Data Visualization Using Rational Trigonometric Spline

Abstract

This paper describes the use of trigonometric spline to visualize the given planar data. The goal of this work is to determine the smoothest possible curve that passes through its data points while simultaneously satisfying the shape preserving features of the data. Positive, monotone, and constrained curve interpolating schemes, by using a 𝐶¹ piecewise rational cubic trigonometric spline with four shape parameters, are developed. Two of these shape parameters are constrained and the other two are set free to preserve the inherited shape features of the data as well as to control the shape of the curve. Numerical examples are given to illustrate the worth of the work.

1. Introduction

Data visualization, the technique of using images to represent information, has its history in the days back to the second century AD. But most of the developments are made in the last couple of centuries, predominantly during the last 30 years. It has extensively been used in industrial design, image processing, computer vision, computer aided geometric design, computer graphics, and many more. Shape preserving interpolation is a powerful tool to visualize the data in the form of curves and surfaces. The problem of curve interpolation to the given data has been studied with various requirements. One may be concerned with the smoothness of the interpolating curves, the preservation of the underlying shape features of the data, the computational complexity, or the fulfillment of certain constraints. Shape preserving signifies preserving the three basic and crucial geometrical features such as positivity, monotonicity, and convexity of the data. These shape characteristics can be easily observed when data arises from a physical experiment. In this case, it becomes vital that the interpolant produces curves more smooth and represent physical reality as close as possible. For this purpose, designers and engineers want such approximation methods that represent such physical situations accurately.

At present, spline methods have become the main tools for solving the majority of problems involving the approximation of functions, which also includes interpolation problems. Many spline functions exist that generate smooth and visually pleasant curves. Sarfraz et al. [1] studied shape preserving curve interpolation for positive, monotone, convex data and data lying above a line, using a piecewise rational cubic function with four shape parameters. Hussain et al. [2] visualized scientific data with shape preserving 𝐶¹ rational cubic interpolation by developing positive, monotone, and constrained data preserving schemes.

In recent years, polynomial splines and NURBS are replaced by trigonometric splines in order to prevail over the difficulties faced in using the former. Polynomial splines are not able to represent circular arcs and conics which are the most basic geometrical entity in almost every modeling system [3]. Also conics find a widespread application in optical and telecommunication instruments. NURBS offers great flexibility and precision for handling both analytic and freeform shapes but at the expense of intuitiveness and ease of implementation. A number of authors have contributed to trigonometric splines to represent curves and surfaces [4–11]. Lately, an alternative approach in shape preserving, using the trigonometric splines is introduced. Liu et al. [10] studied cubic trigonometric polynomial B-spline curves and surfaces with shape parameter. Ibraheem et al. [12] introduced a 𝐶¹ piecewise rational trigonometric cubic function and piecewise rational trigonometric bicubic function with four shape parameters to visualize the positivity of positive curve and surface data. Zhu et al. [13] constructed shape preserving quartic trigonometric interpolation spline curves with shape parameters. The authors obtained 𝐹³ continuous shape preserving spline curves, for any shape parameters satisfying the shape preserving conditions.

In this paper, we present a 𝐶¹ piecewise rational cubic trigonometric interpolating curve scheme for positive, monotone, and constrained data. Four positive shape parameters are used in the description of the interpolating scheme. Two of these parameters are constrained by simple data-dependent conditions to preserve the inherited shape feature of the data while the other two are kept free to modify the shape of the positive, monotone, and constrained curve. The scheme is useful for both equally and unequally spaced data.

This paper is arranged as follows. In Section 2, a 𝐶¹ piecewise rational cubic trigonometric spline with four shape parameters is constructed. Sections 3, 4, and 5 describe positive, monotone, and constrained curve interpolation schemes, respectively, to visualize positive, monotone, and constrained 2D data. Section 5 concludes the paper with some future work.

2. 𝐶¹ Piecewise Rational Cubic Trigonometric Spline

In this section, we develop a 𝐶¹ piecewise rational cubic trigonometric spline with four parameters. Two of these parameters are free that can be used to preserve and control the shape of the interpolating curve while the other two are restricted. Data-dependent constraints are developed for these parameters to preserve the shape characteristics of the data under consideration.

Suppose that for knot spacing 𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑛, given data points are defined as {(𝑡_𝑖,𝑓_𝑖):𝑖=0,1,2,…,𝑛} where 𝑓_𝑖 are the function values at the knots of the function being interpolated. In each subinterval [𝑡_𝑖,𝑡_𝑖+1], 𝑖=0,1,2,…,𝑛−1, a piecewise rational cubic trigonometric spline with four shape parameters is defined as 𝑃(𝑡)≡𝑃_𝑖(𝑡)=∑³_𝑗=0𝜉_𝑗𝑏_𝑗∑³_𝑗=0𝑤^𝑗_𝑖𝑏_𝑗, (1) where 𝑏₀=(1−sin𝑢)²,𝑏₁(),𝑏=2sin𝑢1−sin𝑢₂(),𝑏=2cos𝑢1−cos𝑢₃=(1−cos𝑢)². (2) The spline in (1) is 𝐶¹ continuous if the following interpolating conditions are satisfied: 𝑃𝑡_𝑖=𝑓_𝑖,𝑃𝑡_𝑖+1=𝑓_𝑖+1,𝑃^′𝑡_𝑖=𝑑_𝑖,𝑃^′𝑡_𝑖+1=𝑑_𝑖+1, (3) where 𝑃^′(𝑡) denotes the derivative with respect to “𝑡” and 𝑑_𝑖 are derivative values at given knots 𝑡_𝑖 that are used for the shape control and smoothness of curve. These 𝑑_𝑖 are either given or can be computed by some numerical method [14].

Using conditions (3), the values of unknowns 𝜉_𝑖, 𝑖=0,1,2,3 are 𝜉₀=𝑤⁰_𝑖𝑓_𝑖,𝜉₁=𝜋𝑤¹_𝑖𝑓_𝑖+ℎ_𝑖𝑑_𝑖𝑤⁰_𝑖𝜋,𝜉₂=𝜋𝑤²_𝑖𝑓_𝑖+1−ℎ_𝑖𝑑_𝑖+1𝑤³_𝑖𝜋,𝜉₃=𝑤³_𝑖𝑓_𝑖+1. (4) Thus after using (4), (1) takes the form 𝑃(𝑡)≡𝑃_𝑖(𝑡)=𝑝_𝑖(𝑢)𝑞_𝑖(𝑢), (5) where 𝑝(𝑢)=𝑤⁰_𝑖𝑓_𝑖𝑏₀+𝜋𝑓_𝑖𝑤¹_𝑖+ℎ_𝑖𝑑_𝑖𝑤⁰_𝑖𝜋𝑏₁+𝜋𝑤²_𝑖𝑓_𝑖+1−ℎ_𝑖𝑑_𝑖+1𝑤³_𝑖𝜋𝑏₂+𝑤³_𝑖𝑓_𝑖+1𝑏₃,𝑞(𝑢)=𝑏₀𝑤⁰_𝑖+𝑏₁𝑤¹_𝑖+𝑏₂𝑤²_𝑖+𝑏₃𝑤³_𝑖, (6)𝑢=(𝜋/2)((𝑡−𝑡_𝑖)/ℎ_𝑖),  ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖, and 𝑤^𝑗_𝑖, 𝑗=0,1,2,3 are positive shape parameters. It is to mention that if the values of the shape parameters are chosen on trial basis, the shape characteristics of the data are not always preserved. Thus there arises a need for some conditions to be imposed on these shape parameters.

3. Positive Curve Interpolation

In this section, we utilize 𝐶¹ piecewise rational cubic trigonometric spline developed in Section 1 to generate a positivity preserving curve using a positive data set.

Theorem 1. A 𝐶¹ piecewise rational cubic trigonometric spline defined in (5) preserves the positivity of the positive data in each subinterval [𝑡_𝑖,𝑡_𝑖+1], 𝑖=0,1,2,…,𝑛−1, if the shape parameters satisfy the following conditions: 𝑤⁰_𝑖,𝑤³_𝑖𝑤>0,_1𝑖>max0,−𝑑_𝑖ℎ_𝑖𝑤⁰_𝑖𝜋𝑓_𝑖,𝑤²_𝑖𝑑>max0,^𝑖+1ℎ_𝑖𝑤³_𝑖𝜋𝑓_𝑖+1. (7)

Proof. Consider a data set {(𝑡_𝑖,𝑓_𝑖):𝑖=0,1,2,…,𝑛} such that 𝑡_𝑖<𝑡_𝑖+1,𝑖=0,1,2,…,𝑛−1,𝑓_𝑖>0. (8)𝐶¹ piecewise rational cubic trigonometric spline given in (5) preserves positivity through positive data if 𝑃_𝑖(𝑡)>0.
𝑃_𝑖(𝑡)>0 if 𝑝_𝑖(𝑢),𝑞_𝑖(𝑢)>0. (9) Since 𝑞_𝑖(𝑢)>0, thus the positivity of the interpolant depends on 𝑝_𝑖(𝑢) only.
𝑝_𝑖(𝑢)>0 if all the coefficients are positive. It yields 𝑤¹_𝑖>−𝑑_𝑖ℎ_𝑖𝑤⁰_𝑖𝜋𝑓_𝑖,𝑤²_𝑖>𝑑_𝑖+1ℎ_𝑖𝑤³_𝑖𝜋𝑓_𝑖+1. (10) Thus the sufficient conditions for the interpolant defined in (5) to preserve the positivity of positive data are that the shape parameters satisfy 𝑤⁰_𝑖,𝑤³_𝑖𝑤>0,_1𝑖>max0,−𝑑_𝑖ℎ_𝑖𝑤⁰_𝑖𝜋𝑓_𝑖,𝑤²_𝑖𝑑>max0,^𝑖+1ℎ_𝑖𝑤³_𝑖𝜋𝑓_𝑖+1. (11) This proves the desired result.

These conditions on the shape parameters can also be expressed as 𝑤¹_𝑖=𝜂_𝑖+max0,−𝑑_𝑖ℎ_𝑖𝑤⁰_𝑖𝜋𝑓_𝑖,𝜂_𝑖𝑤>0,_2𝑖=𝜈_𝑖𝑑+max0,^𝑖+1ℎ_𝑖𝑤³_𝑖𝜋𝑓_𝑖+1,𝜈_𝑖>0. (12) The developed scheme is used to demonstrate the positivity preserving of positive data. The curves in Figures 1 and 3 are drawn by using 𝐶¹ piecewise rational cubic trigonometric spline for 2D positive data sets given in Tables 1 and 2, respectively. Random values to the shape parameters are assigned and it is clearly visible that the resulting curves do not preserve the positivity. On the other hand, the positivity preserving curves in Figures 2 and 4 are generated for the same data set by using the scheme developed in Section 2. These figures also delineate the role of free shape parameters in shape control of the curve.

Table 1 

A 2D positive dataset.

Table 2 

A 2D positive dataset.

Figure 1 

Nonpositivity preserving rational cubic trigonometric curve.

Figure 2 

A 𝐶1 positivity preserving curve with different values of free parameters.

Figure 3 

Nonpositive rational cubic trigonometric curve.

Figure 4 

Positive curve by rational cubic trigonometric spline with different values of free parameters.

4. Monotone Curve Interpolation

A 2D data set {(𝑡_𝑖,𝑓_𝑖):𝑖=0,1,2,…,𝑛} with 𝑡_𝑖<𝑡_𝑖+1, 𝑖=0,1,2,…,𝑛−1 is said to be monotonically increasing (monotonically decreasing) if 𝑓_𝑖≤𝑓_𝑖+1𝑓_𝑖≥𝑓_𝑖+1. (13) This section discusses a monotonicity preserving curve interpolating scheme with four parameters. For a given set of monotone data points, we wish to generate a piecewise cubic trigonometric curve that interpolates the data points and preserves monotonicity as well. We drive data-dependent conditions for two shape parameters while leaving the other two for designer’s choice.

Theorem 2. The 𝐶¹ piecewise rational cubic trigonometric spline defined in (5) preserves the monotonicity through monotone data in each subinterval [𝑡_𝑖,𝑡_𝑖+1], 𝑖=0,1,2,…,𝑛−1 if the shape parameters satisfy the following conditions: 𝑤⁰_𝑖,𝑤³_𝑖𝛼>0,_𝑖<𝑤¹_𝑖<𝛽_𝑖,𝛾_𝑖<𝑤¹_𝑖<𝛿_𝑖, (14) where 𝛼_𝑖=max0,𝑤⁰_𝑖,𝑑_𝑖𝑤⁰_𝑖𝜋Δ_𝑖,𝛽_𝑖=𝑑1+^𝑖𝜋Δ_𝑖𝑤⁰_𝑖,𝛾_𝑖=max0,𝑤³_𝑖,𝑑_𝑖+1𝑤³_𝑖𝜋Δ_𝑖,𝛿_𝑖=𝑑1+^𝑖+1𝜋Δ_𝑖𝑤³_𝑖. (15)

Proof. Let {(𝑡_𝑖,𝑓_𝑖):𝑖=0,1,2,…,𝑛} be a monotonically increasing data set, that is, 𝑓_𝑖≤𝑓_𝑖+1orequivalentlyΔ_𝑖=𝑓_𝑖+1−𝑓_𝑖ℎ_𝑖≥0. (16)The case of monotonically decreasing data set can be dealt in a similar fashion.
For monotonicity, the necessary conditions on derivatives are 𝑑_𝑖≥0,𝑖=0,1,2,…,𝑛. (17) There arise the following two cases for the interpolant (5) to preserve the monotonicity of monotone data.
Case 1. 𝑑_𝑖=𝑑_𝑖+1=0  when Δ_𝑖=0. In this case, 𝑃_𝑖(𝑡) reduces to 𝑃_𝑖(𝑡)=𝑓_𝑖,∀𝑡∈𝑡_𝑖,𝑡_𝑖+1. (18) This proves that the interpolant is monotone.
Case 2. When Δ_𝑖≠0, then 𝑃_𝑖(𝑡) is monotonically increasing if and only if 𝑃^′_𝑖(𝑡)>0,∀𝑡∈𝑡_𝑖,𝑡_𝑖+1. (19) For 𝑡∈[𝑡_𝑖,𝑡_𝑖+1], 𝑃^′_𝑖(𝑡) is presented in a simpler form as 𝑃^′_𝑖(𝑡)=𝜋2𝑞_𝑖(𝑢)²×𝐵₁cos𝑢(1−sin𝑢)²+𝐵₂cos𝑢1+sin²𝑢+2𝐵₃cos𝑢+𝐵₄cos𝑢sin𝑢+2𝐵₅sin𝑢+𝐵₆sin𝑢1+cos²𝑢+𝐵₇sin𝑢(1−cos𝑢)²+4𝐵₈ (20) with 𝐵₁=2𝑑_𝑖𝑤⁰_𝑖²𝜋,𝐵₂=2Δ_𝑖𝑤¹_𝑖−𝑤⁰_𝑖2𝑤²_𝑖−𝑤³_𝑖−4𝑑_𝑖+1𝑤³_𝑖𝜋𝑤¹_𝑖−𝑤⁰_𝑖−2𝑑_𝑖𝑤⁰_𝑖𝜋2𝑤²_𝑖−𝑤³_𝑖,𝐵₃=Δ_𝑖𝑤⁰_𝑖−𝑤¹_𝑖+𝑑_𝑖𝑤⁰_𝑖𝜋𝑤³_𝑖,𝐵₄=2Δ_𝑖𝑤⁰_𝑖−2𝑤¹_𝑖2𝑤²_𝑖−𝑤³_𝑖+4𝑑_𝑖𝑤⁰_𝑖𝜋2𝑤²_𝑖−𝑤³_𝑖−4𝑑_𝑖+1𝑤³_𝑖𝜋𝑤⁰_𝑖−2𝑤¹_𝑖+4Δ_𝑖𝑤²_𝑖−𝑤³_𝑖−𝑑_𝑖𝜋+𝑑_𝑖+1𝜋𝑤³_𝑖𝑤⁰_𝑖+4Δ_𝑖𝑤¹_𝑖𝑤³_𝑖,𝐵₅=Δ_𝑖𝑤³_𝑖−𝑤²_𝑖+𝑑_𝑖+1𝑤³_𝑖𝜋𝑤⁰_𝑖,𝐵₆=2Δ_𝑖2𝑤¹_𝑖−𝑤⁰_𝑖𝑤²_𝑖−𝑤³_𝑖−4𝑑_𝑖𝑤⁰_𝑖𝜋𝑤²_𝑖−𝑤³_𝑖−2𝑑_𝑖+1𝑤³_𝑖𝜋2𝑤¹_𝑖−𝑤⁰_𝑖,𝐵₇=2𝑑_𝑖+1𝑤³_𝑖²𝜋,𝐵₈=Δ_𝑖𝑤⁰_𝑖−𝑤¹_𝑖𝑤²_𝑖−𝑤³_𝑖+𝑑_𝑖𝑤⁰_𝑖𝜋𝑤²_𝑖−𝑤³_𝑖−𝑑_𝑖+1𝑤³_𝑖𝜋𝑤⁰_𝑖−𝑤¹_𝑖. (21) The denominator of (20) is always positive. Thus the sufficient conditions for monotonicity preserving curve are 𝐵_𝑘≥0,𝑘=1,…,8, (22) since 𝐵₁,𝐵₇>0.
Also 𝐵_𝑖≥0,  𝑖=2,3,4,5,6,8 if 𝑤¹_𝑖>𝑤⁰_𝑖,𝑤¹_𝑖>𝑑_𝑖𝑤⁰_𝑖𝜋Δ_𝑖,𝑤¹_𝑖<𝑑1+^𝑖𝜋Δ_𝑖𝑤⁰_𝑖, (23)𝑤²_𝑖>𝑤³_𝑖,𝑤²_𝑖>𝑑_𝑖+1𝑤³_𝑖𝜋Δ_𝑖,𝑤²_𝑖<𝑑1+^𝑖+1𝜋Δ_𝑖𝑤³_𝑖, (24) Hence, to preserve the monotonicity of monotone data and control the shape of the curve as per desire, (23) and (24) can be written as 𝑤⁰_𝑖,𝑤³_𝑖𝛼>0,_𝑖<𝑤¹_𝑖<𝛽_𝑖,𝛾_𝑖<𝑤¹_𝑖<𝛿_𝑖 (25) with 𝛼_𝑖=max0,𝑤⁰_𝑖,𝑑_𝑖𝑤⁰_𝑖𝜋Δ_𝑖,𝛽_𝑖=𝑑1+^𝑖𝜋Δ_𝑖𝑤⁰_𝑖,𝛾_𝑖=max0,𝑤³_𝑖,𝑑_𝑖+1𝑤³_𝑖𝜋Δ_𝑖,𝛿_𝑖=𝑑1+^𝑖+1𝜋Δ_𝑖𝑤³_𝑖 (26) as required.
To produce a monotone curve using a monotone data, the restrictions on the shape parameters can be rearranged as 𝑤⁰_𝑖,𝑤³_𝑖𝛼>0,_𝑖+𝑘_𝑖=𝑤¹_𝑖=𝛽_𝑖−𝑙_𝑖,𝑘_𝑖,𝑙_𝑖𝛾>0,_𝑖+𝑚_𝑖=𝑤²_𝑖=𝛿_𝑖−𝑛_𝑖,𝑚_𝑖,𝑛_𝑖>0. (27) To implement the developed scheme, two monotone data sets are given in Tables 3 and 4, respectively. Initially, the curves are generated by assigning arbitrary values to the four shape parameters and nonmonotonicity preserving curves are shown in Figures 5 and 7, respectively. To remedy this deficiency, the scheme developed in Theorem 2 is applied to the same data sets and monotonicity preservation and smoothness of the curves are shown in Figures 6 and 8, respectively.

Table 3 

A 2D monotone dataset.

Table 4 

A 2D monotone dataset.

Figure 5 

Nonmonotonicity preserving curve.

Figure 6 

Monotonicity preserving curve with different values of shape parameters.

Figure 7 

Nonmonotonicity preserving curve.

Figure 8 

Monotone data visualization with specified values of free parameters.

5. Constrained Curve Interpolation

In this section, we generalize the curve scheme for positive data developed in Section 2. It is assumed that the data under consideration lies not only above the line 𝑦=0, but also above any arbitrary line 𝑦=𝑚𝑥+𝑐. We wish to drive a scheme for generating a curve which interpolates this data and lies above the line as well.

Theorem 3. The 𝐶¹ piecewise rational cubic trigonometric spline defined in (5) preserves the shape of data lying above an arbitrary straight line in each subinterval [𝑡_𝑖,𝑡_𝑖+1], 𝑖=0,1,2,…,𝑛−1, if the following conditions are satisfied: 𝑤⁰_𝑖,𝑤³_𝑖𝑤>0,_2𝑖>max0,−𝑑_𝑖ℎ_𝑖𝑤⁰_𝑖𝜋𝑓_𝑖,𝑤−𝑙_2𝑖𝑑>max0,^𝑖+1ℎ_𝑖𝑤³_𝑖𝜋𝑓_𝑖+1.−𝑙 (28)

Proof. Let {(𝑡_𝑖,𝑓_𝑖):𝑖=0,1,2,…,𝑛} be a set of data points lying above a given straight line 𝑦=𝑚𝑥+𝑐; that is, 𝑓_𝑖>𝑚𝑡_𝑖+𝑐. (29) The curve will lie above the straight line if the rational cubic trigonometric spline (5) satisfies the following condition: 𝑃(𝑡)>𝑚𝑡+𝑐,∀𝑡∈𝑡₀,𝑡_𝑛. (30) For each subinterval [𝑡_𝑖,𝑡_𝑖+1], (30) can be expressed as 𝑃_𝑖(𝑡)=𝑝_𝑖(𝑢)𝑞_𝑖(𝑢)>𝛼_𝑖21−𝜋𝑢−2𝜋𝑢𝛽_𝑖 (31) or 𝑝_𝑖(𝑢)−𝑙𝑞_𝑖(𝑢)>0, (32) where 𝑙=𝛼_𝑖(1−(2/𝜋)𝑢)−(2/𝜋)𝑢𝛽_𝑖 with 𝛼_𝑖=𝑚𝑡_𝑖+𝑐 and 𝛽_𝑖=𝑚𝑡_𝑖+1+𝑐. Using the values of 𝑝_𝑖(𝑢) and 𝑞_𝑖(𝑢) as defined in (5), (31) can be written in a simplified form as ³_𝑗=0𝜉_𝑗𝑏_𝑗−𝑙³_𝑗=0𝑤^𝑗_𝑖𝑏_𝑗>0. (33) Using (4), we get ³_𝑗=0𝑏_𝑗𝐶_𝑗>0, (34) where 𝐶₀=𝑓_𝑖−𝑙𝑤⁰_𝑖,𝐶₁=𝜋𝑤¹_𝑖𝑓_𝑖+ℎ_𝑖𝑑_𝑖𝑤⁰_𝑖𝜋−𝑙𝑤¹_𝑖,𝐶₂=𝜋𝑤²_𝑖𝑓_𝑖+1−ℎ_𝑖𝑑_𝑖+1𝑤³_𝑖𝜋−𝑙𝑤²_𝑖,𝐶₄=𝑓_𝑖+1−𝑙𝑤³_𝑖. (35) Since 𝑏_𝑗≥0, 𝑗=0,1,2,3, thus (34) is true if 𝐶_𝑗>0.
As 𝐶₀,𝐶₃>0: 
𝐶₁>0 if 𝑤¹_𝑖>−𝑑_𝑖ℎ_𝑖𝑤⁰_𝑖𝜋𝑓_𝑖.−𝑙 (36) Also 𝐶₂>0 if 𝑤²_𝑖>𝑑_𝑖+1ℎ_𝑖𝑤³_𝑖𝜋𝑓_𝑖+1.−𝑙 (37) Thus for a curve constrained by a line, the parameters must satisfy 𝑤⁰_𝑖,𝑤³_𝑖𝑤>0,_1𝑖>max0,−𝑑_𝑖ℎ_𝑖𝑤⁰_𝑖𝜋𝑓_𝑖,𝑤−𝑙_2𝑖𝑑>max0,^𝑖+1ℎ_𝑖𝑤³_𝑖𝜋𝑓_𝑖+1.−𝑙 (38) Equation (38) can also be expressed as 𝑤⁰_𝑖,𝑤³_𝑖𝑤>0,_1𝑖=𝜇_𝑖+max0,−𝑑_𝑖ℎ_𝑖𝑤⁰_𝑖𝜋𝑓_𝑖−𝑙,𝜇_𝑖𝑤>0,_2𝑖=𝜈_𝑖𝑑+max0,^𝑖+1ℎ_𝑖𝑤³_𝑖𝜋𝑓_𝑖+1−𝑙,𝜈_𝑖>0. (39) The usefulness of the developed scheme is shown by taking data sets lying above a given line. The data set in Table 5 lies above line 𝑦=0.06𝑥+0.02, whereas the data set given in Table 6 lies above the line 𝑦=0.05𝑥+0.23. Figures 9 and 11 are produced by taking the values of the shape parameters on trial and error basis. These figures depict that the curves do not lie above the respective given straight lines. To remove this drawback, curves in Figures 10 and 12 are generated by using the constrained curve scheme developed in the previous theorem. It is clearly shown that the curves not only lie above their same respective lines but also can be made as smooth as required.

Table 5 

2D data set lying above the line 𝑦=0.06𝑥+0.02.

Table 6 

2D data set lying above the line  𝑦=0.05𝑥+0.23.

Figure 9 

Rational cubic trigonometric curve lying below the given line.

Figure 10 

𝐶1 rational cubic trigonometric curve lying above the given line with different values of free parameters.

Figure 11 

Rational cubic trigonometric curve lying below the given line.

Figure 12 

𝐶1 piecewise rational cubic trigonometric curve constrained by a given line.

6. Conclusion and Future Plan

A 𝐶¹ piecewise rational cubic trigonometric spline is discussed in this paper to address the problem of scientific data visualization. Four positive shape parameters are used in the description of positive, monotone, and constrained curve interpolation schemes. Two of these four shape parameters are constrained to preserve the shape of data and the other two are left free for the designer to alter the shape of curves in order to look like as he wants them to be. The presented scheme works well for both equally and unequally spaced data. It is tested for different data sets to show its usefulness in curve construction. In future this scheme will be extended to rational bicubic surface interpolation scheme to generate positive, monotone, and constrained surfaces.

Acknowledgments

The authors are grateful to the anonymous referees for their valuable comments which improved this paper significantly. This work is supported by School of Mathematical Sciences, Universiti Sains Malaysia.