 229-254. [13] Sarfraz, M., Butt, S. and Hussain, M. Z., Visualization of shaped data by a rational cubic spline interpolation, Computers and Graphics, 25(5), (2001), 833-845. [14] Sarfraz, M., Hussain, M. Z. and Chaudhry, F. S., Shape preserving cubic spline for data visualization, Computer Graphics and CAD/CAM, 01, (2005), 189-193. [15] Schmidt, J. W. and Hess, W., Positivity of cubic polynomial on intervals and positive spline interpolation, BIT, 28, (1988), 340-352. [16] Schultz, M. H., Spline Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, (1973).