# Monotone data visualization using rational trigonometric spline interpolation.

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-Bezier rational cubic function. The authors in [3] converted the Bernstein-Bezier rational cubic function to [C.sup.1] cubic Hermite by applying the [C.sup.1] 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 Pena [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 {([x.sub.i], [f.sub.i]), i = 0, 1, 2, ...,n| be the given set of data points defined over the interval [a,b], where a = [x.sub.0] < [x.sub.1] < [x.sub.2] < ... < [x.sub.n] = b. Piecewise rational trigonometric cubic function is defined over each subinterval [I.sub.i] = [[x.sub.i], [x.sub.i+1]] as

[S.sub.i](x) = [p.sub.i]([theta])/[q.sub.i]([theta]), (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

where [theta] = ([pi]/2)((x - [x.sub.i])/[h.sub.i]), [h.sub.i] = [x.sub.i+1] -- [x.sub.i].

The rational trigonometric cubic function (1) is [C.sup.1]; that is, it satisfies the following properties:

S([x.sub.i]) = [f.sub.i], S([x.sub.i+1]) = [f.sub.i+1],

S'([x.sub.i]) = [d.sub.i], S'([x.sub.i+1]) = [d.sub.i+1]. (3)

Here [d.sub.i] and [d.sub.i+1] are derivatives at the end points of the interval [I.sub.i] = [[x.sub.i], [x.sub.i+1]]. The parameters [[alpha].sub.i] and [[delta].sub.i] are real numbers used to modify the shape of the curve.

3. Rational Trigonometric Bicubic PartiallyBlended Function

Let {([x.sub.i],[y.sub.j], [F.sub.i,j]), i = 0,1,2, ..., n-1; j = 0, 1, 2, ..., m-1} be the 3D regular data set defined over the rectangular mesh I = [a, b] x [c, d],let p : a = [x.sub.0] < [x.sub.1] < ... < [x.sub.m] = b be a partition of [a, b], and let q : a = [y.sub.0] < [y.sub.1] < ... < [y.sub.n] be a partition of [c, d]. Rational trigonometric bicubic function which is an extension of rational trigonometric cubic function (1) is defined over each rectangular patch [[x.sub.i], [x.sub.i+1]] x [[y.sub.i], [y.sub.i+1]], where i = 0, 1, 2, ..., n - 1; j = 0, 1, 2, ..., m - 1, as

S (v, y) = -[AFB.sup.T], (4)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

S(x, [y.sub.j]), S(x, [y.sub.j+1]), S([x.sub.i], y), and S([x.sub.i+1], y) are rational trigonometric bicubic functions defined on the boundary of rectangular patch [[x.sub.i], [x.sub.i+1]] x [[y.sub.p], [y.sub.j+1]] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

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 {([x.sub.i], [f.sub.i]), i = 0, l,2, ..., n| be the monotone data defined over the interval [a, b]; that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

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

[S'.sub.i] (x) , [for all] x [member of] [[x.sub.i], [x.sub.i+1]], i = 0, 1,2, ..., n - l. (15)

Now, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

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 [B.sub.i], i = 0,1,2, ..., 11 of the trigonometric basis functions are all positive. This yields the following result:

[[beta].sub.i] > [2[[alpha].sub.i][d.sub.i]/[pi][[DELTA].sub.i]], [[gamma].sub.i] > [2[[delta].sub.i][d.sub.i+1]/[pi][[DELTA].sub.i]]. (18)

The above discussion can be summarized as follows.

Theorem 1. The [C.sup.1] piecewise trigonometric rational cubic function (1) preserves the monotonicity of monotone data if in each subinterval I = [[x.sub.i], [x.sub.i+1]], theparameters [[beta].sub.i] and [[gamma].sub.i] satisfy the following sufficient conditions:

[[beta].sub.i] > [2[[alpha].sub.i][d.sub.i]/[pi][[DELTA].sub.i]], [[gamma].sub.i] > [2[[delta].sub.i][d.sub.i+1]/[pi][[DELTA].sub.i]]. (19)

The above constraints can be rearranged as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

Algorithm 2.

Step 1. Take a monotone data set {([x.sub.i], [f.sub.i]) : i = 0,1,2, ..., n}.

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

Step 3. Compute the values of parameters [[beta].sub.i]'s and [[gamma].sub.i]'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 {([x.sub.i], [y.sub.j], [F.sub.i,j]), i = 0,1,2, ..., n - 1; j = 0,1,2, ..., m - 1} be the monotone data set defined over the rectangular mesh I = [[x.sub.i], [x.sub.i+1]] x [[y.sub.j], [y.sub.j+1]] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

Now, surface patch (4) is monotone if the boundary curves defined in (5)-(12) are monotone.

Now, S(x, [y.sub.j]) is monotone if [S'.sub.i](x, [y.sub.j]) > 0, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

Now the positivity of [S'.sub.i](x,[y.sub.j]) entirely depends on [R.sub.i], i = 0, 1, 2 ..., 11. The denominator in (22) is always positive. Since the parameter [theta] lies in first quadrant therefore the trigonometric basis functions will be positive also. This yields the following constraints on the free parameters:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (24)

S(x, [y.sub.j+1]) is monotone if

[S'.sub.i] (x, [y.sub.j+1]) > 0, (25)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

The denominator in (26) is always positive. Moreover, the trigonometric basis functions are also positive for 0 [less than or equal to] [theta] [less than or equal to] [pi]/2. It follows that the positivity of [S'.sub.i](x, [y.sub.j+1]) entirely depends upon [T.sub.i], i = 0, 1, 2 ..., 11. This yields the following constraints on the free parameters:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (28)

S([x.sub.i], y) is monotone if [S'.sub.i](([x.sub.i], y) > 0. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (29)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30)

Since the denomoinator of (29) is always positive and trigonometric basis functions are positive for so the positivity of 0 [less than or equal to] [phi] [less than or equal to] [pi]/2. It follows that the positivity of [S'.sub.i]([x.sub.i+1], y) entirely depends upon [U.sub.i], i = 0, 1, 2 ..., 11. This yields the following constraints on the free parameters:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

S([x.sub.j+1], y) is monotone if [S'.sub.i]([x.sub.i+1], y) > 0. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (32)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (33)

Finally, [S'.sub.i]([x.sub.i+1], y) is positive if [V.sub.i], i = 0, 1, 2 ..., 11 are positive. This yields the following constraints on the free parameters:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

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 I = [[x.sub.i], [x.sub.j+1]] x [[y.sub.j],[y.sub.j+1]], free parameters [[beta].sub.i,j], [[gamma].sub.i,j], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] satisfy the following constraints:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (35)

The above constraints are rearranged as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (36)

Algorithm 4.

Step 1. Take a 3D monotone data set {([x.sub.i], [y.sub.j], [F.sub.i,j]), i = 0, 1, 2, ..., n; j = 0, 1, 2, ..., m}.

Step 2. Use the Arithmetic Mean Method to estimate the derivatives [F.sup.x.sub.i,j], [F.sup.y.sub.i,j], [F.sup.xy.sub.i,j] at knots (note: Step 2 is only applicable if data is not provided with derivatives).

Step 3. Compute the values of parameters [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 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 C1 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 C1 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.

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

F(x, y) = [square root of ([x.sup.2]/25 + [y.sup.2]/16)],

F(x, y) = log([x.sup.2] + [y.sup.2]). (37)

respectively.

Figures 5 and 7 are produced by interpolating the monotone data sets in Tables 5 and 6, respectively, by [C.sup.1] 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.

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 [I] 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.

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

Conflict of Interests

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

References

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

[3] R. J. Cripps and M. Z. Hussain, "[C.sup.1] monotone cubic Hermite interpolant," Applied Mathematics Letters, vol. 25, no. 8, pp. 1161-1165, 2012.

[4] R. D. Fuhr and M. Kallay, "Monotone linear rational spline interpolation," Computer Aided Geometric Design, vol. 9, no. 4, pp. 313-319, 1992.

[5] M. S. Floater and J. M. Pena, "Monotonicity preservation on triangles," Mathematics of Computation, vol. 69, no. 232, pp. 1505-1519, 2000.

[6] 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.

[7] M. Z. Hussain and M. Hussain, "Visualization of data preserving monotonicity," Applied Mathematics and Computation, vol. 190, no. 2, pp. 1353-1364, 2007.

[8] M. Z. Hussain and M. Sarfraz, "Monotone piecewise rational cubic interpolation," International Journal of Computer Mathematics, vol. 86, no. 3, pp. 423-430, 2009.

[9] 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.

[10] M. Z. Hussain, M. Hussain, and A. Waseem, "Shape-preserving trigonometric functions," Computational and Applied Mathematics, 2013.

[II] 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.

[12] 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

[13] 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.

[14] 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.

[15] 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.

[16] 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.

[17] L. L. Schumaker, "On shape preserving quadratic spline interpolation," SIAM Journal on Numerical Analysis, vol. 20, no. 4, pp. 854-865, 1983.

Farheen Ibraheem, (1) Maria Hussain, (2) and Malik Zawwar Hussain (3)

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

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

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

Correspondence should be addressed to Maria Hussain; mariahussain_1@yahoo.com

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

```
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.

i                    1         2      3     4        5

[d.sub.i]         50.9065   19.6819   0     0      5.7983
[[beta].sub.i]     35.01     7.17     0    0.01      --
[[gamma].sub.i]   0.1890    0.0100    0   0.7871     --

TABLE 4: Numerical results corresponding to Figure 4.

i                   1        2      3     4      5

[d.sub.i]         56.25      0      0     0      15
[[beta].sub.i]    2.8748   9.3179   0    0.01    --
[[gamma].sub.i]    0.01     0.01    0   0.6466   --

TABLE 5: A 3D monotone data set.

y/x     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.

y/x     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

TABLE 7: Numerical values corresponding to Figure 6.

([x.sub.i],      1         2         3         4         5        6
[y.sub.i])

Numerical values of [F.sup.x.sub.i,j]

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 [F.sup.y.sub.i,j]

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 [[beta].sub.i,j]

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 [[gamma].sub.i,j]

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 [[??].sub.i,j]

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 [[??].sub.i,j]

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 [[beta].sub.i,j+1]

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 [[gamma]].sub.i,j+1]

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 [[??].sub.i,j+1]

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 [[??].sub.i,j+1]

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.

([x.sub.i],      1         2         3         4         5        6
[y.sub.i])

Numerical values of [F.sup.x.sub.i,j]

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 [F.sup.y.sub.i,j]

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 [[beta].sub.i,j]

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 [[gamma].sub.i,j]

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 [[??].sub.i,j]

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 [[??].sub.i,j]

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 [[beta].sub.i,j+1]

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 [[gamma]].sub.i,j+1]

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 [[??].sub.i,j+1]

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 [[??].sub.i,j+1]

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               --        --        --        --        --        --
```