Positive Data Visualization Using Trigonometric Function

Abstract

A 𝐶¹ piecewise rational trigonometric cubic function with four shape parameters has been constructed to address the problem of visualizing positive data. Simple data-dependent constraints on shape parameters are derived to preserve positivity and assure smoothness. The method is then extended to positive surface data by rational trigonometric bicubic function. The order of approximation of developed interpolant is 𝑂(ℎ³_𝑖).

1. Introduction

Data visualization is the mechanism to communicate information by means of graphs, images, diagrams, and animations. It is extensively used in interactive simulation, geometrical design, geometric modeling, and computer-aided geometric design. It is an efficacious way to abridge complexity of data and facilitates prompt understanding of data.

The three significant features of data are convexity, monotonicity, and positivity. Either of these features arises in the data whether it is a result of physical process or chemical experiment, and so forth. Plenty of spline functions exist which can produce smooth and visibly pleasant curves but incapable to visualize the inherited shape (convexity, monotonicity, and positivity) of given data. In this paper, the positive data visualization of both curve and surface data is addressed by a rational trigonometric cubic function.

The objective of this paper is to preserve duly emphasized characteristic of data that is, positivity. Asim and Brodlie [1] developed a piecewise rational cubic function to preserve the positivity of positive data. In [1], if the interpolating function did not preserve the positivity in a subinterval, then the authors inserted extra knot to improve this matter. Butt and Brodlie [2] developed a piecewise rational cubic interpolant to preserve the shape of positive data by interval subdivision technique. Goodman et al. [3] constructed nonplanar shape preserving interpolating curve scheme. They obtained a curve through an optimization process involving some fairness criteria, in order to achieve curve by 𝐺² piecewise rational cubic function. Goodman [4] surveyed the shape preserving interpolating algorithms for 2D data. Han [5] presented the cubic trigonometric polynomial curves with shape parameters. It was investigated that partition of knot vector and value of free parameter affect the order of continuity of trigonometric polynomial curves with shape parameters. These cubic trigonometric polynomial curves were approximation of cubic B-spline curves but more convergent to control polygon. Their degree could be reduced to quadratic for a particular value of shape parameter.

Brodlie et al. [6] used piecewise bicubic function to preserve the shape of positive data arranged over the rectangular mesh. They developed sufficient conditions in the term of the first and mixed partial derivatives at the rectangular grid points to preserve positivity. Duan et al. [7] developed a bivariate rational interpolant with four shape parameters in each rectangular patch. The developed interpolant was 𝐶¹ for equally spaced data with a suitable choice of shape parameters. The sufficient restrictions were developed on shape parameters for constrained interpolation of data. M. Z. Hussain and M. Hussain [8] constructed a local positivity preserving scheme for positive data by making constraints on free parameters in the account of rational bicubic partially blended patches. In [8], the authors also developed the constraints on parameters to preserve the shape of data that is lying above the plane. The user did not enjoy the liberty to refine the curves and surface as desired in these schemes. Hussain and Sarfraz [9] developed a piecewise rational cubic function with four families of parameters to preserve the shape of positive data. Further, the authors extended a rational cubic function into rational bicubic function with eight free parameters for the data arranged over the rectangular grid. In [9], simple sufficient conditions were derived on these free parameters to preserve the shape of positive data. The scheme seemed to be computationally expansive.

Duan et al. [10] discussed the rate of convergence of a rational spline with two shape parameters. The range of optimal error coefficient was determined. The jump in the curvature was also studied for uniform data.

In this paper, an alternative cubic trigonometric function is used to preserve positivity of data. The developed scheme has ample useful aspects. It produces 𝐶¹ interpolant. No extra knots are inserted between any two knots to preserve positivity. The developed scheme works for both equally and unequally spaced data. Positivity is attained by imposing the data-dependent constraints on the free parameters rather than assuming certain functional values.

The work in this paper is set up in such a way that Section 2 elucidates the construction of the 𝐶¹ rational trigonometric cubic function to be used in curve scheme. Section 3 discusses the error of approximation. Section 4 extends the 𝐶¹ rational trigonometric cubic function to a 𝐶¹ rational trigonometric bicubic function. Constraints are developed on free parameters in Section 5 to preserve the positive shape of curve and surface data. Finally, Section 6 concludes the paper.

2. 𝐶¹ Rational Trigonometric Cubic Function

In this section, the 𝐶¹ piecewise rational trigonometric cubic function is developed.

The curve data under consideration over the interval [𝑎,𝑏] is {(𝑥_𝑖,𝑦_𝑖),𝑖=0,1,2,…,𝑛}, where the partition of data is 𝑥₀<𝑥₁<𝑥₂<⋯<𝑥_𝑛. The rational trigonometric cubic function over each subinterval 𝐼_𝑖=[𝑥_𝑖,𝑥_𝑖+1] is defined as: 𝑆_𝑖(𝑥)=𝑝_𝑖(𝜃)𝑞_𝑖(𝜃), (2.1) with 𝑝_𝑖(𝜃)=𝛼_𝑖𝑓_𝑖(1−sin(𝜃))³+𝛽_𝑖𝑓_𝑖+2ℎ_𝑖𝛼_𝑖𝑑_𝑖𝜋sin(𝜃)(1−sin(𝜃))²+𝛾_𝑖𝑓_𝑖+1−2ℎ_𝑖𝛿_𝑖𝑑_𝑖+1𝜋cos(𝜃)(1−cos(𝜃))²+𝛿_𝑖𝑓_𝑖+1(1−cos(𝜃))³,𝑞_𝑖(𝜃)=𝛼_𝑖(1−sin(𝜃))³+𝛽_𝑖sin(𝜃)(1−sin(𝜃))²+𝛾_𝑖cos(𝜃)(1−cos(𝜃))²+𝛿_𝑖(1−cos(𝜃))³, (2.2) where 𝜋𝜃=2𝑥−𝑥_𝑖ℎ_𝑖,ℎ_𝑖=𝑥_𝑖+1−𝑥_𝑖. (2.3)

The rational trigonometric cubic function (2.1) has the following properties: 𝑆𝑥_𝑖=𝑓_𝑖,𝑆𝑥_𝑖+1=𝑓_𝑖+1,𝑆⁽¹⁾𝑥_𝑖=𝑑_𝑖,𝑆⁽¹⁾𝑥_𝑖+1=𝑑_𝑖+1. (2.4)

Here, 𝑆⁽¹⁾ and 𝑑_𝑖 are the derivatives with respect to 𝑥 and computed derivatives at knots 𝑥_𝑖. The values of 𝑑_𝑖 can be computed by any numerical scheme if not given with data. 𝑆(𝑥)∈𝐶¹[𝑥₀,𝑥_𝑛] has 𝛼_𝑖 and 𝛿_𝑖 as free parameters.

3. Error Estimation of Interpolation

This section studies the approximation properties of rational trigonometric cubic function (2.1). It is assumed that the data is generated from third-order continuously differentiable function 𝑓(𝑥)∈𝐶³[𝑥₀,𝑥_𝑛].Since the developed interpolation in Section 2 is local, the error of approximation is computed in the subinterval [𝑥_𝑖,𝑥_𝑖+1]. The absolute error is expressed in terms of Peono-Kernel [10] as follows: 𝑓(𝑥)−𝑆_𝑖(𝑥≤1)2𝑓³(𝑥)^𝑥𝑖+1_𝑥𝑖𝑅_𝑥(𝑥−𝜏)²₊𝑑𝜏, (3.1) where 𝑅_𝑥[(𝑥−𝜏)²₊] is the Peono-Kernel. 𝑅_𝑥[(𝑥−𝜏)²₊]=𝑟(𝜏,𝑥) for 𝑥_𝑖<𝜏<𝑥 and 𝑅_𝑥[(𝑥−𝜏)²₊]=𝑠(𝜏,𝑥)for 𝑥<𝜏<𝑥_𝑖+1. Therefore, the integral involved in (3.1) can be expressed as ^𝑥𝑖+1_𝑥𝑖𝑅_𝑥(𝑥−𝜏)²₊𝑑𝜏=^𝑥_𝑥𝑖|𝑟(𝜏,𝑥)|𝑑𝜏+^𝑥𝑖+1_𝑥|𝑠(𝜏,𝑥)|𝑑𝜏. (3.2)

For the 𝐶¹ rational trigonometric cubic function (2.1), 𝑟(𝜏,𝑥) and 𝑠(𝜏,𝑥) have the value 𝑟(𝜏,𝑥)=(𝑥−𝜏)²−1𝑞_𝑖(𝜃)𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃𝑥_𝑖+1−𝜏²−4ℎ_𝑖𝛿_𝑖𝐵₂𝜋𝑥_𝑖+1,𝑠()1−𝜏𝜏,𝑥=−𝑞_𝑖(𝜃)𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃𝑥_𝑖+1−𝜏²−4ℎ_𝑖𝛿_𝑖𝐵₂𝜋𝑥_𝑖+1,𝐵−𝜏₂=cos(𝜃)(1−cos(𝜃))²,𝐵₃=(1−cos(𝜃))³. (3.3)

To compute the integral of absolute values in (3.1), the roots of 𝑟(𝜏,𝑥) and 𝑠(𝜏,𝑥) are calculated. The roots of 𝑟(𝑥,𝑥) in [0,1] are 𝜃=1, 𝜃^∗=1−2𝛿_𝑖/𝜋(𝛾_𝑖−𝛿_𝑖).

The roots of 𝑟(𝜏,𝑥)=0 are 𝜏₁=𝑥+ℎ_𝑖((𝐵−𝐷)/𝐴) and 𝜏₂=𝑥+ℎ_𝑖((𝐵+𝐷)/𝐴), where 𝐴=2𝛼_𝑖𝐵₀+𝛽_𝑖𝐵₁,𝐵=4𝐵₂𝛿_𝑖−2𝜋𝐵₂𝛾_𝑖+𝐵₃𝛿_𝑖(1−𝜃)𝜋,𝐷=4𝛼𝜎−^𝑖𝐵₀+𝛽_𝑖𝐵₁4𝐵₂𝛿_𝑖(1−𝜃)−𝜋𝐵₂𝛾_𝑖+𝐵₃𝛿_𝑖(1−𝜃)²𝜋,𝜎=4𝐵₂𝛿_𝑖−2𝜋𝐵₂𝛾_𝑖+𝐵₃𝛿_𝑖(1−𝜃)𝜋²𝐵₀=(1−sin(𝜃))³,𝐵₁=sin(𝜃)(1−sin(𝜃))². (3.4)𝜎 denotes [(4𝐵₂𝛿_𝑖−2𝜋(𝐵₂𝛾_𝑖+𝐵₃𝛿_𝑖)(1−𝜃))/𝜋]².

The roots of 𝑠(𝜏,𝑥) are 𝜏_3=𝑥𝑖+1 and 𝜏₄=𝑥_𝑖+1−4ℎ_𝑖𝛿_𝑖𝐵₂/𝜋(𝛾_𝑖𝐵₂−𝛿_𝑖𝐵₃).

The following possible cases arise.

Case 1. 0≤𝛾_𝑖≤1,0≤𝜃≤1, (3.1) takes the form 𝑓(𝑥)−𝑆_𝑖(𝑥≤1)2𝑓³(𝜏)ℎ³_𝑖𝜔₁𝛿_𝑖,𝛾_𝑖,𝜔,𝜃₁𝛿_𝑖,𝛾_𝑖,𝜃=^𝑥_𝑥𝑖|𝑟(𝜏,𝑥)|𝑑𝜏+^𝑥𝑖+1_𝑥|𝑠(=1𝜏,𝑥)|𝑑𝜏𝑞_𝑖(𝜃)2𝛾_𝑖𝐵₂𝜃²−𝜋(1−𝜃)𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃𝜃²𝜋+4𝛾_𝑖𝐵₂(1−𝜃)𝜃−𝜋𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃(1−𝜃)²𝜃𝜋+𝛼_𝑖𝐵₀+𝛽_𝑖𝐵₁𝜃³3+(1−𝜃)²6𝛾_𝑖𝐵₂−𝜋𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃(1−𝜃).3𝜋 (3.5)

Case 2. 𝛾_𝑖>1+2/𝜋𝛿_𝑖,0≤𝜃≤𝜃^∗, (3.1) takes the form 𝑓(𝑥)−𝑆_𝑖(𝑥≤1)2𝑓³(𝜏)ℎ³_𝑖𝜔₂𝛿_𝑖,𝛾_𝑖,𝜔,𝜃₂𝛿_𝑖,𝛾_𝑖,𝜃=^𝑥_𝑥𝑖|𝑟(𝜏,𝑥)|𝑑𝜏+^𝑥𝑖+1_𝑥|𝑠(=1𝜏,𝑥)|𝑑𝜏𝑞_𝑖(𝜃)−2𝛼_𝑖𝐵₀+𝛽_i𝐵₁3𝐵−𝐷𝐴³+4𝛿_𝑖𝐵₂−2𝜋(1−𝜃)𝛾_i𝐵₂+𝐵₃𝜋𝐵−𝐷𝐴²−8𝛿_𝑖𝐵₂(1−𝜃)−2𝜋𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃(1−𝜃)²𝜋𝐵−𝐷𝐴−𝛼_𝑖𝐵₀+𝛽_𝑖𝐵₁𝜃³3−2𝛿_𝑖𝐵₂−(1−𝜃)𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃𝜋𝜃²−4𝛿_𝑖𝐵₂(1−𝜃)−𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃(1−𝜃)²𝜋𝜃+64𝛿_𝑖𝐵₂3𝜋³𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃²−2𝛿_𝑖𝐵₂(1−𝜃)²𝜋+(1−𝜃)³𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃3. (3.6)

Case 3. 𝜃^∗≤𝜃≤1, (3.1) takes the form 𝑓(𝑥)−𝑆_𝑖(𝑥≤1)2𝑓³(𝜏)ℎ³_𝑖𝜔₃𝛿_𝑖,𝛾_𝑖,𝜃, (3.7) where 𝜔₃𝛿_𝑖,𝛾_𝑖,𝜃=^𝑥_𝑥𝑖|𝑟(𝜏,𝑥)|𝑑𝜏+^𝑥𝑖+1_𝑥|𝑠(=1𝜏,𝑥)|𝑑𝜏𝑞_𝑖(𝜃)2𝛼_i𝐵₀+𝛽_i𝐵₁3𝐵−𝐷𝐴³+4𝛿_𝑖𝐵₂−2𝜋(1−𝜃)𝛾_i𝐵₂+𝐵₃𝜋𝐵−𝐷𝐴²+8𝛿_𝑖𝐵₂(1−𝜃)−2𝜋𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃(1−𝜃)²𝜋𝐵−𝐷𝐴−2𝛼_𝑖𝐵₀+𝛽_𝑖𝐵₁3𝐵+𝐷𝐴³+4𝛿_𝑖𝐵₂−2𝜋(1−𝜃)𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃𝜋𝐵+𝐷𝐴²+𝛼_i𝐵₀+𝛽_i𝐵₁𝜃³3+8𝛿_𝑖𝐵₂(1−𝜃)−2𝜋𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃(1−𝜃)²𝜋𝐵+𝐷𝐴+2𝛿_𝑖𝐵₂−𝜋(1−𝜃)𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃𝜋𝜃²+4𝛿_𝑖𝐵₂(1−𝜃)−𝜋(1−𝜃)²𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃𝜋+6𝛿_𝑖𝐵₂−𝜋𝛾_𝑖𝐵₂+𝛿_𝑖𝐵₃(1−𝜃)()3𝜋1−𝜃². (3.8)
The above can be summarized as follows.

Theorem 3.1. The error of 𝐶¹ rational trigonometric cubic function (2.1), for𝑓(𝑥)∈𝐶³[𝑥₀,𝑥_𝑛], in each subinterval [𝑥_𝑖,𝑥_𝑖+1] is 𝑓(𝑥)−𝑆_𝑖(𝑥≤1)2𝑓³(𝑥)ℎ³_𝑖𝑐_𝑖,𝑐_𝑖=max_0≤𝜃≤1𝜔𝛿_𝑖,𝛾_𝑖,𝜔𝛿,𝜃_𝑖,𝛾_𝑖,𝜃=max𝜔₁𝛿_𝑖,𝛾_𝑖,𝜃,0≤𝛾_𝑖≤1,0≤𝜃≤1,max𝜔₂𝛿_𝑖,𝛾_𝑖,𝜃,𝛾_𝑖2>1+𝜋𝛿_𝑖,0≤𝜃≤𝜃^∗,max𝜔₃𝛿_𝑖,𝛾_𝑖,𝜃,𝜃^∗≤𝜃≤1. (3.9)

4. Rational Trigonometric Bicubic Function

The rational trigonometric cubic function (2.1) is extended to a rational bicubic function defined over a rectangular mesh 𝐷=[𝑥₀,𝑥_𝑚]×[𝑦₀,𝑦_𝑛]. Let 𝑑∶𝑎=𝑥₀<𝑥₁<⋯<𝑥_𝑚=𝑏 be a partition of [𝑎,𝑏] and 𝑑₁∶𝑐=𝑦₀<𝑦₁<⋯<𝑦_𝑛=𝑑 be a partition of [𝑐,𝑑]. Rational trigonometric bicubic function is defined over each rectangular patch [𝑥_𝑖,𝑥_𝑖+1]×[𝑦_𝑗,𝑦_𝑗+1],  𝑖=0,1,…,𝑚−1;𝑗=0,1,…,𝑛−1 as follows: 𝑆𝑥,𝑦=𝑆_𝑖,𝑗𝑥,𝑦=𝐴_𝑖(𝜃)𝐹𝑖,𝑗𝐴^𝑇_𝑗𝜑, (4.1) where 𝐹=𝐹𝑖,𝑗^𝑖,𝑗𝐹_𝑖,𝑗+1𝐹^𝑦_𝑖,𝑗𝐹^𝑦_𝑖,𝑗+1𝐹_𝑖+1,𝑗𝐹_{𝑖+1,𝑗+1}𝐹^𝑦_𝑖+1,𝑗𝐹^𝑦_{𝑖+1,𝑗+1}𝐹^𝑥_𝑖,𝑗𝐹^𝑥_𝑖,𝑗+1𝐹^𝑥𝑦_𝑖,𝑗𝐹^𝑥𝑦_𝑖,𝑗+1𝐹^𝑥_𝑖+1,𝑗𝐹^𝑥_{𝑖+1,𝑗+1}𝐹^𝑥𝑦_𝑖+1,𝑗𝐹^𝑥𝑦_{𝑖+1,𝑗+1},𝐴_𝑖(𝜃)=𝑎₀(𝜃)𝑎₁(𝜃)𝑎₂(𝜃)𝑎₃(𝜃),𝐴_𝑗𝜑=𝑎₀𝜑𝑎₁𝜑𝑎₂𝜑𝑎₃𝜑,𝑎₀(𝜃)=𝛼_𝑖,𝑗(1−sin(𝜃))³+𝛽_𝑖,𝑗sin(𝜃)(1−sin(𝜃))²𝑞_𝑖(𝜃),𝑎₁(𝜃)=𝛾_𝑖,𝑗cos(𝜃)(1−cos(𝜃))²+𝛿_𝑖,𝑗(1−cos(𝜃))³𝑞_𝑖(𝜃),𝑎₂(𝜃)=2ℎ_𝑖𝛼_𝑖,𝑗sin(𝜃)(1−sin(𝜃))²𝜋𝑞_𝑖(𝜃),𝑎₃(𝜃)=−2ℎ_𝑖𝛿_𝑖,𝑗cos(𝜃)(1−cos(𝜃))²𝜋𝑞_𝑖(𝜃),𝑞_𝑖(𝜃)=𝛼_𝑖,𝑗(1−sin(𝜃))³+𝛽_𝑖,𝑗sin(𝜃)(1−sin(𝜃))²+𝛾_𝑖,𝑗cos(𝜃)(1−cos(𝜃))²+𝛿_𝑖,𝑗(1−cos(𝜃))³,𝑎₀𝜑=𝛼_𝑖,𝑗1−sin(𝜑)³+𝛽_𝑖,𝑗sin𝜑1−sin(𝜑)²𝑞_𝑗𝜑,𝑎₁𝜑=𝛾_𝑖,𝑗cos𝜑1−cos𝜑²+𝛿_𝑖,𝑗1−cos(𝜑)³𝑞_𝑗𝜑,𝑎₂𝜑=2ℎ_𝑗𝛼_𝑖,𝑗sin𝜑1−sin𝜑²𝜋𝑞_𝑗𝜑,𝑎₃𝜑=−2ℎ_𝑗𝛿_𝑖,𝑗cos𝜑1−cos𝜑²𝜋𝑞_𝑗𝜑,𝑞_𝑗𝜑=𝛼_𝑖,𝑗1−sin𝜑³+𝛽_𝑖,𝑗sin𝜑1−sin𝜑²+𝛾_𝑖,𝑗cos𝜑1−cos𝜑²+𝛿_𝑖,𝑗1−cos𝜑³,𝜋𝜑=2𝑦−𝑦_𝑗ℎ_𝑗. (4.2)

The entries of 𝐹(𝑖,𝑗) are the first and mixed partial derivatives at the four corner positions of the cubic patch.

4.1. 𝐶¹ Rational Trigonometric Bicubic Function

The rational trigonometric function defined in (3.2) interpolates the data values 𝐹_𝑖,𝑗 and partial derivatives 𝐹^𝑥_𝑖,𝑗, 𝐹^𝑦_𝑖,𝑗, 𝐹^𝑥𝑦_𝑖,𝑗 defined at four corners of rectangular patch, that is, 𝑆𝑥_𝑖,𝑦_𝑗=𝐹_𝑖,𝑗,𝜕𝑆𝑥_𝑖,𝑦_𝑗𝜕𝑥=𝐹^𝑥_𝑖,𝑗,𝜕𝑆𝑥_𝑖,𝑦_𝑗𝜕𝑦=𝐹^𝑦_𝑖,𝑗,𝜕𝑆𝑥_𝑖,𝑦_𝑗𝜕𝑥𝜕𝑦=𝐹^𝑥𝑦_𝑖,𝑗. (4.3)

Since each rectangular patch is bounded by four boundary curves so to blend the rectangular patches to generate a 𝐶¹ continuous surface, following sufficient conditions must be satisfied along the four boundaries of each rectangular patch: 𝜕𝑆_𝑖,𝑗𝑥_𝑖+1,𝑦𝜕𝑥_𝜃=𝜋/2−𝜕𝑆_𝑖+1,𝑗𝑥_𝑖+1,𝑦𝜕𝑥_𝜃=0=0,𝜕𝑆_{𝑖−1,𝑗}𝑥_𝑖,𝑦𝜕𝑥_𝜃=𝜋/2−𝜕𝑆_𝑖,𝑗𝑥_𝑖,𝑦𝜕𝑥_𝜃=0=0,𝜕𝑆_𝑖,𝑗𝑥,𝑦_𝑗+1𝜕𝑦_𝜑=𝜋/2−𝜕𝑆_𝑖,𝑗+1𝑥,𝑦_𝑗+1𝜕𝑦_𝜑=0=0,𝜕𝑆_{𝑖,𝑗−1}𝑥,𝑦_𝑗𝜕𝑦_𝜑=𝜋/2−𝜕𝑆_𝑖,𝑗𝑥,𝑦_𝑗𝜕𝑦_𝜑=0=0. (4.4)

The entities ℎ_𝑖, ℎ_𝑗, 𝐹_𝑖,𝑗, 𝐹^𝑥_𝑖,𝑗, 𝐹^𝑦_𝑖,𝑗,𝐹^𝑥𝑦_𝑖,𝑗, 𝑖=0,1,2,…,𝑚;𝑗=0,1,2,…,𝑛 are assumed fixed then, we have the following observations: 𝜕𝑆_𝑖,𝑗(𝑥_𝑖+1,𝑦)/𝜕𝑥|_𝜃=𝜋/2−(𝜕𝑆_𝑖+1,𝑗(𝑥_𝑖+1,𝑦))/𝜕𝑥|_𝜃=0=0 if 𝜋ℎ_𝑖𝛼_𝑖+1,𝑗𝛽_𝑖+1,𝑗𝛿²_𝑖,𝑗𝐹_𝑖+1,𝑗𝛼_𝑖,𝑗−𝛼_𝑖+1,𝑗=0,2𝜋ℎ_𝑖ℎ_𝑖+1𝛼²_𝑖+1,𝑗𝛿²_𝑖,𝑗𝐹^𝑥_𝑖+1,𝑗𝛽_𝑖,𝑗−𝛽_𝑖+1,𝑗+4ℎ_𝑖ℎ_𝑖+1ℎ_𝑗𝛼²_𝑖+1,𝑗𝛿²_𝑖,𝑗𝐹^𝑥𝑦_𝑖+1,𝑗𝛼_𝑖,𝑗−𝛼_𝑖+1,𝑗=0,2𝜋ℎ_𝑖ℎ_𝑖+1𝛼²_𝑖+1,𝑗𝛿²_𝑖,𝑗𝐹^𝑥_{𝑖+1,𝑗+1}𝛾_𝑖,𝑗−𝛾_𝑖+1,𝑗+4ℎ_𝑖ℎ_𝑖+1ℎ_𝑗𝛼²_𝑖+1,𝑗𝛿²_𝑖,𝑗𝐹^𝑥𝑦_{𝑖+1,𝑗+1}𝛿_𝑖,𝑗−𝛿_𝑖+1,𝑗=0,2𝛼²_𝑖+1,𝑗ℎ_𝑖ℎ_𝑖+1𝛿²_𝑖,𝑗𝐹^𝑥_{𝑖+1,𝑗+1}𝛿_𝑖,𝑗−𝛿_𝑖+1,𝑗=0, (4.5) then(1)𝜋ℎ_𝑖𝛼_𝑖+1,𝑗𝛽_𝑖+1,𝑗𝛿²_𝑖,𝑗𝐹_𝑖+1,𝑗(𝛼_𝑖,𝑗−𝛼_𝑖+1,𝑗)=0 if 𝛼_𝑖,𝑗=𝛼_𝑖+1,𝑗,(2)2𝜋ℎ_𝑖ℎ_𝑖+1𝛼²_𝑖+1,𝑗𝛿²_𝑖,𝑗𝐹^𝑥_𝑖+1,𝑗(𝛽_𝑖,𝑗−𝛽_𝑖+1,𝑗)+4ℎ_𝑖ℎ_𝑖+1ℎ_𝑗𝛼²_𝑖+1,𝑗𝛿²_𝑖,𝑗𝐹^𝑥𝑦_𝑖+1,𝑗(𝛼_𝑖,𝑗−𝛼_𝑖+1,𝑗)=0 if 𝛼_𝑖,𝑗=𝛼_𝑖+1,𝑗, 𝛽_𝑖,𝑗=𝛽_𝑖+1,𝑗,(3)2𝜋ℎ_𝑖ℎ_𝑖+1𝛼²_𝑖+1,𝑗𝛿²_𝑖,𝑗𝐹^𝑥_{𝑖+1,𝑗+1}(𝛾_𝑖,𝑗−𝛾_𝑖+1,𝑗)+4ℎ_𝑖ℎ_𝑖+1ℎ_𝑗𝛼²_𝑖+1,𝑗𝛿²_𝑖,𝑗𝐹^𝑥𝑦_{𝑖+1,𝑗+1}(𝛿_𝑖,𝑗−𝛿_𝑖+1,𝑗)=0 if 𝛾_𝑖,𝑗=𝛾_𝑖+1,𝑗, 𝛿_𝑖,𝑗=𝛿_𝑖+1,𝑗,(4)2𝛼²_𝑖+1,𝑗ℎ_𝑖ℎ_𝑖+1𝛿²_𝑖,𝑗𝐹^𝑥_{𝑖+1,𝑗+1}(𝛿_𝑖,𝑗−𝛿_𝑖+1,𝑗)=0 if 𝛿_𝑖,𝑗=𝛿_𝑖+1,𝑗. Consider that 𝜕𝑆_{𝑖−1,𝑗}𝑥_𝑖,𝑦𝜕𝑥_𝜃=𝜋/2−𝜕𝑆_𝑖,𝑗𝑥_𝑖,𝑦𝜕𝑥_𝜃=0=0 (4.6)if 2𝛼²_𝑖,𝑗ℎ_𝑖ℎ_𝑖−1𝛿²_{𝑖−1,𝑗}𝐹^𝑥_𝑖,𝑗𝛼_{𝑖−1,𝑗}−𝛼_𝑖,𝑗=0,2𝜋𝛼²_𝑖,𝑗ℎ_𝑖ℎ_𝑖−1𝛿²_{𝑖−1,𝑗}𝐹^𝑥_𝑖,𝑗𝛽_{𝑖−1,𝑗}−𝛽_𝑖,𝑗+4𝛼²_𝑖,𝑗ℎ_𝑖ℎ_𝑖−1𝛿²_{𝑖−1,𝑗}𝐹^𝑥_𝑖,𝑗𝛼_{𝑖−1,𝑗}−𝛼_𝑖,𝑗=0,4𝛼²_𝑖,𝑗ℎ_𝑖ℎ_𝑖−1𝛿²_{𝑖−1,𝑗}𝐹^𝑥𝑦_𝑖,𝑗+1𝛿_{𝑖−1,𝑗}−𝛿_𝑖,𝑗+2𝜋𝛼²_𝑖,𝑗ℎ_𝑖ℎ_𝑖−1𝛿²_{𝑖−1,𝑗}𝐹^𝑥_𝑖,𝑗𝛾_{𝑖−1,𝑗}−𝛾_𝑖,𝑗=0,2𝛼²_𝑖,𝑗ℎ_𝑖ℎ_𝑖−1𝛿²_{𝑖−1,𝑗}𝐹^𝑥_𝑖,𝑗+1𝛿_{𝑖−1,𝑗}−𝛿_𝑖,𝑗=0, (4.7) then(1)2𝛼²_𝑖,𝑗ℎ_𝑖ℎ_𝑖−1𝛿²_{𝑖−1,𝑗}𝐹^𝑥_𝑖,𝑗(𝛼_{𝑖−1,𝑗}−𝛼_𝑖,𝑗)=0 if 𝛼_{𝑖−1,𝑗}=𝛼_𝑖,𝑗,(2)2𝜋𝛼²_𝑖,𝑗ℎ_𝑖ℎ_𝑖−1𝛿²_{𝑖−1,𝑗}𝐹^𝑥_𝑖,𝑗(𝛽_{𝑖−1,𝑗}−𝛽_𝑖,𝑗)+4𝛼²_𝑖,𝑗ℎ_𝑖ℎ_𝑖−1𝛿²_{𝑖−1,𝑗}𝐹^𝑥_𝑖,𝑗(𝛼_{𝑖−1,𝑗}−𝛼_𝑖,𝑗))=0 if 𝛽_{𝑖−1,𝑗}=𝛽_𝑖,𝑗 and 𝛼_{𝑖−1,𝑗}=𝛼_𝑖,𝑗,(3)4𝛼²_𝑖,𝑗ℎ_𝑖ℎ_𝑖−1𝛿²_{𝑖−1,𝑗}𝐹^𝑥𝑦_𝑖,𝑗+1(𝛿_{𝑖−1,𝑗}−𝛿_𝑖,𝑗)+2𝜋𝛼²_𝑖,𝑗ℎ_𝑖ℎ_𝑖−1𝛿²_{𝑖−1,𝑗}𝐹^𝑥_𝑖,𝑗(𝛾_{𝑖−1,𝑗}−𝛾_𝑖,𝑗)=0 if 𝛿_{𝑖−1,𝑗}=𝛿_𝑖,𝑗 and 𝛾_{𝑖−1,𝑗}=𝛾_𝑖,𝑗,(4)2𝛼²_𝑖,𝑗ℎ_𝑖ℎ_𝑖−1𝛿²_{𝑖−1,𝑗}𝐹^𝑥_𝑖,𝑗+1(𝛿_{𝑖−1,𝑗}−𝛿_𝑖,𝑗)=0 if 𝛿_{𝑖−1,𝑗}=𝛿_𝑖,𝑗. Consider that 𝜕𝑆_𝑖,𝑗𝑥,𝑦_𝑗+1𝜕𝑦_𝜑=𝜋/2−𝜕𝑆_𝑖,𝑗+1𝑥,𝑦_𝑗+1𝜕𝑦_𝜑=0=0 (4.8)if 𝛼²_𝑖,𝑗+1𝛿²_𝑖,𝑗𝐹^𝑦_𝑖,𝑗+1𝛼_𝑖,𝑗−𝛼_𝑖,𝑗+1=0,2ℎ_𝑖𝛿²_𝑖,𝑗𝐹^𝑥𝑦_𝑖,𝑗+1𝛼_𝑖,𝑗−𝛼_𝑖,𝑗+1𝜋𝛼=0,_2𝑖,𝑗+1𝛿²_𝑖,𝑗𝐹^𝑦_{𝑖+1,𝑗+1}𝛾_𝑖,𝑗−𝛾_𝑖,𝑗+1+2ℎ_𝑖𝛼²_𝑖,𝑗+1𝛿²_𝑖,𝑗𝐹^𝑥𝑦_{𝑖+1,𝑗+1}𝛿_𝑖,𝑗−𝛿_𝑖,𝑗+1𝛿=0,_2𝑖,𝑗𝛼²_𝑖,𝑗+1𝐹^𝑦_{𝑖+1,𝑗+1}𝛿_𝑖,𝑗−𝛿_𝑖,𝑗+1=0, (4.9) then(1)𝛼²_𝑖,𝑗+1𝛿²_𝑖,𝑗𝐹^𝑦_𝑖,𝑗+1(𝛼_𝑖,𝑗−𝛼_𝑖,𝑗+1)=0 if 𝛼_𝑖,𝑗=𝛼_𝑖,𝑗+1,(2)2ℎ_𝑖𝛿²_𝑖,𝑗𝐹^𝑥𝑦_𝑖,𝑗+1(𝛼_𝑖,𝑗−𝛼_𝑖,𝑗+1)=0 if 𝛼_𝑖,𝑗=𝛼_𝑖,𝑗+1,(3)𝜋𝛼²_𝑖,𝑗+1𝛿²_𝑖,𝑗𝐹^𝑦_{𝑖+1,𝑗+1}(𝛾_𝑖,𝑗−𝛾_𝑖,𝑗+1)+2ℎ_𝑖𝛼²_𝑖,𝑗+1𝛿²_𝑖,𝑗𝐹^𝑥𝑦_{𝑖+1,𝑗+1}(𝛿_𝑖,𝑗−𝛿_𝑖,𝑗+1)=0 if 𝛾_𝑖,𝑗=𝛾_𝑖,𝑗+1, 𝛿_𝑖,𝑗=𝛿_𝑖,𝑗+1,(4)𝛿²_𝑖,𝑗𝛼²_𝑖,𝑗+1𝐹^𝑦_{𝑖+1,𝑗+1}(𝛿_𝑖,𝑗−𝛿_𝑖,𝑗+1)=0 if 𝛿_𝑖,𝑗=𝛿_𝑖,𝑗+1. Consider that 𝜕𝑆_{𝑖,𝑗−1}𝑥,𝑦_𝑗𝜕𝑦_𝜑=𝜋/2−𝜕𝑆_𝑖,𝑗𝑥,𝑦_𝑗𝜕𝑦_𝜑=0=0 (4.10)if 𝛼²_𝑖,𝑗𝛿²_{𝑖,𝑗−1}𝐹^𝑦_𝑖,𝑗𝛼_{𝑖,𝑗−1}−𝛼_𝑖,𝑗𝜋𝛼=0,_2𝑖,𝑗𝛿²_{𝑖,𝑗−1}𝐹^𝑦_𝑖,𝑗𝛽_{𝑖,𝑗−1}−𝛽_𝑖,𝑗+2ℎ_𝑖𝛼²_𝑖,𝑗𝛿²_{𝑖,𝑗−1}𝐹^𝑥𝑦_𝑖,𝑗𝛼_{𝑖,𝑗−1}−𝛼_𝑖,𝑗𝜋𝛼=0,_2𝑖,𝑗𝛿²_{𝑖,𝑗−1}𝐹^𝑦_𝑖+1,𝑗𝛾_{𝑖,𝑗−1}−𝛾_𝑖,𝑗+2ℎ_𝑖𝛼²_𝑖,𝑗𝛿²_{𝑖,𝑗−1}𝐹^𝑥𝑦_𝑖,𝑗𝛿_{𝑖,𝑗−1}−𝛿_𝑖,𝑗𝛼=0,_2𝑖,𝑗𝛿²_{𝑖,𝑗−1}𝐹^𝑦_𝑖+1,𝑗𝛿_{𝑖,𝑗−1}−𝛿_𝑖,𝑗=0, (4.11) then(1)𝛼²_𝑖,𝑗𝛿²_{𝑖,𝑗−1}𝐹^𝑦_𝑖,𝑗(𝛼_{𝑖,𝑗−1}−𝛼_𝑖,𝑗)=0 if 𝛼_{𝑖,𝑗−1}=𝛼_𝑖,𝑗,(2)𝜋𝛼²_𝑖,𝑗𝛿²_{𝑖,𝑗−1}𝐹^𝑦_𝑖,𝑗(𝛽_{𝑖,𝑗−1}−𝛽_𝑖,𝑗)+2ℎ_𝑖𝛼²_𝑖,𝑗𝛿²_{𝑖,𝑗−1}𝐹^𝑥𝑦_𝑖,𝑗(𝛼_{𝑖,𝑗−1}−𝛼_𝑖,𝑗)=0 if 𝛽_{𝑖,𝑗−1}=𝛽_𝑖,𝑗 and 𝛼_{𝑖,𝑗−1}=𝛼_𝑖,𝑗,(3)𝜋𝛼²_𝑖,𝑗𝛿²_{𝑖,𝑗−1}𝐹^𝑦_𝑖+1,𝑗(𝛾_{𝑖,𝑗−1}−𝛾_𝑖,𝑗)+2ℎ_𝑖𝛼²_𝑖,𝑗𝛿²_{𝑖,𝑗−1}𝐹^𝑥𝑦_𝑖,𝑗(𝛿_{𝑖,𝑗−1}−𝛿_𝑖,𝑗)=0 if 𝛾_{𝑖,𝑗−1}=𝛾_𝑖,𝑗 and 𝛿_{𝑖,𝑗−1}=𝛿_𝑖,𝑗,(4)𝛼²_𝑖,𝑗𝛿²_{𝑖,𝑗−1}𝐹^𝑦_𝑖+1,𝑗(𝛿_{𝑖,𝑗−1}−𝛿_𝑖,𝑗)=0 if 𝛿_{𝑖,𝑗−1}=𝛿_𝑖,𝑗.

The above discussion is summarized as follows.

Theorem 4.1. The piecewise bivariate rational cubic trigonometric function 𝑆(𝑥,𝑦) defined in (4.1) is 𝐶¹ over the whole domain if the shape design parameters satisfy the following relations:(1)𝛼_𝑖,𝑗=𝛼_𝑖, 𝛽_𝑖,𝑗=𝛽_𝑖, 𝛾_𝑖,𝑗=𝛾_𝑖, 𝛿_𝑖,𝑗=𝛿_𝑖, 𝑖=0,1,2,…,𝑚−1 and for all values of 𝑗.(2)𝛼_𝑖,𝑗=𝛼_𝑗, 𝛽_𝑖,𝑗=𝛽_𝑗, 𝛾_𝑖,𝑗=𝛾_𝑗, 𝛿_𝑖,𝑗=𝛿_𝑗, 𝑗=0,1,2,…,𝑛−1 and for all values of 𝑖.

5. Positivity Preserving Techniques

Preserving positivity is the essence of data visualization in many fields of study. There are many physical situations where the entities only have meaning when their values are positive. For instance, population density, probability distribution, amount of rain fall, and resistance of an electric circuit are some of the areas where data values can not be negative. Similarly, terrain modeling, formation of geological crust movement to predict earth quake and volcanic eruptions, fluid dynamic, and carbon dating are few of the fields in which the resulting surface is required to be positive. Therefore, it is customary that interpolating curve and surface inherent the positivity of data. The subsequent subsections aim at developing constraints on the shape parameters in the description of rational trigonometric function and rational trigonometric bicubic function so that the resultant curve and surface are positive for a positive data and visibly eye catching.

5.1. Positive Curve Interpolation

The positive curve data is data set {(𝑥_𝑖,𝑓_𝑖),𝑖=0,1,2,…,𝑛} satisfying the condition 𝑓_𝑖>0,𝑖=0,1,2,…,𝑛−1. (5.1)

The piecewise rational trigonometric cubic function defined in (2.1) inherits the positive shape in curve if in each subinterval 𝐼_𝑖=[𝑥_𝑖,𝑥_𝑖+1] the following relation is true 𝑆_𝑖(𝑥)>0,𝑖=0,1,2,…,𝑛−1. (5.2)

The central idea is to impose conditions on free parameters to assure positivity. It can be observed that strictly positive denomination 𝑞_𝑖(𝜃) can be guaranteed if 𝛼_𝑖>0,𝛽_𝑖>0,𝛾_𝑖>0,𝛿_𝑖>0. (5.3)

Thus, the positivity of 𝑆_𝑖(𝑥)=𝑝_𝑖(𝜃)/𝑞_𝑖(𝜃) depends on the positivity of the trigonometric cubic polynomial 𝑝_𝑖(𝜃), and the difficulty level only reduces to the determination of suitable values of 𝛼_𝑖, 𝛽_𝑖, 𝛾_𝑖 and 𝛿_𝑖; for which the polynomial 𝑝_𝑖(𝜃)>0.

𝑝_𝑖(𝜃)>0 if the coefficients 𝐴_𝑖,𝑖=0,1,2,3, of trigonometric basis function are positive. Positivity of these coefficients yields the following result: 𝛽_𝑖>−2ℎ_𝑖𝑑_𝑖𝛼_𝑖𝜋𝑓_𝑖,𝛾_𝑖>2ℎ_𝑖𝑑_𝑖+1𝛿_𝑖𝜋𝑓_𝑖+1. (5.4)

The above discussion can be summarized as follows.

Theorem 5.1. The 𝐶¹ piecewise rational trigonometric cubic function preserve the positivity of positive data if in each subinterval 𝐼_𝑖=[𝑥_𝑖,𝑥_𝑖+1], the parameters 𝛽_𝑖 and 𝛾_𝑖 satisfy the following sufficient conditions: 𝛽_𝑖>Max0,−2ℎ_𝑖𝑑_𝑖𝛼_𝑖𝜋𝑓_𝑖,𝛾_𝑖>Max0,2ℎ_𝑖𝑑_𝑖+1𝛿_𝑖𝜋𝑓_𝑖+1. (5.5)

The above constraints can be rearranged as follows: 𝛽_𝑖=𝑢_𝑖+Max0,−2ℎ_𝑖𝑑_𝑖𝛼_𝑖𝜋𝑓_𝑖,𝑢_𝑖𝛾>0,_𝑖=𝑣_𝑖+Max0,2ℎ_𝑖𝑑_𝑖+1𝛿_𝑖𝜋𝑓_𝑖+1,𝑣_𝑖>0. (5.6)

Proof . The Theorem 5.1 can be easily established by combining (5.3) and (5.4).

5.2. 𝐶¹ Positive Surface Interpolation

Let {𝐹_𝑖,𝑗∶𝑖=0,1,2,…,𝑚;𝑗=0,1,2,…,𝑛}be the positive data, that is, 𝐹_𝑖,𝑗>0,𝑖=0,1,2,…,𝑚;𝑗=0,1,2,…,𝑛. The aim is to construct a 𝐶¹ rational trigonometric bicubic function 𝑆(𝑥,𝑦) on 𝐷=[𝑎,𝑏]×[𝑐,𝑑] such that 𝑆𝑥_𝑖,𝑦_𝑗=𝐹_𝑖,𝑗,𝑆𝑥,𝑦>0forall𝑥,𝑦∈𝐷. (5.7)

The 𝐶¹ rational trigonometric bicubic function 𝑆(𝑥,𝑦) is simplified to the following form 𝑆=𝑎𝑥,𝑦_𝑖,𝑗(1−sin(𝜃))³+𝑏_𝑖,𝑗sin(𝜃)(1−sin(𝜃))²+𝑐_𝑖,𝑗cos(𝜃)(1−cos(𝜃))²+𝑑_𝑖,𝑗(1−cos(𝜃))³𝛼_𝑖(1−sin(𝜃))³+𝛽_𝑖sin(𝜃)(1−sin(𝜃))²+𝛾_𝑖cos(𝜃)(1−cos(𝜃))²+𝛿_𝑖(1−cos(𝜃))³, (5.8) with 𝑎_𝑖,𝑗=(1−sin(𝜃))³𝑞_𝑗𝜑1−sin𝜑³𝐴₀+sin𝜑1−sin𝜑²𝐴₁+cos𝜑1−cos𝜑²𝐴₂+1−cos𝜑³𝐴₃,𝐴₀=𝛼_𝑖𝛼_𝑗𝐹_𝑖,𝑗,𝐴₁=𝛼_𝑖𝛽_𝑗𝐹_𝑖,𝑗+2ℎ_𝑗𝛼_𝑖𝛼_𝑗𝐹^𝑦_𝑖,𝑗,𝐴₂=𝛼_𝑖𝛾_𝑗𝐹_𝑖,𝑗+1−2ℎ_𝑗𝛼_𝑖𝛿_𝑗𝐹^𝑦_𝑖,𝑗+1,𝐴₃=𝛼_𝑖𝛿_𝑗𝐹_𝑖,𝑗+1,𝑏_𝑖,𝑗=sin(𝜃)(1−sin(𝜃))²𝑞_𝑗𝜑1−sin𝜑³𝐸₀+sin𝜑1−sin𝜑²𝐸₁+cos𝜑1−cos𝜑²𝐸₂+1−cos𝜑³𝐸₃,𝐸₀=𝛽_𝑖𝛼_𝑗𝐹_𝑖,𝑗+2ℎ_𝑖𝛼_𝑖𝛼_𝑗𝐹^𝑥_𝑖,𝑗,𝐸₁=𝛽_𝑖𝛽_𝑗𝐹_𝑖,𝑗+1+2ℎ_𝑖𝛼_𝑖𝛽_𝑗𝐹^𝑥_𝑖,𝑗+2ℎ_𝑗𝛽_𝑖𝛿_𝑗𝐹^𝑦_𝑖,𝑗+2ℎ_𝑖𝛼_𝑖𝛼_𝑗𝐹^𝑥𝑦_𝑖,𝑗,𝐸₂=𝛽_𝑖𝛾_𝑗𝐹_𝑖,𝑗+1+2ℎ_𝑖𝛼_𝑖𝛾_𝑗𝐹^𝑥_𝑖,𝑗+1−2ℎ_𝑗𝛽_𝑖𝛿_𝑗𝐹^𝑦_𝑖,𝑗+1+2ℎ_𝑖𝛼_𝑖𝛿_𝑗𝐹^𝑥𝑦_𝑖,𝑗+1,𝐸₃=𝛽_𝑖𝛿_𝑗𝐹_𝑖,𝑗+1+2ℎ_𝑖𝛼_𝑖𝛿_𝑗𝐹^𝑥_𝑖,𝑗+1,𝑐_𝑖,𝑗=cos(𝜃)(1−cos(𝜃))²𝑞_𝑗𝜑1−sin𝜑³𝐶₀+sin𝜑1−sin𝜑²𝐶₁+cos𝜑1−cos𝜑²𝐶₂+1−cos𝜑³𝐶₃,𝐶₀=𝛽_𝑖𝛼_𝑗𝐹_𝑖,𝑗+2ℎ_𝑖𝛼_𝑖𝛼_𝑗𝐹^𝑥_𝑖,𝑗,𝐶₁=𝛾_𝑖𝛽_𝑗𝐹_{𝑖+1,𝑗+1}−2ℎ_𝑖𝛿_𝑖𝛽_𝑗𝐹^𝑥_{𝑖+1,𝑗+1}+2ℎ_𝑗𝛾_𝑖𝛼_𝑗𝐹^𝑦_𝑖+1,𝑗−2ℎ_𝑖𝛿_𝑖𝛼_𝑗𝐹^𝑥𝑦_𝑖+1,𝑗,𝐶₂=𝛾_𝑖𝛾_𝑗𝐹_{𝑖+1,𝑗+1}−2ℎ_𝑖𝛿_𝑖𝛾_𝑗𝐹^𝑥_{𝑖+1,𝑗+1}−2ℎ_𝑗𝛾_𝑖𝛿_𝑗𝐹^𝑦_{𝑖+1,𝑗+1}−2ℎ_𝑖𝛿_𝑖𝛿_𝑗𝐹^𝑥𝑦_{𝑖+1,𝑗+1},𝐶₃=𝛾_𝑖𝛿_𝑗𝐹^𝑦_{𝑖+1,𝑗+1}−2ℎ_𝑖𝛿_𝑖𝛿_𝑗𝐹^𝑥𝑦_{𝑖+1,𝑗+1},𝑑_𝑖,𝑗=(1−cos(𝜃))³𝑞_𝑗𝜑1−sin𝜑³𝐷₀+sin𝜑1−sin𝜑²𝐷₁+cos𝜑1−cos𝜑²𝐷₂+1−cos𝜑³𝐷₃,𝐷₀=𝛿_𝑖𝛼_𝑗𝐹_𝑖+1,𝑗,𝐷₁=𝛿_𝑖𝛽_𝑗𝐹_𝑖+1,𝑗+2ℎ_𝑗𝛿_𝑖𝛼_𝑗𝐹^𝑦_𝑖+1,𝑗,𝐷₂=𝛿_𝑖𝛾_𝑗𝐹_{𝑖+1,𝑗+1}−2ℎ_𝑗𝛿_𝑖𝛿_𝑗𝐹^𝑦_{𝑖+1,𝑗+1},𝐷₃=𝛿_𝑖𝛿_𝑗𝐹_{𝑖+1,𝑗+1}. (5.9)

Since 𝛼_𝑖,𝛽_𝑖,𝛾_𝑖,𝛿_𝑖, 𝛼_𝑗,𝛽_𝑗,𝛾_𝑗,𝛿_𝑗 are assumed positive real numbers and 0≤𝜃≤𝜋/2, 0≤𝜑≤𝜋/2, so 𝑆(𝑥,𝑦)>0if 𝑎_𝑖,𝑗>0,𝑏_𝑖,𝑗>0,𝑐_𝑖,𝑗>0,𝑑_𝑖,𝑗>0. (5.10)𝑎_𝑖,𝑗>0 if 𝐴_𝑖>0,𝑖=0,1,2,3. (5.11)𝐴_𝑖>0,𝑖=0,1,2,3, if 𝛽_𝑗>−2ℎ_𝑗𝛼_𝑗𝐹^𝑦_𝑖,𝑗𝜋𝐹_𝑖,𝑗,𝛾_𝑗>2ℎ_𝑗𝛿_𝑗𝐹^𝑦_𝑖,𝑗+1𝜋𝐹_𝑖,𝑗+1. (5.12) Similarly, 𝑏_𝑖,𝑗>0 if 𝐸_𝑖>0,𝑖=0,1,2,3. (5.13)𝐸_𝑖>0,𝑖=0,1,2,3, if 𝛾_𝑗>2ℎ_𝑗𝛿_𝑗𝐹^𝑦_𝑖,𝑗+1𝜋𝐹_𝑖,𝑗+1,𝛽_𝑖>Max𝐶𝑑₀,𝐶𝑑₁,𝐶𝑑₂,𝐶𝑑₃,𝐶𝑑₄, (5.14) with 𝐶𝑑₀=−2ℎ_𝑗𝛼_𝑗𝐹^𝑦_𝑖,𝑗𝜋𝐹_𝑖,𝑗,𝐶𝑑₁=−2ℎ_𝑖𝛼_𝑖𝐹^𝑥_𝑖,𝑗𝜋𝐹_𝑖,𝑗,𝐶𝑑₂=−2ℎ_𝑖𝛼_𝑖𝐹^𝑥_𝑖,𝑗+1𝜋𝐹_𝑖,𝑗+1,𝐶𝑑₃=−2ℎ_𝑖𝛼_𝑖𝜋𝛽_𝑗𝐹^𝑥_𝑖,𝑗+2ℎ_𝑗𝛼_𝑗𝐹^𝑥𝑦_𝑖,𝑗𝜋²𝐹_𝑖,𝑗𝛽_𝑗+2𝜋ℎ_𝑗𝛼_𝑗𝐹^𝑦_𝑖,𝑗,𝐶𝑑₄=−2ℎ_𝑖𝛼_𝑖𝜋𝛾_𝑗𝐹^𝑥_𝑖,𝑗+1−2ℎ_𝑗𝛿_𝑗𝐹^𝑥𝑦_𝑖,𝑗+1𝜋²𝐹_𝑖,𝑗+1𝛾_𝑗−2𝜋ℎ_𝑗𝛿_𝑗𝐹^𝑦_𝑖,𝑗+1. (5.15) Similarly, 𝑐_𝑖,𝑗>0, implies the following constraints: 𝐶_𝑖>0,𝑖=0,1,2,3. (5.16)𝐶_𝑖>0,𝑖=0,1,2,3, if 𝛽_𝑗>−2ℎ_𝑗𝛼_𝑗𝐹^𝑦_𝑖+1,𝑗𝜋𝐹_𝑖+1,𝑗,𝛾_𝑖>Max𝐶𝑑₅,𝐶𝑑₆,𝐶𝑑₇,𝐶𝑑₈,𝐶𝑑₉, (5.17) where 𝐶𝑑₅=2ℎ_𝑗𝛿_𝑗𝐹^𝑦_{𝑖+1,𝑗+1}𝜋𝐹_{𝑖+1,𝑗+1},𝐶𝑑₆=−2ℎ_𝑖𝛿_𝑖𝐹^𝑥_𝑖+1,𝑗𝜋𝐹_𝑖+1,𝑗,𝐶𝑑₇=2ℎ_𝑖𝛿_𝑖𝐹^𝑥_{𝑖+1,𝑗+1}𝜋𝐹_{𝑖+1,𝑗+1},𝐶𝑑₈=2ℎ_𝑖𝛿_𝑖𝜋𝛽_𝑗𝐹^𝑥_𝑖+1,𝑗+2ℎ_𝑗𝛼_𝑗𝐹^𝑥𝑦_𝑖+1,𝑗𝜋²𝐹_𝑖+1,𝑗𝛽_𝑗+2𝜋ℎ_𝑗𝛼_𝑗𝐹^𝑦_𝑖+1,𝑗,𝐶𝑑₉=2ℎ_𝑖𝛿_𝑖𝜋𝛾_𝑗𝐹^𝑥_{𝑖+1,𝑗+1}−2ℎ_𝑗𝛿_𝑗𝐹^𝑥𝑦_{𝑖+1,𝑗+1}𝜋²𝐹_{𝑖+1,𝑗+1}𝛾_𝑗−2𝜋ℎ_𝑗𝛿_𝑗𝐹^𝑦_{𝑖+1,𝑗+1}, (5.18) and finally, 𝑑_𝑖,𝑗>0 if 𝐷_𝑖>0,𝑖=0,1,2,3. (5.19)𝐷_𝑖>0,𝑖=0,1,2,3, if 𝛽_𝑗>−2𝛼_𝑗ℎ_𝑗𝐹^𝑦_𝑖+1,𝑗𝜋𝐹_𝑖+1,𝑗,𝛾_𝑗>2𝛿_𝑗ℎ_𝑗𝐹^𝑦_{𝑖+1,𝑗+1}𝜋𝐹_{𝑖+1,𝑗+1}. (5.20)

The above conditions can be summarized as follows.

Theorem 5.2. The 𝐶¹ piecewise rational trigonometric bicubic interpolant 𝑆(𝑥,𝑦) defined over the rectangular mesh 𝐷=[𝑥₀,𝑥_𝑛]×[𝑦₀,𝑦_𝑚], in (5.8), is positive if the following sufficient conditions are satisfied 𝑆𝑥_𝑖,𝑦_𝑗=𝐹_𝑖,𝑗,𝛼𝑖=0,1,2,…,𝑚;𝑗=0,1,2…,𝑛,_𝑖>0,𝛼_𝑗>0,𝛿_𝑖>0,𝛿_𝑗𝛽>0,_𝑗>Max0,−2ℎ_𝑗𝛼_𝑗𝐹^𝑦_𝑖,𝑗𝜋𝐹_𝑖,𝑗,−2ℎ_𝑗𝛼_𝑗𝐹^𝑦_𝑖+1,𝑗𝜋𝐹_𝑖+1,𝑗,𝛾_𝑗>Max0,2ℎ_𝑗𝛿_𝑗𝐹^𝑦_𝑖,𝑗+1𝜋𝐹_𝑖,𝑗+1,2ℎ_𝑗𝛿_𝑗𝐹^𝑦_{𝑖+1,𝑗+1}𝜋𝐹_{𝑖+1,𝑗+1},𝛽_𝑖>Max0,𝐶𝑑₀,𝐶𝑑₁,𝐶𝑑₂,𝐶𝑑₃,𝐶𝑑₄,𝛾_𝑖>Max0,𝐶𝑑₅,𝐶𝑑₆,𝐶𝑑₇,𝐶𝑑₈,𝐶𝑑₉, (5.21) where 𝐶𝑑₀,𝐶𝑑₁,𝐶𝑑₂,𝐶𝑑₃,𝐶𝑑₄,𝐶𝑑₅,𝐶𝑑₆,𝐶𝑑₇,𝐶𝑑₈, and 𝐶𝑑₉ have been defined in Section 5.

Proof. By combining (5.12)–(5.20), Theorem 5.2 can be easily obtained.

6. Numerical Examples

This section exemplifies the curve and surface scheme for positive data developed in Sections 4 and 5. The 𝐶¹ rational trigonometric cubic function (2.1) is first used to interpolate the positive data sets taken in Tables 1 and 2, respectively. Arbitrary values are assigned to free parameters and resulting curves are shown in Figures 1 and 3, respectively. It is clear from Figures 1 and 3 that rational trigonometric cubic function does not preserve the shape of data for arbitrary values of free parameters. The positive curves for the same data sets are produced in Figures 2 and 4 by the rational positivity preserving surface scheme developed in Section 5.1.

Table 1 

A 2D positive data set.

Table 2 

A 2D positive data set.

Figure 1 

𝐶1 rational trigonometric cubic function with 𝛼𝑖=0.5,𝛽𝑖=1.0,𝛾𝑖=0.5,and𝛿𝑖=1.0.

Figure 2 

𝐶1 positive rational trigonometric cubic function with 𝛼𝑖=0.5,𝛿𝑖=1.0.

Figure 3 

𝐶1 rational trigonometric cubic function with 𝛼𝑖=0.5,𝛽𝑖=1.0,𝛾𝑖=0.5,and𝛿𝑖=1.0.

Figure 4 

𝐶1 rational trigonometric cubic function with 𝛼𝑖=0.5,𝛿𝑖=1.0.

The 3D positive data set in Tables 3 and 4 are generated from the functions 𝐹(𝑥,𝑦)=𝑥²+𝑥𝑦+𝑦²+0.1 and 𝐹(𝑥,𝑦)=𝑥²+𝑦², respectively.

Table 3 

A 3D positive data set.

Table 4 

A 3D positive data set.

Figure 5 and Figure 7 are produced by interpolating the positive data sets in Tables 3 and 4, respectively by 𝐶¹ rational trigonometric bicubic function for arbitrary values of free parameter. Positive surfaces in Figures 6 and 8 are produced by interpolating the same data by the positivity preserving scheme developed in Section 5.2.

Figure 5 

𝐶1 rational trigonometric bicubic function with 𝛼𝑖=8,𝛽𝑖=8,𝛾𝑖=8,𝛿𝑖=8,𝛼𝑗=4,𝛽𝑗=4,𝛾𝑗=4,and𝛿𝑗=4.

Figure 6 

𝐶1 positive rational trigonometric bicubic function with 𝛼𝑖=8,𝛿𝑖=8,𝛼𝑗=8,and𝛿𝑗=8.

Figure 7 

𝐶1 rational trigonometric bicubic function with 𝛼𝑖=7,𝛽𝑖=7,𝛾𝑖=7,𝛿𝑖=7,𝛼𝑗=3,𝛽𝑗=3,𝛾𝑗=3,and𝛿𝑗=3.

Figure 8 

𝐶1 positive rational trigonometric bicubic function with 𝛼𝑖=6,𝛿𝑖=6,𝛼𝑗=6,and𝛿𝑗=6.

7. Conclusion

In recent years, the trigonometric curves are attaining considerable importance due to their trigonometric basis functions which provide them the opportunity to construct conics, cylinders, surface of revolution, catenary, and so on. Han [5] used B-spline representation of cubic trigonometric curve with shape parameter for shape designing. In this paper, a Hermite form of 𝐶¹ rational trigonometric function is introduced. The developed interpolant involves four free parameters in each subintervals. Constraints are developed on two of these parameters to preserve positive shape of data, while the remaining was free for shape modification. The developed 𝐶¹ interpolant is extended to 𝐶¹ rational trigonometric bicubic interpolant for the treatment positive surface data. It is observed that error of developed interpolant is of order three same as the polynomial interpolant. The surface interpolation scheme developed in this paper is 𝐶¹, whereas in [8, 9] it was 𝐶⁰ only. Moreover, the 𝐶¹ continuity of curve and surface data interpolant does constrain step length.