C1 Rational Cubic/Linear TrigonometricInterpolation Spline with Positivity-preserving

PropertyXiangbin Qin and Qingsong Xu

Abstract—A class of C1 rational cubic/linear trigonometricinterpolation spline with two local parameters is proposed. Sim-ple sufficient conditions for constructing positivity-preservinginterpolation curves are developed. By using the boolean sumof quadratic trigonometric interpolating operators to blendtogether the proposed rational cubic/linear trigonometric in-terpolation splines as four boundary functions, a kind of C1

blending rational cubic/linear interpolation surface with fourfamilies of local parameters is constructed. Simple sufficientdata dependent conditions are also deduced for generatingC1 positivity-preserving interpolation surfaces on rectangulargrids.

Index Terms—Data visualization, Trigonometric interpolationspline, Positivity-preserving, Local parameter.

I. INTRODUCTION

SPline has a wide range of applications in Engineering,such as data fitting [1], principal components analy-

sis [2], signal restoration [3] and so on. In visualizingscientific data, when the data are arising from some complexfunction or from some scientific phenomena, it becomescrucial that the resulting spline can preserve the shapefeatures of the data. Positivity is one of the essential shapefeatures of data. Many physical situations have entities thatgain meaning only when their values are positive, such as aprobability distribution function, monthly rainfall amounts,speed of winds at different intervals of time, and half-life ofa radioactive substance and so on. For given positive data,ordinary interpolation spline methods such as the classicalcubic interpolation spline usually ignore positive charac-teristic. Thus constructing positivity-preserving interpolationspline is an essential problem and many methods have beenproposed, such as the cubic interpolation spline methods [4],the rational polynomial interpolation spline methods [5], [6],[7], [8], [9], [10], [11].

In the last years, for generating positivity-preserving in-terpolation curves, some trigonometric interpolation splinemethods have been proposed, see for example [12], [13],[14], [15], [16], [17], and the references quoted therein.In [12], a kind of rational cubic/cubic trigonometric in-terpolation spline with four families of parameters wasdeveloped, and sufficient conditions were given for con-structing positivity-preserving interpolation curves. In [13],a positivity-preserving interpolation spline was presented byusing rational quadratic/quadratic trigonometric interpolationspline with two families of parameters. Latter, in [14],

Manuscript received October 5, 2016; revised December 27, 2016.Xiangbin Qin, Qingsong Xu are with the Department of School of

Mathematics and Statistics, Central South University, Changsha 410083,PR China (e-mail: xiangbinqin@163.com; qsxu@csu.edu.cn).

the rational quadratic/quadratic trigonometric interpolationspline with two families of parameters was further extendedto four families of parameters. In [16], a new rationalquadratic/quadratic trigonometric interpolation spline withfour families of parameters was constructed for generatingpositivity-preserving interpolation curves. The smoothnessof the resulting positivity-preserving interpolation curves bythe above rational trigonometric interpolation spline methodsattains C1 continuity. Recently, in [15], [17], two kinds ofquadratic trigonometric interpolation splines were proposedfor generating GC1 continuous positivity-preserving interpo-lation curves.

For constructing C1 positivity-preserving interpolationsurfaces, the well known Coons surface technique [18] hasbeen widely used, see for example [19], [20], [21], [22],[23] and the references quoted therein. In [19], [20], byexchanging the cubic Hermite blending functions used forthe classical bi-cubic Coons surface with two different kindsof rational cubic blending functions, two classes of C1

rational bi-cubic were presented. And constrains concerningthe local free parameters were given for visualizing 3Dpositive data on rectangular grids. Like the classical bi-cubicCoons surface technique, these schemes need to providethe twists on the grid lines for generating interpolationsurfaces. In [21], [22], [23], based upon the boolean sum ofcubic interpolating operators, by blending different rationalcubic interpolation splines as the boundary functions, simplerschemes without making use of twists for constructing C1

positive interpolation of gridded data were given. These ra-tional bi-cubic partially blended interpolation spline methodsare convenience since they are possible to control the shapeof the interpolation surfaces by using the boundary functions,though they have to pay the price that the generated surfaceshave zero twist vectors at the data points.

The sufficient conditions for generating positivity-preserving interpolation surfaces developed in [21], [22], [23]were based on the claim given in [24]: bi-cubic partiallyblended interpolation surface patch inherits all the propertiesof network of boundary curves. Thus, these methods have acommon point that the positivity of the global interpolationsurfaces are determined by the positivity of the four boundarycurves of each local interpolation surface patch respectively.However, as it was pointed out in [20] that, these methodsdid not depict the positive surfaces due to the coon patchesbecause they conserved the shape of data only on theboundaries of patch not inside the patch.

There are also some positivity-preserving interpolationsurface schemes developed by using trigonometric inter-polation spline methods. In [12], a kind of rational bi-

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cubic trigonometric interpolation spline was constructed forgenerating C1 continuous positivity-preserving interpolationsurfaces. In [13], a class of rational bi-quadratic trigonomet-ric interpolation spline was developed for constructing C1

continuous positivity-preserving interpolation surfaces. Thedisadvantages of these two methods lie in that the parametersdo not have local control property for generating interpo-lation surfaces. Recently, in [25], a kind of bi-quadratictrigonometric interpolation spline was developed for gen-erating GC1 continuous positivity-preserving interpolationsurfaces with local control parameters.

In this paper, we propose a kind of C1 rational cubic/lineartrigonometric interpolation spline with two local parame-ters. By using the boolean sum of quadratic trigonomet-ric interpolating operators to blend together the proposedrational cubic/linear trigonometric interpolation splines asfour boundary functions, a class of C1 blending rationalcubic/linear interpolation surface with four families of localparameters is constructed. Simple sufficient data dependentconditions are also deduced for generating C1 positivity-preserving interpolation curves and surfaces. The developedschemes improve on the existing methods in some ways:(1) The smoothness of bi-quadratic interpolant given in [8]is C0 and the quadratic trigonometric interpolation splinesgiven in [15], [17], [25] are GC1 continuous while in thispaper it is C1.(2) The rational trigonometric interpolation schemes devel-oped in [12], [13] do not allow the designer to locally refinethe positive surface as per consumers demand. Whereas, thegiven method is done by introducing local free parameterswhich they are used in the description of interpolation curvesand surfaces.(3) In [21], [22], [23], the authors claimed that the rationalbi-cubic partially blended functions (coon patches) generateda positive surface but they conserved the shape of dataonly on the boundaries of patch not inside the patch andthey did not provide proof that the conditions given [21],[22], [23] will be always sufficient to generate positivity-preserving interpolation surfaces everywhere in the domain.In contrast, we develop new constrain conditions on theboundary curves of each local interpolation surface patchand the given conditions are sufficient to generate positivity-preserving interpolation surfaces everywhere in the domainwith theory proving.

The rest of this paper is organized as follows. Section IIgives the construction of C1 rational cubic/linear interpo-lation spline with two local free parameters. The sufficientconditions for generating positivity-preserving interpolationcurves are discussed. In section III, a kind of C1 blendingrational cubic/linear interpolation surfaces with four familiesof local free parameters is described. Simple sufficient datadependent constraints are derived on the local free parametersto preserve the shape of 3D positive data on rectangular grids.Several numerical examples are also given to prove the worthof the new developed schemes. Conclusion is given in thesection IV.

II. C1 POSITIVITY-PRESERVING INTERPOLATION CURVES

In this section, we firstly construct a new kind of C1

rational cubic/linear trigonometric interpolation spline with

two families of free parameters, and then we developed sim-ple sufficient conditions for generating positivity-preservinginterpolation curves. Several numerical examples and com-parisons are also given.

A. Rational cubic/linear trigonometric interpolation spline

Let fi ∈ R, i = 1, ..., n, be data given at the distinct knotsxi ∈ R, i = 1, ..., n, with interval spacing hi = xi+1 −xi > 0, and let di ∈ R denote the first derivative valuesdefined at the knots. For x ∈ [xi, xi+1], a piecewise rationalcubic/linear trigonometric interpolation spline with two localparameters αi and βi is defined as follows

T (x) = B0(t;αi)fi +B1(t;αi)[fi +

2hi

π(1+αi)di

]+B2(t;βi)

[fi+1 − 2hi

π(1+βi)di+1

]+B3(t;βi)fi+1,

(1)where t = π(x − xi)/(2hi), αi, βi ∈ [0,+∞), i =1, 2, ..., n−1, and the four rational cubic/linear trigonometricbasis functions Bj(t;αi) and B3−j(t;βi), j = 0, 1 are givenby

B0 (t;αi) =1−sin t

1+αi sin t ,

B1 (t;αi) =sin t(1−sin t)(1+αi+αi sin t)

1+αi sin t ,

B2 (t;βi) =cos t(1−cos t)(1+βi+βi cos t)

1+βi cos t,

B3 (t;βi) =1−cos t

1+βi cos t.

The spline given in (1) is a C1 interpolation spline as itsatisfies the following interpolation properties{

T (xi) = fi, T (xi+1) = fi+1,T ′(xi) = di, T ′(xi+1) = di+1.

For any t ∈ [0, π/2] and αi, βi ∈ [0,+∞), it is easy tocheck that the four rational cubic/linear trigonometric basisfunctions possess the properties of partition of unity andnonnegativity, that is{

B0 (t;αi) +B1 (t;αi) +B2 (t;βi) +B3 (t;βi) = 1,Bi (t;αi) ≥ 0, B3−i (t;βi) ≥ 0, i = 0, 1.

From the expression of the interpolation spline T (x) givenin (1), we have

limαi,βi→+∞

T (x) = ficos2t+ fi+1sin

2t

which implies that the parameters αi, βi serve as tensionparameters.

In applications, the first derivative values di, i =1, 2, . . . , n are not known and should be specified in advance.In this paper, they are computed by using the followingArithmetic mean method

d1 = ∆1 − h1

h1+h2(∆2 −∆1) ,

di =∆i−1+∆i

2 , i = 2, 3, . . . , n− 1,

dn = ∆n−1 +hn−1

hn−2+hn−1(∆n−1 −∆n−2) ,

where ∆i = (fi+1−fi)/hi. This Arithmetic mean method isthe three-point difference approximation based on arithmeticcalculation, which is computationally economical and suit-able for visualization of shaped data, see for example [21].

For convenience, in the following discussion, for x ∈[xi, xi+1], we will also denote the interpolation spline T (x)given in (1) as T (t; fi, fi+1; di, di+1;αi, βi).

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B. Positivity-preserving condition

For given positivity data set {(xi, fi), i = 1, 2, . . . , n},since the four rational cubic/linear trigonometric basis func-tions are nonnegative on [0, π/2] and strict positive in(0, π/2), it is obvious that the interpolation spline T (x) givenin (1) is positive on each subinterval Ii = [xi, xi+1] if{

αi ≥ 0, βi ≥ 0,

fi +2hidi

π(1+αi)≥ 0, fi+1 − 2hidi+1

π(1+βi)≥ 0.

From these, we can immediately obtain the following suffi-cient conditions for T (x) (x ∈ [x1, xn]) preserving positivity αi = max

{−1− 2hidi

πfi, 0}+ ai, ai ≥ 0,

βi = max{−1 + 2hidi+1

πfi+1, 0}+ bi, bi ≥ 0,

(2)

where i = 1, 2, . . . , n− 1 and ai, bi serve as free parametersfor the users to interactively adjust the shape of the obtainedpositivity-preserving interpolation curves.

C. Numerical examples and comparisons

TABLE IPOSITIVE DATA SET GIVEN IN [21].

i 1 2 3 4 5 6 7xi 0 2 4 10 28 30 32fi 20.8 8.8 4.2 0.5 3.9 6.2 9.6

TABLE IITHE POSITIVE DATA SET GIVEN IN [19].

i 1 2 3 4 5 6 7xi 2 3 7 8 9 13 14fi 10 2 3 7 2 4 10

TABLE IIITHE POSITIVE DATA SET GIVEN IN [14].

i 1 2 3 4 5 6 7 8xi 0 0.04 0.05 0.06 0.07 0.08 0.12 0.13fi 0.82 1.2 0.978 0.6 0.3 0.1 0.15 0.48

Fig. 1 shows the positivity-preserving interpolation curvesgenerated by using the conditions (2) with different freeparameters ai and bi for the positive data set given in Tab. Iand the graphics of their first derivatives, respectively. Fromthe results, we can see that the interpolation curves preservethe shape of the positive data set given in Tab. I nicely andthey all achieve C1 continuity.

Fig. 2 shows the positivity-preserving interpolation curvesgenerated by using the conditions (2) with different freeparameters ai and bi for the positive data set given in Tab. IIand the graphics of their first derivatives, respectively. In canbe seen from 2 that the interpolation curves clearly preservethe shape of the positive data set given in Tab. II and theyall reach C1 continuity.

Fig. 3 and Fig. 4 show the positivity-preserving interpo-lation curves generated by the methods given in [12], [13],[14], [15], [16] and our method with a set of appropriateparameters for the positive data sets given in Tab. III and

Tab. IV, respectively. From the results, it can be seen that ourpiecewise rational cubic/linear trigonometric interpolationspline describes the positive data set more fairly than themethods given in [12], [13], [14], [15], [16].

III. C1 POSITIVITY-PRESERVING INTERPOLATIONSURFACES

In this section, by using the boolean sum of quadratictrigonometric interpolating operators to blend together theconstructed rational cubic/linear trigonometric interpolationsplines as the four boundary functions, we shall constructa class of C1 blending rational cubic/linear trigonometricinterpolation surface with four families of local parameters.By developing new constrains on the boundary functions,we will also theoretically deduce simple sufficient datadependent conditions on the local parameters to generate C1

positivity-preserving interpolation surfaces for positive dataon rectangular grids.

A. Blending rational cubic/linear trigonometric interpola-tion surfaces

Let {(xi, yi, Fij), i = 1, 2, . . . , n; j = 1, 2, . . . ,m} be agiven set of data points defined over the rectangular domainD = [x1, xn] × [y1, ym], where πx : x1 < x2 < . . . < xn

is the partition of [x1, xn] and πy : y1 < y2 < . . . < ym isthe partition of [y1, ym]. For (x, y) ∈ [xi, xi+1]× [yj , yj+1],by using the boolean sum of quadratic trigonometric inter-polating operators to blend together the rational cubic/lineartrigonometric interpolation splines (1) as the boundary func-tions, a new blending rational cubic/linear trigonometricinterpolation surface is given as follows

F (x, y) = −[−1 b0(t) b1(t)

]H

−1b0(s)b1(s)

, (3)

where hxi = xi+1 − xi, h

yj = yj+1 − yj , t = π(x −

xi)/(2hxi ), s = π(y − yj)/(2h

yj ) and

H =

0 F (x, yj) F (x, yj+1)F (xi, y) Fi,j Fi,j+1

F (xi+1, y) Fi+1,j Fi+1,j+1,

and

b0(z) := cos2z, b1(z) := sin2z,F (x, yj) := T (t;Fi,j , Fi+1,j ;D

xi,j , D

xi+1,j ;α

xi,j , β

xi,j),

F (xi, y) := T (s;Fi,j , Fi,j+1;Dyi,j , D

yi,j+1;α

yi,j , β

yi,j).

Here, Dxi,j , D

yi,j are known as the first partial derivatives

at the grid point (xi, yj) and(αxi,j

)(n−1)×m

,(βxi,j

)(n−1)×m

,(αyi,j

)n×(m−1)

,(βyi,j

)n×(m−1)

are called four families oflocal parameters. From the interpolation surface T (x, y)given in (1), we can see that the changes of a local parameterαxi,j will affect the shape of two neighboring patches F (x, y)

defined in the domain (x, y) ∈ (xi, xi+1) × (yj−1, yj+1).And the changes of a local parameter αy

i,j will affect theshape of two neighboring patches F (x, y) defined in thedomain (x, y) ∈ (xi−1, xi+1) × (yj , yj+1). And the localparameters βx

i,j and βyi,j have the same effect region on the

shape of the generated interpolation surface F (x, y) as thatof the local parameters αx

i,j and αyi,j , respectively. Since the

four boundary functions F (x, yj), F (x, yj+1), F (xi, y) and

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0 10 20 300

5

10

15

20

(A) Interpolant T1(x) with a

i=b

i=0.4.

0 10 20 300

5

10

15

20

(B) Interpolant T2(x) with a

i=b

i=0.8.

0 10 20 300

5

10

15

20

(C) Interpolant T3(x) with a

i=0.8,b

i=0.

0 10 20 30−8

−6

−4

−2

0

2

4

(D) The first derivative of T1(x).

0 10 20 30−8

−6

−4

−2

0

2

4

(E) The first derivative of T2(x).

0 10 20 30−8

−6

−4

−2

0

2

4

(F) The first derivative of T3(x).

Fig. 1. Positivity-preserving interpolation curves generated by using the conditions (2) with different free parameters ai and bi for the positive data setgiven in Tab. I and the graphics of their first derivatives.

2 4 6 8 10 12 140

2

4

6

8

10

(A) Interpolant T4(x) with a

i=b

i=0.

2 4 6 8 10 12 140

2

4

6

8

10

(B) Interpolant T5(x) with a

i=b

i=1.

2 4 6 8 10 12 140

2

4

6

8

10

(C) Interpolant T6(x) with a

i=1,b

i=0.

2 4 6 8 10 12 14−10

−5

0

5

10

(D) The first derivative of T4(x).

2 4 6 8 10 12 14−10

−5

0

5

10

(E) The first derivative of T5(x).

2 4 6 8 10 12 14−10

−5

0

5

10

(F) The first derivative of T6(x).

Fig. 2. Positivity-preserving interpolation curves generated by using the conditions (2) with different free parameters ai and bi for the positive data setgiven in Tab. II and the graphics of their first derivatives.

0 0.02 0.04 0.06 0.08 0.1 0.120

0.5

1

1.5

(A) Method [12].0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

(B) Method [13].0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

(C) Method [14].

0 0.02 0.04 0.06 0.08 0.1 0.120

0.5

1

1.5

(D) Method [15].0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

(E) Method [16].0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.5

1

1.5

(F) Our method.

Fig. 3. C1 positivity-preserving curves generated by different methods for the positive data set given in Tab. III. Their corresponding parameters for allsegments are set as method [12]: (w0

i = 0.2, w1i = 0.6, w2

i = 0.3, w3i = 0.1); method [13]: (µi = 4.1, ηi = 4.5); method [14]: (w0

i = 0.2, w1i =

0.8, w2i = 0.8, w3

i = 0.3); method [15]: (αi = 2, βi = 1.5); method [16]: (w0i = 0.2, w1

i = 0.5, w2i = 0.5, w3

i = 0.2); our method: (αi = 1, βi = 0).

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TABLE IVTHE POSITIVE DATA SET GIVEN IN [14].

i 1 2 3 4 5 6 7 8 9 10xi 0 3.25 15 26.5 30 32 37 40 42.5 44fi 8.8 3 0.025 3.1 6.2 9.6 20 22.5 21.519 20

0 10 20 30 400

5

10

15

20

25

(A) Method [12].0 10 20 30 40

0

5

10

15

20

25

(B) Method [13].0 10 20 30 40

0

5

10

15

20

25

(C) Method [14].

0 10 20 30 400

5

10

15

20

25

(D) Method [15].0 10 20 30 40

0

5

10

15

20

25

(E) Method [16].0 10 20 30 40

0

5

10

15

20

25

(F) Our Method.

Fig. 4. C1 positivity-preserving curves generated by different methods for the positive data set given in Tab. IV. Their corresponding parameters for allsegments are set as method [12]: (w0

i = 0.5, w1i = 4, w2

i = 4, w3i = 0.5); method [13]: (µi = 3, ηi = 3); method [14]: (w0

i = 0.2, w1i = 0.9, w2

i =0.9, w3

i = 0.2); method [15]: (αi = 2, βi = 2); method [16]: (w0i = 2, w1

i = 4, w2i = 4, w3

i = 2); our method: (αi = 0.2, βi = 0.1).

F (xi+1, y) are all C1 continuous, we can easily concludethat the given blending rational cubic/linear trigonometricinterpolation surface F (x, y) is global C1 continuous overthe rectangular domain [x1, xn]× [y1, ym].

In most applications, the first partial derivatives Dxi,j and

Dyi,j are not given and hence must be determined either from

given data or by some other means. In this paper, we use thefollowing arithmetic mean method to compute the first partialderivatives

Dx1,j = ∆x

1,j +(∆x

1,j −∆x2,j

) hx1

hx1+hx

2,

Dxn,j = ∆x

n−1,j +(∆x

n−1,j −∆xn−2,j

) hxn−1

hxn−2

+hxn−1

,

Dxi,j =

∆xi−1,j+∆x

i,j

2 , i = 2, 3, . . . , n− 1; j = 1, 2, . . . ,m,

Dyi,1 = ∆y

i,1 +(∆y

i,1 −∆yi,2

) hy1

hy1+hy

2,

Dyi,m = ∆y

i,m−1 +(∆y

i,m−1 −∆yi,m−2

) hym−1

hym−2

+hym−1

,

Dyi,j =

∆yi,j−1

+∆yi,j

2 , i = 1, 2, . . . , n; j = 2, 3, . . . , n− 1,

where ∆xi,j = (Fi+1,j − Fi,j)/h

xi and ∆y

i,j =

(Fi,j+1 − Fi,j)/hyj . This arithmetic mean method is

computationally economical and suitable for visualizationof shaped data [21].

B. Positivity-preserving conditions

In this subsection, we want to develop simply schemes sothat the C1 interpolation surface F (x, y) given in (3) canpreserve the shape of 3D positive data on rectangular grides.

Let {(xi, yi, Fi,j)} be a positive data set defined over therectangular grid [xi, xi+1] × [yj , yj+1], i = 1, 2, · · · , n − 1,j = 1, 2, · · · ,m − 1 such that Fi,j > 0,∀i, j. The interpo-lation surface F (x, y) given in (3) preserves the shape ofpositive data if

F (x, y) > 0, ∀(x, y) ∈ [x1, xn]× [y1, ym].

For (x, y) ∈ [xi, xi+1]× [yj , yj+1], we want to rewrite theexpression of the interpolation surface F (x, y) given in 3 asthe following form

F (x, y) = b0(s)F (x, yj) + b1(s)F (x, yj+1) + b0(t)F (xi, y)+b1(t)F (xi+1, y)− b0(t)b0(s)Fi,j − b0(t)b1(s)Fi,j+1

−b1(t)b0(s)Fi+1,j − b1(t)b1(s)Fi+1,j+1

= b0(s)[F (x, yj)− 1

2b0(t)Fi,j − 12b1(t)Fi+1,j

]+b1(s)

[F (x, yj+1)− 1

2b0(t)Fi,j+1 − 12b1(t)Fi+1,j+1

]+b0(t)

[F (xi, y)− 1

2b0(s)Fi,j − 12b1(s)Fi,j+1

]+b1(t)

[F (xi+1, y)− 1

2b0(s)Fi+1,j − 12b1(s)Fi+1,j+1

].(4)

Without loss of generality, for any (x, y) ∈ [xi, xi+1] ×[yj , yj+1], since b0(z) and b1(z) are strict positive for anyz ∈ (0, 1), from (4), we can see that the interpolation surfaceF (x, y) is positive everywhere in the domain [xi, xi+1] ×[yj , yj+1] if the following constrains hold

F (x, yj)− 12b0(t)Fi,j − 1

2b1(t)Fi+1,j > 0,F (x, yj+1)− 1

2b0(t)Fi,j+1 − 12b1(t)Fi+1,j+1 > 0,

F (xi, y)− 12b0(s)Fi,j − 1

2b1(s)Fi,j+1 > 0,F (xi+1, y)− 1

2b0(s)Fi+1,j − 12b1(s)Fi+1,j+1 > 0.

(5)For F (x, yj) − 1

2b0(t)Fi,j − 12b1(t)Fi+1,j , notice that

B0(t;αxi,j) + B1(t;α

xi,j) = b0(t) and B2(t;β

xi,j) +

B3(t;βxi,j) = b1(t), from (1), we have

F (x, yj)− 12b0(t)Fi,j − 1

2b1(t)Fi+1,j

= 12B0(t;α

xi,j)Fi,j +B1(t;α

xi,j)

[Fi,j

2 +2hx

i

π(1+αxi,j)

Dxi,j

]+B2(t;β

xi,j)

[Fi+1,j

2 − 2hxi

π(1+βxi,j)

Dxi+1,j

]+1

2B3(t;βxi,j)Fi+1,j ,

Thus we can see that the following constrains are sufficient

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to ensure F (x, yj)− 12b0(t)Fi,j − 1

2b1(t)Fi+1,j > 0{αxi,j ≥ 0, βx

i,j ≥ 0,Fi,j

2 +2hx

i

π(1+αxi,j)

Dxi,j ≥ 0,

Fi+1,j

2 − 2hxi

π(1+βxi,j)

Dxi+1,j ≥ 0,

which bring forth the following sufficient conditions αxi,j ≥ max

{−1− 4hx

i Dxi,j

πFi,j, 0},

βxi,j ≥ max

{−1 +

4hxi D

xi+1,j

πFi+1,j, 0},

(6)

In the same manner, we can also derive similar sufficientconditions for F (x, yj+1)− 1

2b0(t)Fi,j+1− 12b1(t)Fi+1,j+1 >

0, F (xi, y) − 12b0(s)Fi,j − 1

2b1(s)Fi,j+1 > 0 andF (xi+1, y)− 1

2b0(s)Fi+1,j− 12b1(s)Fi+1,j+1 > 0. In conclu-

sion, for a positive data set, we can obtain the following suffi-cient conditions for F (x, y) > 0,∀(x, y) ∈ [x1, xn]×[y1, yn]

αxi,j = max

{−1− 4hx

i Dxi,j

πFi,j, 0}+ axi,j ,

βxi,j = max

{−1 +

4hxi D

xi+1,j

πFi+1,j, 0}+ bxi,j ,

αyi,j = max

{−1− 4hy

jDy

i,j

πFi,j, 0}+ ayi,j ,

βyi,j = max

{−1 +

4hyjDy

i,j+1

πFi,j+1, 0}+ byi,j ,

(7)

where i = 1, 2, · · · , n − 1, j = 1, 2, · · · ,m − 1 andaxi,j ,ayi,j ,bxi,j and byi,j are arbitrary nonnegative real numbersand serve as free parameters.

C. Numerical examples and comparisons

We shall give several numerical examples to show that theproposed C1 interpolation surface F (x, y) given in (3) canbe used to nicely visualize the shape of 3D positive dataon rectangular grids. In the following figures, the given datapoints have been marked with solid black dots.

Fig. 5 shows the positivity-preserving interpolation sur-faces F1(x, y) and F2(x, y) for the 3D positive data set givenin Tab. V. From the results, we can see that both the twovisually pleasing interpolation surfaces preserve the shape ofthe 3D positive data set given in Tab. V genially.

Fig. 6 shows the positivity-preserving interpolation sur-faces F3(x, y) and F4(x, y) for the 3D positive data setgiven in Tab. VI. As can be seen from Fig. 6, both the twointerpolation surfaces visualize the shape of the 3D positivedata set given in Tab. IV well.

Fig. 7 shows the positivity-preserving interpolation sur-faces F5(x, y) and F6(x, y) for the 3D positive data set givenin Tab. VII. As can be seen from the Fig. 7, both the twointerpolation surfaces visualize the positive shape of the datagiven in Tab. VII nicely and the shape of the interpolationsurfaces can be adjusted conveniently by using the local freeparameters.

Fig. 8 show the positivity-preserving interpolation surfacesgenerated by the method given in [12] and our method with aset of appropriate parameters for the positive data sets givenin Tab. VIII. From the results, it can be seen that our blendingcubci/linear trigonometric interpolation spline describes thepositive data set more fairly than the method given in [12].

IV. CONCLUSION

As stated above, the constructed C1 rational cubic/lineartrigonometric interpolation spline is suitable for constructingpositivity-preserving interpolation curves. To demonstrate

TABLE VIITHE 3D POSITIVE DATA SET GIVEN IN [19].

y/x −3 −2 −1 1 2 3−3 0.0124 0.0238 0.0404 0.0404 0.0238 0.0124−2 0.0238 0.0635 0.1667 0.1667 0.0635 0.0238−1 0.0404 0.1667 1.3333 1.3333 0.1667 0.04041 0.0404 0.1667 1.333 1.3333 0.1667 0.04042 0.0238 0.0635 0.1667 0.1667 0.0635 0.02383 0.0124 0.0238 0.0404 0.0404 0.0238 0.0124

TABLE VIIITHE 3D POSITIVE DATA SET GIVEN IN [12].

y/x −3 −2 −1 0 1 2 3−3 18 13 10 9 10 13 18−2 13 8 5 4 5 8 13−1 10 5 2 1 2 5 100 9 4 1 0 1 4 91 10 5 2 1 2 5 102 13 8 5 4 5 8 133 18 13 10 9 10 13 18

our method, comparisons with the existing methods [12],[13], [14], [15], [16] are provided. The proposed blendingrational cubic/linear trigonometric interpolation surfaces withfour families of local free parameters can be C1 continuouswithout making use of the first mixed partial derivatives atthe data points. For 3D positive data on rectangular grids, bydeveloping new constrain conditions on the boundary curvesof each local interpolation surface patch, we theoreticallydeduce simple sufficient data dependent conditions on thelocal free parameters to generate positivity-preserving inter-polation surfaces everywhere in the domains. The developedpositivity-preserving schemes are not only local and compu-tationally economical but also visually pleasant. The givenmethod also allows extensions to generate C1 positivity-preserving interpolation surfaces for 3D non-gridded data.These will be our future work.

REFERENCES

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TABLE VTHE 3D POSITIVE DATA SET [20].

y/x −3 −2 −1 0 1 2 3−3 0.0401 0.0404 0.1755 1.0401 0.1755 0.0404 0.0401−2 0.0583 0.0586 0.1936 1.0583 0.1936 0.0586 0.0583−1 0.4078 0.4082 0.5432 1.4079 0.5432 0.4082 0.40780 1.04 1.0403 1.1753 2.04 1.1753 1.0403 1.041 0.4078 0.4082 0.5432 1.4079 0.5432 0.4082 0.40782 0.0583 0.0586 0.1936 1.0583 0.1936 0.0586 0.05833 0.0401 0.0404 0.1755 1.0401 0.1755 0.0404 0.0401

Fig. 5. C1 positivity-preserving interpolation surfaces for the 3D positive data set given in Tab. V.

Fig. 6. C1 positivity-preserving interpolation surfaces for the 3D positive data set given in Tab. VI.

TABLE VITHE 3D POSITIVE DATA SET GIVEN IN [21].

y/x 0.0001 1.5 3 4.5 6 7.5 90.0001 0.6667 0.5 0.5 0.5 0.5 0.5 0.5

1.5 0.4422 0.4807 0.4936 0.4970 0.4982 0.4989 0.49923 0.0022 0.1681 0.3341 0.4095 0.4447 0.4631 0.4738

4.5 0.0472 0.1295 0.2603 0.3491 0.4006 0.4309 0.44976 0.0022 0.0575 0.1681 0.2657 0.3341 0.3793 0.4095

7.5 0.0156 0.0515 0.1331 0.2184 0.2876 0.3385 0.37529 0.0021 0.0283 0.0926 0.1681 0.2364 0.2916 0.3340

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Fig. 7. C1 positivity-preserving interpolation surfaces for the 3D positive data set given in Tab. VII.

2x0-2-2

(A) Method [12].

0y2

0

10

20

z

-5

0

5

10

15

20

-3 -2 -1 0 1 2 3(B) The xz-view of method [12].

x

z

-5

0

5

10

15

20

-3 -2 -1 0 1 2 3(C) The yz-view of method [12].

y

Fig. 8. C1 positivity-preserving interpolation surfaces generated by different methods for the 3D positive data set given in Tab. VIII. Their correspondingparameters for all segments are set as method [12]: (ai = 1, βi = 3, γi = 3, δi = 1, aj = 3, βj = 3, γj = 3, δj = 3); our method: (αx

i,j = βxi,j =

αyi,j = βy

i,j = 0).

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