The Scientific World Journal

Volume 2014 (2014), Article ID 602453, 14 pages

http://dx.doi.org/10.1155/2014/602453

Research Article

Monotone Data Visualization Using Rational Trigonometric Spline Interpolation

Farheen Ibraheem,¹ Maria Hussain,² and Malik Zawwar Hussain³

¹National University of Computer and Emerging Sciences, Lahore, Pakistan

²Department of Mathematics, Lahore College for Women University, Lahore 54600, Pakistan

³Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan

Received 3 January 2014; Accepted 5 February 2014; Published 3 April 2014

Academic Editors: A. Bellouquid and A. A. Elsadany

Copyright © 2014 Farheen Ibraheem et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Rational cubic and bicubic trigonometric schemes are developed to conserve monotonicity of curve and surface data, respectively. The rational cubic function has four parameters in each subinterval, while the rational bicubic partially blended function has eight parameters in each rectangular patch. The monotonicity of curve and surface data is retained by developing constraints on some of these parameters in description of rational cubic and bicubic trigonometric functions. The remaining parameters are kept free to modify the shape of curve and surface if required. The developed algorithm is verified mathematically and demonstrated graphically.

1. Introduction

The technique or algorithm employed in creating images, diagrams, or animations for imparting a piece of information is termed as visualization. It has a key role to play in different fields like science, engineering, education, and medicine as it can aid experts in identifying and interpreting different patterns and artifacts in their data and provide a three-dimensional display of data for the solution of a wide range of problems.

The methods used to obtain visual representations from abstract data have been in practice for a long time. However, physical quantities often emanate distinctive features (such as positivity, convexity, and monotonicity) and it becomes imperative that the visual model must contain the shape feature to fathom the physical phenomenon, the scientific experiment, and the idea of the designer. Spline interpolating functions play elemental role in visualizing shaped data. This paper specifically addresses the problem of visualizing monotone curve and surface data.

Monotonicity is an indispensable characteristic of data stemming from many physical and scientific experiments. The relationship between the partial pressure of oxygen and percentage dissociation of hemoglobin, consumption function in economics, concentration of atrazine and nitrate in shallow ground waters, and approximation of couples and quasi couples are few phenomena which exhibit monotone trend.

Efforts have been put in by many researchers and a variety of approaches has been proposed to solve this eminent issue [1–17]. Cripps and Hussain [3] visualized the 2D monotone data by Bernstein-Bézier rational cubic function. The authors in [3] converted the Bernstein-Bézier rational cubic function to cubic Hermite by applying the continuity conditions at the end points of interval. The lower bounds of weights functions were determined to visualize monotone data as monotone curve. Hussain and Sarfraz [8] have conserved monotonicity of curve data by rational cubic function with four shape parameters, two of which were set free and two were shape parameters. Data dependent constraints on shape parameters were developed which assure the monotonicity but one shape parameter is dependent on the other which makes it economically very expensive. Rational cubic function with two shape parameters suggested by Sarfraz [13] sustained monotonicity of curves but lacked the liberty to amend the curve which makes it inappropriate for interactive design. Piecewise rational cubic function was used by M. Z. Hussain and M. Hussain [7] to visualize 2D monotone data by developing constraints on the free parameters in the specification of rational cubic function. The authors also extended rational cubic function to rational bicubic partially blended function. Simple constraints were derived on the free parameters in the description of rational bicubic partially blended patches to visualize the 3D monotone data. Three kinds of monotonicity preservation of systems of bivariate functions on triangle were defined and studied by Floater and Peña [5]. Sarfraz et al. [12] developed constraints in the specification of a bicubic function to visualize the shape of 3D monotone data.

This paper is a noteworthy addition in the field of shape preservation when the data under consideration admits monotone trend. The suggested algorithm offers numerous advantages over the prevailing ones. Orthogonality of sine and cosine function compels much smoother visual results as compared to algebraic spline. Derivative of the trigonometric spline is much lower than that of algebraic spline. Moreover, trigonometric splines play an instrumental role in robotic manipulator path planning.

The remainder of the paper is structured as follows. Section 2 is devoted to reviewing the rational trigonometric cubic function developed in [11]. In Section 3, rational trigonometric cubic function is extended to rational trigonometric bicubic function. Section 4 aims to develop monotonicity preserving constraints for 2D data. Section 5 submits a solution to shape preservation of 3D monotone data. In Section 6, numerical examples have been demonstrated. Section 7 draws the conclusion and significance of this research.

2. Rational Trigonometric Cubic Function

In this section, rational trigonometric cubic function [11] is reviewed.

Let be the given set of data points defined over the interval , where . Piecewise rational trigonometric cubic function is defined over each subinterval as

where , .

The rational trigonometric cubic function (1) is ; that is, it satisfies the following properties:

Here and are derivatives at the end points of the interval . The parameters and are real numbers used to modify the shape of the curve.

3. Rational Trigonometric Bicubic Partially Blended Function

Let be the 3D regular data set defined over the rectangular mesh , let be a partition of , and let be a partition of . Rational trigonometric bicubic function which is an extension of rational trigonometric cubic function (1) is defined over each rectangular patch , where , as

where

, , , and are rational trigonometric bicubic functions defined on the boundary of rectangular patch as

where

where

where

where

4. Monotone Curve Interpolation

Monotonicity is a crucial shape property of data and it emanates from many physical phenomenon, engineering problems, scientific applications, and so forth, for instance, dose response curve in biochemistry and pharmacology, approximation of couples and quasi couples in statistics, empirical option pricing model in finance, consumption function in economics, and so forth. Therefore, it is customary that the resulting interpolating curve must retain the monotone shape of data.

In this section, constraints on shape parameters in the description of rational trigonometric cubic function (1) have been developed to preserve 2D monotone data.

Let be the monotone data defined over the interval ; that is,

The curve will be monotone if the rational trigonometric cubic function (1) satisfies the condition

Now, we have

where

The denominator in (16) is a squared quantity, thus, positive. Hence, monotonicity of rational trigonometric cubic spline depends upon the positivity of numerator which can be attained if the coefficients of the trigonometric basis functions are all positive. This yields the following result:

The above discussion can be summarized as follows.

Theorem 1. The piecewise trigonometric rational cubic function (1) preserves the monotonicity of monotone data if in each subinterval , the parameters and satisfy the following sufficient conditions:

The above constraints can be rearranged as

Algorithm 2.

Step 1. Take a monotone data set .

Step 2. Use the Arithmetic Mean Method [11] to estimate the derivatives ’s at knots ’s (note: Step 2 is only applicable if data is not provided with derivatives).

Step 3. Compute the values of parameters ’s and ’s using Theorem 1.

Step 4. Substitute the values of variables from Steps 1–3 in rational trigonometric cubic function (1) to visualize monotone curve through monotone data.

5. Monotone Surface Interpolation

Let be the monotone data set defined over the rectangular mesh such that

Now, surface patch (4) is monotone if the boundary curves defined in (6)–(12) are monotone.

Now, is monotone if , where

with

Now the positivity of entirely depends on . The denominator in (22) is always positive. Since the parameter lies in first quadrant therefore the trigonometric basis functions will be positive also. This yields the following constraints on the free parameters:

is monotone if

where

with

The denominator in (26) is always positive. Moreover, the trigonometric basis functions are also positive for . It follows that the positivity of entirely depends upon . This yields the following constraints on the free parameters:

is monotone if . We have

where

Since the denomoinator of (29) is always positive and trigonometric basis functions are positive for so the positivity of . It follows that the positivity of entirely depends upon . This yields the following constraints on the free parameters:

is monotone if . We have

where

Finally, is positive if are positive. This yields the following constraints on the free parameters:

The above discussion can be put forward as the following theorem.

Theorem 3. The bicubic partially blended rational trigonometric function defined in (4) visualizes monotone data in view of the monotone surface if in each rectangular grid , free parameters satisfy the following constraints:

The above constraints are rearranged as

Algorithm 4.

Step 1. Take a 3D monotone data set .

Step 2. Use the Arithmetic Mean Method to estimate the derivatives at knots (note: Step 2 is only applicable if data is not provided with derivatives).

Step 3. Compute the values of parameters using Theorem 3.

Step 4. Substitute the values of variables from Steps 1–3 in rational trigonometric cubic function (4) to visualize monotone surface through monotone data.

6. Numerical Example

This section illustrates the monotonicity preserving schemes developed in Sections 4 and 5 with the help of examples. The data in Table 1 is observed by exposing identical samples of hemoglobin to different partial pressures of oxygen which results in varying degree of saturation of hemoglobin with oxygen. The sample obtaining the highest amount is said to be saturated. The amount of oxygen combined with the remaining samples is taken as percentage of this maximum value. At a low partial pressure of oxygen, the percentage saturation of hemoglobin is very low; that is, hemoglobin is combined with only a very little oxygen. At high partial pressure of oxygen, the percentage saturation of hemoglobin is very high; that is, hemoglobin is combined with large amounts of oxygen, that is, a monotone relation, so the resulting curve must exhibit the same behavior. Figure 1 represents the curve created by assigning random values to free parameters in description of rational trigonometric cubic function (1) which does not retain the monotone nature of the data. This impediment is removed by applying monotonicity preserving schemes developed in Section 4 and is shown in Figure 2. It is evident from the figure that this curve preserves the monotone shape of hemoglobin dissociation curve. Similar investigation in Table 2 displays a series of results for percentage saturation of myoglobin and partial pressure of oxygen. Figure 3 is produced by assigning random values to free parameters in description of rational trigonometric cubic function (1) which fails to conserve the monotone trend of data. Algorithm 2 developed in Section 4 is applied to remove this drawback and Figure 4 displays the required result. Numerical results corresponding to Figures 2 and 4 are shown in Tables 3 and 4.

Table 1: The varying ability of hemoglobin to carry oxygen.

Partial pressure of oxygen (kPa)
0
2
8
10
18

Saturation of hemoglobin (%)
0
70
91
91
110

Table 2: The varying ability of myoglobin.

Partial pressure of oxygen (kPa)
0
4
6
8
10

Saturation of myoglobin (%)
0
100
100
100
115

Table 3: Numerical results corresponding to Figure 2.

1
2
3
4
5

50.9065
19.6819
0
0
5.7983

35.01
7.17
0
0.01
—

0.1890
0.0100
0
0.7871
—

Table 4: Numerical results corresponding to Figure 4.

1
2
3
4
5

56.25
0
0
0
15

2.8748
9.3179
0
0.01
—

0.01
0.01
0
0.6466
—

Figure 1: rational trigonometric cubic function with , .

Figure 2: monotone rational trigonometric cubic function with .

Figure 3: rational trigonometric cubic function with .

Figure 4: monotone rational trigonometric cubic function with .

The 3D monotone data set in Tables 5 and 6 are generated from the following functions:

respectively.

Table 5: A 3D monotone data set.

1
2
3
4
5
6

1
0.3202
0.5385
0.7762
1.0198
1.2659
1.5133

2
0.4717
0.6403
0.8500
1.0770
1.3124
1.5524

3
0.6500
0.7810
0.9605
1.1662
1.3865
1.6155

4
0.8382
0.9434
1.0966
1.2806
1.4841
1.7000

5
1.0308
1.1180
1.2500
1.4142
1.6008
1.8028

6
1.2258
1.3000
1.4151
1.5620
1.7328
1.9209

Table 6: A 3D monotone data set.

1
2
3
4
5
6

1
0.6931
1.6094
2.3026
2.8332
3.2581
3.6109

2
1.6094
2.0794
2.5649
2.9957
3.3673
3.6889

3
2.3026
2.5649
2.8904
3.2189
3.5264
3.8067

4
2.8332
2.9957
3.2189
3.4657
3.7136
3.9512

5
3.2581
3.3673
3.5264
3.7136
3.9120
4.1109

6
3.6109
3.6889
3.8067
3.9512
4.1109
4.2767

Figures 5 and 7 are produced by interpolating the monotone data sets in Tables 5 and 6, respectively, by rational trigonometric bicubic function for arbitrary values of free parameter. Monotone surfaces in Figures 6 and 8 are produced by interpolating the same data by the monotonicity preserving scheme developed in Section 5. Tables 7 and 8 enclose numerical results against Figures 6 and 8.

Table 7: Numerical values corresponding to Figure 6.

1
2
3
4
5
6

Numerical values of

1
0.1382
0.0823
0.0555
0.0413
0.0328
0.0271

2
0.1649
0.1213
0.0921
0.0732
0.0603
0.0511

3
0.1832
0.1515
0.1233
0.1018
0.0858
0.0738

4
0.1904
0.1685
0.1448
0.1240
0.1071
0.0936

5
0.1938
0.1783
0.1593
0.1407
0.1243
0.1105

6
0.1962
0.1856
0.1709
0.1550
0.1396
0.1259

Numerical values of

1
0.2087
0.2280
0.2406
0.2448
0.2467
0.2480

2
0.1481
0.1892
0.2184
0.2312
0.2377
0.2423

3
0.1068
0.1552
0.1926
0.2130
0.2247
0.2333

4
0.0813
0.1292
0.1686
0.1937
0.2097
0.2221

5
0.0649
0.1096
0.1481
0.1754
0.1943
0.2097

6
0.0538
0.0947
0.1310
0.1588
0.1794
0.1969

Numerical values of

1
10.9406
9.7062
9.0177
8.6526
8.4470
—

2
11.0996
10.3406
10.0079
9.8513
9.7684
—

3
11.6858
11.1996
10.8694
10.6747
10.5583
—

4
11.8607
11.5787
11.3235
11.1397
11.0149
—

5
11.9272
11.7583
11.5754
11.4218
11.3048
—

6
—
—
—
—
—
—

Numerical values of

1
13.0594
14.2938
14.9823
15.3474
15.5530
—

2
12.3315
12.9236
13.3931
13.7011
13.8977
—

3
12.1426
12.4531
12.7625
13.0043
13.1785
—

4
12.0737
12.2518
12.4569
12.6399
12.7863
—

5
12.0728
12.2417
12.4246
12.5782
12.6952
—

6
—
—
—
—
—
—

Numerical values of

1
11.4688
11.5120
11.8546
11.9391
11.9689
—

2
10.5384
10.8247
11.5416
11.7866
11.8858
—

3
9.7828
10.3810
11.2336
11.6016
11.7732
—

4
9.2668
10.1222
10.9942
11.4274
11.6537
—

5
8.9258
9.9673
10.8217
11.2811
11.5418
—

6
—
—
—
—
—
—

Numerical values of

1
12.5312
12.1490
12.0616
12.0312
12.0311
—

2
13.4616
12.4963
12.2213
12.1165
12.1142
—

3
14.2172
12.8787
12.4267
12.2357
12.2268
—

4
14.7332
13.2084
12.6331
12.3675
12.3463
—

5
15.0742
13.4662
12.8168
12.4961
12.4582
—

6
—
—
—
—
—
—

Numerical values of

1
11.3239
10.5207
10.0947
9.8548
9.7100
—

2
12.0640
11.6759
11.4932
11.3965
11.3401
—

3
13.0662
12.6810
12.4538
12.3180
12.2329
—

4
13.5085
13.2108
12.9963
12.8508
12.7519
—

5
13.7180
13.5047
13.3254
13.1890
13.0885
—

6
—
—
—
—
—
—

Numerical values of

1
15.4850
16.2308
16.6264
16.8491
16.9836
—

2
14.0006
14.5092
14.8428
15.0559
15.1949
—

3
13.4909
13.8260
14.0880
14.2768
14.4104
—

4
13.2728
13.4950
13.6932
13.8518
13.9731
—

5
13.2618
13.4600
13.6264
13.7531
13.8464
—

6
—
—
—
—
—
—

Numerical values of

1
9.4938
11.1409
12.5497
13.1538
13.4520
—

2
8.8689
10.3644
11.8756
12.6696
13.1076
—

3
8.6842
10.0813
11.4747
12.3097
12.8189
—

4
8.6336
10.0176
11.2653
12.0687
12.5971
—

5
8.6325
10.0429
11.1705
11.9192
12.4370
—

6
—
—
—
—
—
—

Numerical values of

1
14.5834
13.5377
13.2398
13.1262
13.1238
—

2
15.4020
13.9519
13.4623
13.2553
13.2457
—

3
15.9609
14.3091
13.6858
13.3981
13.3751
—

4
16.3304
14.5884
13.8848
13.5375
13.4964
—

5
16.5779
14.7990
14.0514
13.6641
13.6025
—

6
—
—
—
—
—
—

Table 8: Numerical values corresponding to Figure 8.

1
2
3
4
5
6

Numerical values of

1
1.0279
0.4623
0.2308
0.1322
0.0843
0.0581

2
0.8047
0.4778
0.2939
0.1928
0.1341
0.0979

3
0.6119
0.4581
0.3270
0.2350
0.1731
0.1312

4
0.4778
0.4012
0.3180
0.2473
0.1928
0.1521

5
0.3889
0.3466
0.2939
0.2428
0.1987
0.1627

6
0.3168
0.2966
0.2667
0.2326
0.1991
0.1689

Numerical values of

1
1.0279
0.8047
0.6119
0.4778
0.3889
0.3168

2
0.4623
0.4778
0.4581
0.4012
0.3466
0.2966

3
0.2308
0.2939
0.3270
0.3180
0.2939
0.2667

4
0.1322
0.1928
0.2350
0.2473
0.2428
0.2326

5
0.0843
0.1341
0.1731
0.1928
0.1987
0.1991

6
0.0581
0.0979
0.1312
0.1521
0.1627
0.1689

Numerical values of

1
13.4612
11.8021
10.5579
9.7618
9.2601
—

2
13.9316
11.8084
10.8374
10.3699
10.1191
—

3
13.8377
12.7622
11.9437
11.4236
11.0979
—

4
13.4933
12.9563
12.4102
11.9764
11.6602
—

5
13.2255
12.9325
12.5819
12.2566
11.9879
—

6
—
—
—
—
—
—

Numerical values of

1
10.5388
12.1979
13.4421
14.2382
14.7399
—

2
10.5932
11.3237
12.0568
12.6377
13.0617
—

3
10.8043
11.1752
11.6161
12.0237
12.3602
—

4
10.9824
11.1929
11.4696
11.7539
12.0121
—

5
10.7745
11.0675
11.4181
11.7434
12.0121
—

6
—
—
—
—
—
—

Numerical values of

1
13.4612
13.9316
13.8377
13.4933
13.2255
—

2
11.8021
11.8084
12.7622
12.9563
12.9325
—

3
10.5579
10.8374
11.9437
12.4102
12.5819
—

4
9.7618
10.3699
11.4236
11.9764
12.2566
—

5
9.2601
10.1191
11.0979
11.6602
11.9879
—

6
—
—
—
—
—
—

Numerical values of

1
10.5388
10.5932
10.8043
10.9824
10.7745
—

2
12.1979
11.3237
11.1752
11.1929
11.0675
—

3
13.4421
12.0568
11.6161
11.4696
11.4181
—

4
14.2382
12.6377
12.0237
11.7539
11.7434
—

5
14.7399
13.0617
12.3602
12.0121
12.0121
—

6
—
—
—
—
—
—

Numerical values of

1
13.7691
12.3176
11.3888
10.8035
10.4245
—

2
13.7765
12.6436
12.0982
11.8056
11.6334
—

3
14.8893
13.9343
13.3275
12.9476
12.7025
—

4
15.1157
14.4785
13.9724
13.6036
13.3401
—

5
15.0879
14.6788
14.2994
13.9859
13.7398
—

6
—
—
—
—
—
—

Numerical values of

1
13.2144
14.5622
15.4247
15.9682
16.3201
—

2
12.2673
13.0616
13.6908
14.1502
14.4789
—

3
12.1065
12.5841
13.0257
13.3902
13.6766
—

4
12.1257
12.4254
12.7334
13.0131
13.2509
—

5
11.9898
12.3697
12.7220
13.0131
13.2416
—

6
—
—
—
—
—
—

Numerical values of

1
7.0627
9.6496
12.0876
13.2188
13.7521
—

2
6.8759
8.4746
10.6260
11.9816
12.7945
—

3
7.0547
8.2958
10.0152
11.2619
12.1246
—

4
7.2590
8.4154
9.8191
10.8928
11.7015
—

5
7.4425
8.6142
9.8100
10.7305
11.4556
—

6
—
—
—
—
—
—

Numerical values of

1
13.2144
12.2673
12.1065
12.1257
11.9898
—

2
14.5622
13.0616
12.5841
12.4254
12.3697
—

3
15.4247
13.6908
13.0257
12.7334
12.7220
—

4
15.9682
14.1502
13.3902
13.0131
13.0131
—

5
16.3201
14.4789
13.6766
13.2509
13.2416
—

6
—
—
—
—
—
—

Figure 5: rational trigonometric bicubic function with , , ,  ,  ,  ,  , ,  , ,  ,  , ,  ,  ,  .

Figure 6: monotone rational trigonometric bicubic function with ,  , ,  ,  ,  ,  , .

Figure 7: rational trigonometric bicubic function with ,  ,  ,   ,  ,  ,  ,  ,  , ,  ,  ,  ,  ,  , .

Figure 8: monotone rational trigonometric bicubic function with ,  ,  ,  ,  ,  ,  ,  .

7. Conclusion

In this paper, monotonicity of data is retained by developing constraints on free parameters in the specification of rational trigonometric function and bicubic blended function. Authors in [7, 8] used algebraic function while the proposed algorithm applies trigonometric function which gives much smoother result due to orthogonality of sine and cosine function. Shape preserving techniques of Butt and Brodlie [1] required insertion of additional knots. In [12], developed scheme failed to maintain smoothness. The proposed technique is local, affirms smoothness, works well for data with derivatives, and does not require insertion of extra knots. Derivative of trigonometric spline is much lower than that of polynomial spline.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

S. Butt and K. W. Brodlie, “Preserving positivity using piecewise cubic interpolation,” Computers & Graphics, vol. 17, no. 1, pp. 55–64, 1993.

P. Costantini and F. Fontanella, “Shape-preserving bivariate interpolation,” SIAM Journal on Numerical Analysis, vol. 27, no. 2, pp. 488–506, 1990.

R. J. Cripps and M. Z. Hussain, “C¹ monotone cubic Hermite interpolant,” Applied Mathematics Letters, vol. 25, no. 8, pp. 1161–1165, 2012.

R. D. Fuhr and M. Kallay, “Monotone linear rational spline interpolation,” Computer Aided Geometric Design, vol. 9, no. 4, pp. 313–319, 1992.

M. S. Floater and J. M. Peña, “Monotonicity preservation on triangles,” Mathematics of Computation, vol. 69, no. 232, pp. 1505–1519, 2000.

T. N. T. Goodman, B. H. Ong, and K. Unsworth, “Constrained interpolation using rational cubic splines,” in NURBS for Curve and Surface Design, G. Farin, Ed., pp. 59–74, 1991.

M. Z. Hussain and M. Hussain, “Visualization of data preserving monotonicity,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1353–1364, 2007.

M. Z. Hussain and M. Sarfraz, “Monotone piecewise rational cubic interpolation,” International Journal of Computer Mathematics, vol. 86, no. 3, pp. 423–430, 2009.

M. Z. Hussain and S. Bashir, “Shape preserving surface data visualization using rational bi-cubic functions,” Journal of Numerical Mathematics, vol. 19, no. 4, pp. 267–307, 2011.

M. Z. Hussain, M. Hussain, and A. Waseem, “Shape-preserving trigonometric functions,” Computational and Applied Mathematics, 2013.

F. Ibraheem, M. Hussain, M. Z. Hussain, and A. A. Bhatti, “Positive data visualization using trigonometric function,” Journal of Applied Mathematics, vol. 2012, Article ID 247120, 19 pages, 2012.

M. Sarfraz, S. Butt, and M. Z. Hussain, “Surfaces for the visualization of scientific data preserving monotonicity,” in Proceedings of the 7th IMA Mathematics for Surfaces Conference, T. N. T. Goodman and R. Martin, Eds., pp. 479–495, Dundee, UK, September 1997.

M. Sarfraz, “A rational cubic spline for the visualization of monotonic data: an alternate approach,” Computers & Graphics, vol. 27, no. 1, pp. 107–121, 2003.

M. Sarfraz and M. Z. Hussain, “Data visualization using rational spline interpolation,” Journal of Computational and Applied Mathematics, vol. 189, no. 1-2, pp. 513–525, 2006.

M. Sarfraz, M. Z. Hussain, and M. Hussain, “Shape-preserving curve interpolation,” International Journal of Computer Mathematics, vol. 89, no. 1, pp. 35–53, 2012.

M. Sarfraz, M. Z. Hussain, and M. Hussain, “Modeling rational spline for visualization of shaped data,” Journal of Numerical Mathematics, vol. 21, no. 1, pp. 63–87, 2013.

L. L. Schumaker, “On shape preserving quadratic spline interpolation,” SIAM Journal on Numerical Analysis, vol. 20, no. 4, pp. 854–865, 1983.