Egyptian Informatics Journal (2013) 14, 211–220

Cairo University

Egyptian Informatics Journal

www.elsevier.com/locate/eijwww.sciencedirect.com

C1 rational quadratic trigonometric spline

Maria Hussain *, Sidra Saleem

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

Received 9 February 2013; revised 28 August 2013; accepted 22 September 2013

Available online 17 October 2013

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KEYWORDS

Quadratic trigonometric

spline;

Shape parameters;

Bezier function;

Bivariate trigonometric

function

Corresponding author. Tel.:mail address: mariahussain_

er review under responsib

formation, Cairo University.

Production an

10-8665 � 2013 Production

tp://dx.doi.org/10.1016/j.eij.2

+92 0331@yahoo

ility of

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and hosti

013.09.00

Abstract A Bezier like C1 rational quadratic trigonometric polynomial spline is developed. It

defines two shape parameters in each subinterval. The approximation and geometric properties

are investigated. The curvature continuity is established. The developed rational quadratic trigono-

metric polynomial spline is extended to C1 piecewise rational bi-quadratic function with four shape

parameters in each rectangular patch. Data dependent constraints are developed on the shape

parameters in the description of piecewise rational quadratic and bi-quadratic trigonometric poly-

nomial spline for shape preservation of curve and regular surface data. The developed shape pre-

serving schemes provide tangent continuity in quadratic form and does not restrict interval

length, derivatives or data.� 2013 Production and hosting by Elsevier B.V. on behalf of Faculty of Computers and Information,

Cairo University.

1. Introduction

Rational trigonometric interpolating splines with ten-

sion(shape parameters) are preferred because these can inter-polate the same data in the form of straight line and curve.These tension parameters do not affect the order of continuity

of spline. Moreover, the rational structure of these splines al-lows coping with singularities. Data gathered, whether, physi-cally or experimentally has at least one of the shape properties,positivity, monotonicity and convexity. The amount of rain-

fall, gas discharge and exponential functions are a few positivedata producing sources. The path inscribed by reboots arm

1 4930071..com (M. Hussain).

Faculty of Computers and

g by Elsevier

ng by Elsevier B.V. on behalf of F

2

movement and designing of its circuits are few examples ofmonotone and convex data.

Bao et al. [2] presented a rational blended interpolant with

free parameters. The developed interpolant was used for valuecontrol and shape control, and the range of parameters wasdetermined by minimizing the bending energy. Brodlie et al.

[3] extended the cubic Hermite to bi-cubic Hermite for positiveand constrained data interpolation. The developed scheme wasC1. Delgado and Pena in [6] investigated the rational Bezier

surfaces for shape preservation of data and established that ra-tional Bezier surfaces are not monotonicity preserving. Duanet al. [7] revisited the rational spline [8] which was C1 only if

slope of secant line coincided with slope of tangent line. Theboundedness, value control, inflection point control and con-vexity at a point of the interpolant were studied for uniformdata. Duan et al. [9] developed a bivariate rational interpolant

with four shape parameters in each rectangular patch. Thedeveloped interpolant was C1 for equally spaced data with asuitable choice of shape parameters. The sufficient restrictions

were developed on shape parameters for constrained

aculty of Computers and Information, Cairo University.

212 M. Hussain, S. Saleem

interpolation of data. Duan et al. [10] discussed the rate of con-vergence of a rational spline with two shape parameters.

Han [14] presented the cubic trigonometric polynomial

curves with shape parameters. The partition of knot vectorand value of free parameter affected the order of continuity.Han et al. [16] presented a cubic trigonometric Bezier curve

with two shape parameters and compared it to the cubic Beziercurve. The effect of shape parameters on the shape of the curvewas analyzed found that it was closer to control polygon than

the cubic Bezier curves. Hussain et al. [17] used rational qua-dratic function with a shape parameter to preserve the shapeof curve data. The developed scheme of this paper only assuredposition continuity. Lamberti and Manni [18] utilized the cubic

Hermite in its parametric form to preserve the shape of data.The authors constrained the knot vector to introduce the ten-sion effect and shape preservation. C2-continuity was estab-

lished by a set of restrictions on first order derivatives at theknots. Manni and Sablonniere [19] developed a C1 parametricHermite interpolation technique for comonotone (monotone

w.r.t. x and/or y) regular data using piecewise quadratic andbi-quadratic components. The shape preserving scheme im-posed a restriction on knot vector and derivatives. Zhang

et al. [21] constructed a bivariate rational function with shapeparameters. The developed bivariate rational interpolant wasC1 if the derivative was equal to slope of tangent line. A posi-tive convex discriminant function was constructed for the suit-

able choice of shape parameters to assure convex surfacethrough convex data.

The study in this research paper is motivated by Bao et al.

[2], Delgado and Pena [6] and Hussain et al. [17]. The aim is todevelop a rational interpolating spline to ensure local controlon a single interval and C1 continuity in quadratic structure.

The remainder of the paper is organized as follows. Sec-tion 2 presents a new rational quadratic trigonometric functionwith two tension parameters in each subinterval. Section 3

transforms this rational quadratic trigonometric function totrigonometric spline and discusses its approximation proper-ties and shape properties. Section 4 extends trigonometricspline to C1 bivariate rational quadratic trigonometric func-

tion. Section 5 is of numerical examples and Section 6 con-cludes the paper.

2. Bezier like rational quadratic trigonometric function

Let the data under consideration be {(xi, fi), i= 0,1,2, . . . ,n}.The usual increasing partition of domain is adopted. The Be-

zier like rational quadratic trigonometric function S(x) is de-fined over the interval Ii = [xi, xi+1], i= 0,1,2, . . . ,n � 1 as:

SðxÞ ¼X3k¼0

RkðxÞPk; ð1Þ

Rk(x), k = 0, 1, 2, 3 are the rational quadratic trigonometricbasis functions defined as

R0ðxÞ ¼B0ðxÞRðxÞ ; R1ðxÞ ¼

liB1ðxÞRðxÞ ;

R2ðxÞ ¼giB2ðxÞRðxÞ ; R3ðxÞ ¼

B3ðxÞRðxÞ ;

B0ðxÞ ¼ ð1� sinðdiÞÞ2; B1ðxÞ ¼ ð1� sinðdiÞÞ sinðdiÞ;B2ðxÞ ¼ ð1� cosðdiÞÞ cosðdiÞ;

B3ðxÞ ¼ ð1� cosðdiÞÞ2;RðxÞ ¼ B0ðxÞ þ liB1ðxÞ þ giB2ðxÞ þ B3ðxÞ;

di ¼pðx� xiÞ

2hi; hi ¼ xiþ1 � xi; i ¼ 0; 1; 2; . . . ; n� 1:

Pk, k= 0, 1, 2, 3 are the control points so the Bezier like ra-tional quadratic trigonometric function (1) is termed as controlpoint form. The parameters li and gi are the shape designparameters or tension parameters in each subinterval Ii = [xi,

xi+1], i= 0,1,2, . . . ,n � 1.

Theorem 1. The rational quadratic trigonometric function (1)satisfies the following properties:

1. Endpoint interpolation property: S(di = 0) = P0 andS di ¼ p

2

� �¼ P 3.

2. Convex Hull property: The curve always lies in the con-vex hull of control points Pk, k = 0, 1, 2, 3.

3. The rational quadratic trigonometric function is invariantunder the affine transformations.

4. The rational quadratic trigonometric function representsconic section under certain restrictions on shape designparameters.

Proof.

1. It can be easily established by substituting di = 0 and di ¼ p2

in Eq. (1).2. Since Bk(x) P 0, k= 0, 1, 2, 3 for di 2 0; p

2

� �and shape

design parameters (li, gi) are assumed positive real num-

bers, so Rk (x), k= 0, 1, 2, 3 are non-negative. The simplesummation establishes that

P3k¼0 RkðxÞ ¼ 1. It asserts con-

vex hull property, i.e. the curve will always lie in the convex

hull of control points.3. Let T be an affine transformation defined as

T(X) = AX + T1, X is the vector to be transformed, A istransformation matrix and T1 is the translation vector.

Applying the affine transformation T to the rational qua-dratic trigonometric function (1) we have

TðSðxÞÞ ¼ TX3k¼0

RkðxÞPk

!¼ A

X3k¼0

RkðxÞPk þ T1: ð2Þ

SinceP3

k¼0 RkðxÞ ¼ 1, so the expression (2) takes the form:

TðSðxÞÞ ¼ AX3k¼0

RkðxÞPk þX3k¼0

RkðxÞT1 ¼X3k¼0

RkðxÞðAPk þ T1Þ

¼X3k¼0

RkðxÞTðPkÞ:

Ellipse is finite over whole domain, parabola tends to infinityat most at a single point of domain and hyperbola tends toinfinity at most two points of the domain. While interpolating

a conic segment (hyperbola, parabola and ellipse) by the

C1 rational quadratic trigonometric spline 213

rational quadratic trigonometric function S(x) defined in (1) it

is necessary to choose those values of shape design parameters(li gi) which fulfill the aforesaid restrictions. The values ofshape design parameters (li gi) are computed as follows:

Substituting the values of rational quadratic trigonometricbasis functions Rk(x), k = 0, 1, 2, 3 in Eq. (1) it takes theform

SðxÞ ¼ B0ðxÞ þ liB1ðxÞ þ giB2ðxÞ þ B3ðxÞRðxÞ : ð3Þ

Hence S(x) is infinite if R(x) = 0. To simplify the computationwe shall restrict to special case that is li = gi. Substituting the

values of Bk(x), k= 0, 1, 2, 3; li = giin R(x) = B0

(x) + liB1(x) + giB2(x) + B3(x) = 0 and after some simplifi-cation it reduces to

RðxÞ ¼ 2ðgi � 2Þ2 sin2ðdiÞ þ 2ðgi � 2Þð3� giÞ sinðdiÞþ ð5� 2giÞ ¼ 0: ð4Þ

Eq. (4) is quadratic in sin(di). The discriminant of R(x) = 0

w.r.t sin(di) is D = 4(gi � 2)2 {(3 � gi)2 � 2(5 � 2gi)}. It can

be easily computed from D that roots of R(x) = 0 will be

(a) real and distinct if gi 2 �1; 1�ffiffiffi2p� �[ 1þ

ffiffiffi2p

;1� �

;(b) single real root if gi 2 1�

ffiffiffi2p

; 1þffiffiffi2p� �

;(c) no real root(roots are imaginary) if

gi 2 1�ffiffiffi2p

; 1þffiffiffi2p� �

.

S(x) represents hyperbola if gi 2 �1; 1�ffiffiffi2p� �[

1þffiffiffi2p

;1� �

, parabola if gi 2 1�ffiffiffi2p

; 1þffiffiffi2p� �

and ellipse if

gi 2 1�ffiffiffi2p

; 1þffiffiffi2p� �

. h

3. Rational quadratic trigonometric spline

The rational quadratic trigonometric function (3) istransformed to spline by applying the following

C1-continuity conditions at the end points of the intervalIi = [xi, xi+1]

SðxiÞ ¼ fi; Sðxiþ1Þ ¼ fiþ1; S0ðxiÞ ¼ di; S0ðxiþ1Þ ¼ diþ1:

The rational quadratic trigonometric spline over the subin-terval Ii = [xi, xi+1] is defined as

SðxÞ ¼ B0ðxÞA0 þ B1ðxÞA1 þ B2ðxÞA2 þ B3ðxÞA3

RðxÞ ; ð5Þ

A0 ¼ fi; A1 ¼ lifi þ2hidi

p; A2 ¼ gifiþ1 �

2hidiþ1p

;A3 ¼ fiþ1:

Theorem 2. For g(x) 2 C3[x0, xn], let S(x) be a rationalquadratic trigonometric spline (5) interpolating g(x) in [xi,

xi+1], then for the positive parameters li and gi, the error ofinterpolating function satisfies jgðxÞ � SðxÞj 6 kgð3ÞðkÞkh3i c

^i,

with c^i ¼ max06h61 -ðli; gi; hÞ, where

-ðli; gi; hÞ ¼max -1ðli; gi; hÞ; 0 6 gi 6 1; 0 6 h 6 1;

max -2ðli; gi; hÞ; gi > 1þ 2p ; 0 6 h 6 h�;

max -3ðli; gi; hÞ; gi > 1þ 2p ; h� 6 h 6 1:

8><>:

-1 ¼h3i

RðxÞ2B2h

2 � pð1� hÞðgiB2 þ B3Þh2

p

þ 4B2ð1� hÞh� pðgiB2 þ B3Þð1� hÞ2h

pþ ðB0 þ liB1Þh3

3

þð1� hÞ2ð6B2 � pðgiB2 þ B3Þð1� hÞÞ3p

);

-2 ¼h3i

RðxÞ �2ðB0 þ liB1Þ

3

E�D

C

�3(

þð4B2 � 2pð1� hÞðgiB2 þ B3ÞÞp

E�D

C

�2

� 8B2ð1� hÞ � 2pðgiB2 þ B3Þð1� hÞ2

p

!E�D

C

�

�ðB0 þ liB1Þh3

3� 2B2 � ð1� hÞðgiB2 þ B3Þ

p

�h2

� 4B2ð1� hÞ � ðgiB2 þ B3Þð1� hÞ2

p

!h

� 2B2 � ð1� hÞðgiB2 þ B3Þp

�h2 þ 64B3

2

3p3ðgiB2 þ B3Þ2

� 2B2ð1� hÞ2

pþ ð1� hÞ3ðgiB2 þ B3Þ

3

);

-3 ¼h3i

RðxÞ2ðB0 þ liB1Þ

3

E�D

C

�3(

� 4B2 � 2pð1� hÞðgiB2 þ B3Þp

�E�D

C

�2

þ 8B2ð1� hÞ � 2pðgiB2 þ B3Þð1� hÞ2

p

!E�D

C

�

� 2ðB0 þ giB1Þ3

EþD

C

�3

þ 4B2 � 2pð1� hÞðgiB2 þ B3Þp

�EþD

C

�2

þ 8B2ð1� hÞ � 2pðgiB2 þ B3Þð1� hÞ2

p

!EþD

C

�

þðB0 þ liB1Þh3

3þ 4B2ð1� hÞ � pðgiB2 þ B3Þð1� hÞ2

p

!h

þ 6B2 � pð1� hÞðgiB2 þ B3Þ3p

�ð1� hÞ2

þ 2B2 � pð1� hÞðgiB2 þ B3Þp

�h2

�;

C ¼ 2ðB0 þ liB1Þ;

D¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4B2�2ðgiB2þB3Þð1�hÞ

p

�2

�4ðB0þliB1Þð1�hÞð4B2�ðgiB2þB3Þð1�hÞÞ

s;

E ¼ 4B2 � 2ðgiB2 þ B3Þð1� hÞp

;

h ¼ x� xi

hi; h� ¼ 1� 2

pðgi � 1Þ :

214 M. Hussain, S. Saleem

Proof. Let {(xi, fi), i= 0,1,2, . . . ,n} be the plane data defined

over the interval [a, b]. If we interpolate this data by rationalquadratic trigonometric spline (5) then S(xi) = fi, S

0 (xi) = di,i= 0,1,2, . . . ,n but S(x) approximates the data in between the

knots. To measure the accuracy of this approximation supposethat the data are generated from a function g(x) 2 C3[x0, xn].The error of approximation over an arbitrary subintervalIi = [xi, xi+1] is defined by Peano Kernel Theorem [20] as:

F½g� ¼ gðxÞ � SðxÞ ¼ 1

2

Z xiþ1

xi

gð3ÞðkÞFxðwÞdk;

where Fx is known as Peano Kernel and w ¼ ðx� kÞ2þ is thetruncated power function. The absolute error of approxima-tion over the subinterval Ii = [xi, xi+1] is defined as:

jgðxÞ � SðxÞj 6 1

2kgð3ÞðkÞk

Z xiþ1

xi

jFxðwÞjdk: ð6Þ

Due to truncated power function, Fx(w) is partitioned into twosubintervals as follows:

Fx(w) = f1(x, k), for k 2 (xi, x) and Fx(w) = f2(x, k), fork 2 (x, xi+1), where

f1ðx; kÞ ¼ ðx� kÞ2 �ðgiB2 þ B3Þðxiþ1 � kÞ2 � 4hi

p B2ðxiþ1 � kÞRðxÞ ;

f2ðx; kÞ ¼ �ðgiB2 þ B3Þðxiþ1 � kÞ2 � 4hi

p B2ðxiþ1 � kÞRðxÞ :

Bk are the Bk(x), k = 0, 1, 2, 3 are already defined in Section 2.The integral involved in Eq. (6) is expressedR xiþ1

xijFxðxÞjdk ¼

R xxijf1ðx; kÞjdkþ

R xiþ1x jf2ðx; kÞjdk.

It is observed by simple computation that if

gi < min 1; 1þ 2p

� �¼ 1, then the roots of f1 (x, x)and f2(x, x)

in [0,1] are h = 0, h = 1. If gi > max 1; 1þ 2p

� �¼ 1þ 2

p, then

the roots of f1(x, x) and f2(x, x) in [0,1] are h = 0, h = 1 and

h = h*, where h� ¼ 1� 2pðgi�1Þ

.

To locate the roots of f1(x, k), it is rearranged as:

f1ðx; kÞ ¼1

RðxÞ ðx� kÞ2ðB0 þ liB1Þn

þðx� kÞhi4

pB2 � 2ðgiB2 þ B3Þð1� hÞ

�þh2i

4

pB2ð1� hÞ � ðgiB2 þ B3Þð1� hÞ2

��:

The roots of f1(x, k) are k�1 ¼ xþ hiE�DC

� �and k�2 ¼ xþ hi

EþDC

� �,

where

C¼ 2ðB0þliB1Þ;

D¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið4B2�2pðgiB2þB3Þð1�hÞÞ

p

�2

�4ðB0þliB1Þð4B2ð1�hÞ�ðgiB2þB3Þð1�hÞ2Þ

s;

E¼ð4B2�2pðgiB2þB3Þð1�hÞÞp

:

The roots of f2(x, k) are k* = xi+1 and k� ¼ xiþ1 � 4hiB2

pðgiB2þB3Þ.

1. For h 2 [0,1] , h* R [0,1] and gi < 1, jgðxÞ � SðxÞj 6 12kgð3Þ

ðkÞkh3i -1ðli; gi; hÞ,

-1 ¼Z x

xi

jf1ðx; kÞjdkþZ xiþ1

x

jf2ðx; kÞjdk ¼Z x

xi

f1ðx; kÞdk

þZ xiþ1

x

f2ðx; kÞdk ¼h3i

RðxÞ2B2

2 � pð1� hÞðgiB2 þ B3Þh2

p

þ 4B2ð1� hÞh� pðgiB2 þ B3Þð1� hÞ2h

pþ ðB0 þ liB1Þh3

3

þð1� hÞ2 6B2 � p giB2 þ B3ð Þð1� hÞð Þ3p

):

2. For h 6 h*, h* 2 [0,1] and gi > 1þ 2p ; jgðxÞ � SðxÞj

612

gð3ÞðkÞ h3

i -2 li; gi; hð Þ,

-2 ¼Z x

xi

jf1ðx; kÞjdkþZ xiþ1

x

jf2ðx; kÞjdk ¼ �Z k�1

xi

f1ðx; kÞdk

þZ x

k�1

f1ðx; kÞdk�Z k�

x

f2ðx; kÞdkþZ xiþ1

k�f2ðx; kÞdk

¼ h3iRðxÞ �

2ðB0 þ liB1Þ3

E�D

C

�3(

þ 4B2 � 2pð1� hÞðgiB2 þ B3Þð Þp

E�D

C

�2

� 8B2ð1� hÞ � 2pðgiB2 þ B3Þð1� hÞ2

p

!E�D

C

��ðB0 þ liB1Þh3

3� 2B2 � ð1� hÞðgiB2 þ B3Þ

p

�h2

� 4B2ð1� hÞ � ðgiB2 þ B3Þð1� hÞ2

p

!hþ 64B3

2

3p3ðgiB2 þ B3Þ2

� 2B2ð1� hÞ2

pþ ð1� hÞ3 giB2 þ B3ð Þ

3

):

3. For h P h*, h* 2 [0,1] and gi > 1þ 2p ,

jgðxÞ � SðxÞj 6 12kgð3ÞðkÞkh3

i -3ðli; gi; hÞ,

-3 ¼Z x

xi

jf1ðx; kÞjdkþZ xiþ1

x

jf2ðx; kÞjdk ¼Z k�1

xi

f1ðx; kÞdk

�Z k�2

k�1

f2ðx; kÞdkþZ x

k�2

f1ðx; kÞdkþZ xiþ1

x

f2ðx; kÞdk

¼ h3iRðxÞ

2ðB0 þ liB1Þ3

E�D

C

�3(

� 4B2 � 2pð1� hÞðgiB2 þ B3Þp

�E�D

C

�2

þ 8B2ð1� hÞ � 2pðgiB2 þ B3Þð1� hÞ2

p

!E�D

C

�� 2ðB0 þ giB1Þ

3

EþD

C

�3

þ 4B2 � 2pð1� hÞðgiB2 þ B3Þp

�EþD

C

�2

þ 8B2ð1� hÞ � 2pðgiB2 þ B3Þð1� hÞ2

p

!

� EþD

C

�þ ðB0 þ liB1Þh3

3

þ 4B2ð1� hÞ � pðgiB2 þ B3Þð1� hÞ2

p

!h

þ 6B2 � pð1� hÞðgiB2 þ B3Þ3p

�ð1� hÞ2

þ 2B2 � pð1� hÞðgiB2 þ B3Þp

�h2

�: �

215

Remark 1. It is of great interest that for suitably selected

C1 rational quadratic trigonometric spline

parameters li and gi, the piecewise rational quadratictrigonometric spline S(x) is curvature continuous at each break

point if S(2) (xi�) = S(2)(xi+) or Sð2Þi�1ðxi�Þ

���di¼p

2

¼ Sð2Þi

ðxiþÞjdi¼0.

The above condition gives the following system of

equations:

2hidi�1 þ fð4gi�1 � 4Þhi þ ð4li � 4Þhi�1gdi þ 2hi�1diþ1

¼ pfli�1hiDi�1 þ gihi�1Dig; i ¼ 1; 2; 3; . . . ; n� 1: ð7Þ

Eq. (7) represents system of (n � 1) equations. If the derivativeparameters di, i = 0,1,2, . . . ,n are unknowns and shape designparameters (li, gi, i= 0,1,2, . . . ,n) are known then it repre-

sents system of (n � 1) equations in (n + 1) unknowns di,i= 0,1,2, . . . ,n. If the shape design parameters are also un-knowns, then there are (3n+ 1) unknowns. The unique solu-

tion can be determined by applying the end conditions.

Theorem 3. For a positive data, the rational quadratic trigono-metric spline defined in Eq. (5) preserves positivity if the shapeparameters li and gi in each subinterval [xi, xi+1] satisfy the fol-

lowing conditions:

li > max 0;� 2hidipfi

�and gi > max 0;

2hidiþ1pfiþ1

�:

Proof. Assume that the given set of positive data be {(x0,f0), (x1, f1), . . . , (xn, fn)}, xi < xi+1, i= 0,1,2, . . . ,n � 1 andfi > 0, "i. The curve produced by the interpolation of positive

Uðx; yÞ ¼ð1� sinðujÞÞ

2D0 þ ð1� sinðujÞÞ sinðujÞD1 þ ð1� cosðujÞÞbQi;jðujÞ

D0 ¼ Uðx; yjÞ; D1 ¼ li;jUðx; yjÞ þ hjUxðx; yjÞ; D2 ¼ gi;jUðx; yjþ1

D3 ¼ Uðx; yjþ1Þ; uj ¼pðy� yjÞ

2hj; hj ¼ yjþ1 � yj;

bQi;jðujÞ ¼ 1� sinðujÞ� �2 þ li;j 1� sinðujÞ

� �sinðujÞ þ gi;jð1� cosðuj

Uðx; yjÞ ¼1� sinðdiÞð Þ2E0 þ 1� sinðdiÞð Þ sinðdiÞE1 þ ð1� cosðdiÞÞ

Qi;jðdiÞ

E0 ¼ Fi;j; E1 ¼ li;jFi;j þ hiFxi:j; E2 ¼ gi;jFiþ1;j � hiF

xiþ1;j; E3 ¼ Fi

Qi;jðdiÞ ¼ ð1� sinðdiÞÞ2 þ li;jð1� sinðdiÞÞ sinðdiÞ þ gi;jð1� cosðdiÞÞ c

di ¼pðx� xiÞ

2hiand hi ¼ xiþ1 � xi:

Uyðx; yjÞ ¼ð1� sinðdiÞÞ2G0 þ ð1� sinðdiÞÞ sinðdiÞG1 þ ð1� cosðdiÞÞ

Qi;jðdiÞG0 ¼ Fy

i;j;G1 ¼ li;jFyi;j þ hiF

xyi:j ;G2 ¼ gi;jF

yiþ1;j � hiF

xyiþ1:j;G3 ¼ Fy

iþ1;j:

data will be positive if S(x) > 0, for all x 2 [x0, xn]. Since each

of the Bi(x), k = 0, 1, 2, 3 is positive for di 2 0; p2

� �and for all

li > 0, gi > 0, so, R(x) is positive over the whole domain.Now the problem of positivity of S(x) has been reduced to

the positivity of

B0ðxÞfi þ B1ðxÞ lifi þ2hidi

p

�þ B2ðxÞ gifiþ1 �

2hidiþ1p

�þ B3ðxÞfiþ1;

B0ðxÞfi þ B1ðxÞ lifi þ2hidi

p

�þ B2ðxÞ gifiþ1 �

2hidiþ1p

�þ B3ðxÞfiþ1

> 0 only if li > �2hidipfi

and gi >2hidiþ1pfiþ1

:

Combining li > 0, gi > 0, li > � 2hidipfi

and gi >2hidiþ1pfiþ1

, we havethe required result.

4. Bivariate rational quadratic trigonometric function

Let {(xi, yj,Fi, j), i = 0,1,2, . . . ,n1; j= 0,1,2, . . . ,n2} be the givenset of regular data arranged over the rectangular grid [xi,xi+1] · [yj, yj+1], i= 0,1,2, . . . ,n1 � 1; j = 0,1,2, . . . ,n2 � 1.

We wish to interpolate these data by using rational quadratictrigonometric spline (5). Since each rectangle is bounded by fourboundary curves. The final surface patch is obtained by blending

these boundary curves. Interpolation and blending of boundarycurves by rational quadratic trigonometric spline defined in Eq.(5) give birth to the following bivariate rational quadratictrigonometric function over each rectangular patch.

cosðujÞD2 þ ð1� cosðujÞÞ2D3

; ð8Þ

Þ � hjUxðx; yjþ1Þ;

ÞÞ cosðujÞ þ ð1� cosðujÞÞ2;

cosðdiÞE2 þ ð1� cosðdiÞÞ2E3;

ð9Þ

þ1;j;

osðdiÞ þ ð1� cosðdiÞÞ2;

cosðdiÞG2 þ ð1� cosðdiÞÞ2G3;

ð10Þ

216 M. Hussain, S. Saleem

U(x, yj+1) andUy(x, yj+1) are obtained by replacing j by j+ 1 inEqs. (8) and (10).

li, j and gi, j are free parameters along x-axis and li;j and gi;j

are free parameters along y-axis. Fxk;l;F

yk;l;F

xyk;l : k ¼ i;

niþ 1; l ¼ j; jþ 1g are the partial derivatives at four corners of(i, j)th-patch. The bivariate rational quadratic trigonometricfunction (8)) has the following properties:

Uðxi; yjÞ ¼ Fi;j;@Uðxi; yjÞ

@x¼ Fx

i;j;

@Uðxi; yjÞ@y

¼ Fyi;j;

@2Uðxi; yjÞ@x@y

¼ Fxyi;j :

Theorem 4. The piecewise bivariate rational quadratic trigono-

metric function U(x, y) is C1 over the whole domain if the shapedesign parameters satisfy the following relation:

(i) li, j = li and gi, j = gi, i = 0,1,2, . . . , n1 � 1 and for all

values of j.(ii) li;j ¼ lj and gi;j ¼ gj, j = 0,1,2, . . . , n2 � 1 and for all

values of i.

Proof. The rational quadratic trigonometric function (8) inter-

polates the data values Fi, j and partial derivatives Fxi;j;F

yi;j, F

xyi;j

defined at four corners of rectangular patch, i.e.

Uðxi; yjÞ ¼ Fi;j;@Uðxi; yjÞ

@x¼ Fx

i;j;@Uðxi; yjÞ

@y¼ Fy

i;j;

@2Uðxi; yjÞ@x@y

¼ Fxyi;j :

Since each rectangular patch is bounded by four boundary

curves so to blend the rectangular patches to generate aC1 continuous surface following sufficient conditions mustbe satisfied along the four boundaries of each rectangular

patch:

@Ui;jðxiþ1; yÞ@x

����di¼p

2

� @Uiþ1;jðxiþ1; yÞ@x

����diþ1¼0

¼ 0;

@Ui�1;jðxi; yÞ@x

����di�1¼p

2

� @Ui;jðxi; yÞ@x

����di¼0¼ 0;

@Ui;jðx; yjþ1Þ@y

����uj¼p

2

�@Ui;jþ1ðx; yjþ1Þ

@y

����ujþ1¼0

¼ 0;

@Ui;j�1ðx; yjÞ@y

����uj�1¼p

2

�@Ui;jðx; yjÞ

@y

����uj¼0¼ 0:

After some simple computation it is observed that@Ui;jðxiþ1 ;yÞ

@x

���di¼p

2

� @Uiþ1;jðxiþ1 ;yÞ@x

���diþ1¼0

¼ 0,

if liþ1;j ¼ li;j; giþ1;j

¼ gi;j;Fxiþ1;j liþ1;jgi;j � giþ1;jli;j

� �þ 2hj

pFxyiþ1;j li;j � liþ1;j� �

¼ 0: ð11Þ

@Ui�1;jðxi; yÞ@x

����di�1¼p

2

� @Ui;j xi; yð Þ@x

����di¼0¼ 0

if li;j ¼ li�1;j; gi;j ¼ gi�1;j; Fxi;j li;jgi�1;j � gi;jli�1;j� �

þ 2hjp

Fxyi;j li�1;j � li;j

� �¼ 0:

ð12Þ

@Ui;jðx; yjþ1Þ@y

����uj¼p

2

�@Ui;jþ1ðx; yjþ1Þ

@y

����ujþ1¼0

¼ 0

if li;jþ1 ¼ li;j; gi;jþ1 ¼ gi;j; Fyi;jþ1 li;jþ1gi;j � gi;jþ1li;j

� �þ 2hi

pFxyi;jþ1 li;j � li;jþ1� �

¼ 0;

Fyiþ1;jþ1ðli;jþ1gi;j � gi;jþ1li;jÞ þ

2hip

Fxyiþ1;jþ1ðli;j � li;jþ1Þ ¼ 0:

ð13Þ

@Ui;j�1ðx; yjÞ@y

����uj�1¼p

2

�@Ui;j x; yj

� �@y

����uj¼0¼ 0 if

li;j ¼ li;j�1; gi;j ¼ gi;j�1; Fyi;jðli;jgi;j�1 � gi;jli;j�1Þ

þ 2hip

Fxyi;j ðli;j�1 � li;jÞ ¼ 0;

Fyiþ1;j li;jgi;j�1 � gi;jli;j�1� �

þ 2hip

Fxyiþ1;jðli;j�1 � li;jÞ0:

ð14Þ

The system of Eqs. (11)–(14) is satisfied only if li, j = li and gi,j = gi, i = 0,1,2, . . . ,n1 � 1 and for all values of j. li;j ¼ lj and

gi;j ¼ gj, j= 0,1,2, . . . ,n2 � 1 and for all values of i.

Theorem 5. The C1 bivariate rational quadratic trigonometricfunction U (x, y) is positive over the whole domain if the follow-ing sufficient conditions are satisfied:

Uðxi; yjÞ ¼ Fi;j; 8i ¼ 0; 1; 2; . . . ; n1; j ¼ 0; 1; 2; . . . ; n2;

li > max 0;�2hiF

xi;j

pFi;j

;�2hiF

xi;jþ1

pFi;jþ1

�;

gi > max 0;2hiF

xiþ1;j

pFiþ1;j;2hiF

xiþ1;jþ1

pFiþ1;jþ1

�

lj > max 0;�2hjF

yi;j

pFi;j

;�2hjF

yiþ1;j

pFiþ1;j;�

2hj pliFyi;j þ 2hiF

xyi;j

� �p liFi;j þ 2hiF

xi;j

� � ;

8<:�2hj pgiF

yiþ1;j � 2hiF

xyiþ1;j

� �p pgiFiþ1;j � 2hiF

xiþ1;j

� �9=;;

gj > max 0;2hjF

yi;jþ1

pFi;jþ1;2hjF

yiþ1;jþ1

pFiþ1;jþ1;2hj pliF

yi;jþ1 þ 2hiF

xyi;jþ1

� �p pliFi;jþ1 þ 2hiF

xi;jþ1

� � ;

8<:2hj pgiF

yiþ1;jþ1 � 2hiF

xyiþ1;jþ1

� �p pgiFiþ1;jþ1 � 2hiF

xiþ1;jþ1

� �9=;:

Proof. Let {(xi, yj, Fi, j): i= 0,1,2, . . . ,n1; j= 0,1,2, . . . ,n2} bethe given set of positive regular data defined over the domain

D= [c, d] · [e, f]. The requirement is to develop an interpolat-ing C1 bivariate positive rational quadratic trigonometric func-tion, i.e. U(xi, yj) = Fi, j, i= 0,1,2, . . . ,n1; j = 0,1,2, . . . ,n2,and U(x, y) > 0, for all (x, y) 2 D. The C1 bivariate rationalquadratic trigonometric function (8) is rearranged as:

Figure 2 Rational quadratic trigonometric spline with li = 0.4

and gi = 0.6.

C1 rational quadratic trigonometric spline 217

Uðx;yÞ¼

�ð1� sinðdiÞÞ2P0þð1� sinðdiÞÞsinðdiÞP1þð1� cosðdiÞÞcosðdiÞP2þð1� cosðdiÞÞ2P3

1� sinðdiÞð Þ2þli 1� sinðdiÞð ÞsinðdiÞþgið1� cosðdiÞÞcosðdiÞþð1� cosðdiÞÞ2;

P0 ¼ ð1� sinðujÞÞ2T0;0 þ ð1� sinðujÞÞ sinðujÞT0;1

þ ð1� cosðujÞÞ cosðujÞT0;2 þ ð1� cosðujÞÞ2T0;3;

P1 ¼ 1� sinðujÞ� �2

T1;0 þ 1� sinðujÞ� �

sinðujÞT1;1

þ ð1� cosðujÞÞ cosðujÞT1;2 þ ð1� cosðujÞÞ2T1;3;

P2 ¼ ð1� sinðujÞÞ2T2;0 þ ð1� sinðujÞÞ sinðujÞ

T2;1 þ ð1� cosðujÞÞ cosðujÞT2;2 þ ð1� cosðujÞÞ2T2;3

P3 ¼ ð1� sinðujÞÞ2T3;0 þ ð1� sinðujÞÞ sinðujÞ

T3;1 þ ð1� cosðujÞÞ cosðujÞT3;2 þ ð1� cosðujÞÞ2T3;3

T0;0 ¼ Fi;j; T0;1 ¼ ljFi;j þ2hjp

Fyi;j;

T0;2 ¼ gjFi;jþ1 �2hjp

Fyi;jþ1; T0;3 ¼ Fi;jþ1;

T1;0 ¼ liFi;j þ2hip

Fxi;j;

T1;1 ¼ lj liFi;j þ2hip

Fxi;j

�þ 2hj

pliF

yi;j þ

2hip

Fxyi;j

�;

T1;2 ¼ gj liFi;jþ1 þ2hip

Fxi;jþ1

�� 2hj

pliF

yi;jþ1 þ

2hip

Fxyi;jþ1

�;

Table 1 A 2D positive data set.

x 1.0 2.0 3.0 8.0 10.0 11.0 12.0 14.0

y 14.0 8.0 2.0 0.8 0.5 0.25 0.4 0.37

Figure 1 Linear interpolation of data.

Figure 3 Positive rational quadratic trigonometric spline.

T1;3 ¼ liFi;jþ1 þ2hip

Fxi;jþ1;T2;0 ¼ giFiþ1;j �

2hip

Fxiþ1;j;

T2;1 ¼ lj giFiþ1;j �2hip

Fxiþ1;j

�þ 2hj

pgiF

yiþ1;j �

2hip

Fxyiþ1;j

�;

T2;2¼ gj giFiþ1;jþ1�2hipFxiþ1;jþ1

��2hj

pgiF

yiþ1;jþ1�

2hipFxyiþ1;jþ1

�;

T2;3¼ giFiþ1;jþ1�2hipFxiþ1;jþ1;

T3;0 ¼ Fiþ1;j; T3;1 ¼ ljFiþ1;j þ2hjp

Fyiþ1;j;

T3;2 ¼ gjFiþ1;jþ1 �2hjp

Fyiþ1;jþ1; T3;3 ¼ Fiþ1;jþ1:

Table 2 A 3D positive data set.

y/x �3 �2 �1 0 1 2 3

�3 0.0123 0.0236 0.0399 0.0493 0.0399 0.0236 0.0123

�2 0.0236 0.0624 0.1599 0.2499 0.1599 0.0624 0.0236

�1 0.0399 0.1599 0.9999 3.9996 0.9999 0.1599 0.0399

0 0.4444 0.2499 3.9996 40,000 3.9996 0.2499 0.4444

1 0.0399 0.1599 0.9999 3.9996 0.9999 0.1599 0.0399

2 0.0236 0.0624 0.1599 0.2499 0.1599 0.0624 0.0236

3 0.0123 0.0236 0.0399 0.0493 0.03999 0.0236 0.0123

Figure 4 Linear interpolation of positive data.

Figure 5 C1 bivariate rational quadratic trigonometric function

with (li = 0.6, gi = 2, lj ¼ 0:8 and gj ¼ 2:5).

Figure 6 xz-view of Fig. 5.

218 M. Hussain, S. Saleem

Since the shape design parameters li and gi are assumed as po-sitive real numbers. Moreover, the parameters di and uj are re-

stricted to subinterval 0; p2

� �. The denominator of U(x, y) is

always positive. Thus, positivity of U(x, y) depends upon the

positivity of Pk, k= 0, 1, 2, 3.

P0 > 0 is positive if T0,k > 0, k= 0, 1, 2, 3.

T0,k > 0, k = 0, 1, 2, 3 if lj > �2hjF

yi;j

pFi;jand gj >

2hjFyi;jþ1

pFi;jþ1.

Similarly, P1 > 0 if

li > �2hiF

xi;j

pFi;j

; li > �2hiF

xi;jþ1

pFi;jþ1;

lj > �2hj pliF

yi;j þ 2hiF

xyi;j

� �p pliFi;j þ 2hiF

xi;j

� � ; gj >2hj pliF

yi;jþ1 þ 2hiF

xyi;jþ1

� �p pliFi;jþ1 þ 2hiF

xi;jþ1

� � :

Similarly for P2 and P3 are positive if

gi >2hiF

xiþ1;j

pFiþ1;j; gi >

2hiFxiþ1;jþ1

pFiþ1;jþ1; gj >

2hjFyiþ1;jþ1

pFiþ1;jþ1;

lj > �2hjF

yiþ1;j

pFiþ1;j; lj > �

2hj pgiFyiþ1;j � 2hiF

xyiþ1;j

� �p pgiFiþ1;j � 2hiF

xiþ1;j

� � ;

gj >2hj pgiF

yiþ1;jþ1 � 2hiF

xyiþ1;jþ1

� �p pgiFiþ1;jþ1 � 2hiF

xiþ1;jþ1

� � :

5. Numerical examples

Example 1. Here rational quadratic trigonometric splineinterpolation of positive data set {(x, y): (1.0,14.0), (2.0,8.0),

Figure 7 yz-view of Fig. 5.

C1 rational quadratic trigonometric spline 219

(3.0,2.0), (8.0,0.8), (10.0,0.5), (11.0,0.25), (12.0,0.4),(14.0,0.37)} is discussed. The tabular form of this data set is

(see Table 1)

This data is linearly interpolated by MATLAB build in

function plot in Fig. 1 which shows that its shape is positiveover whole domain. In Fig. 2, the same data are interpolatedby rational quadratic trigonometric spline with li = 0.4 and

gi = 0.6. This figure clearly indicates that rational quadratictrigonometric spline is unable to preserve the positive shapeof data for arbitrary values of shape design parameters. The

strength of Theorem 3 is checked by implementing it on thesedata shown in Fig. 3 which reveals that rational quadratic trig-onometric spline interpolates positive shape of data positivelyif the shape design parameters obey the restrictions developed

in Theorem 4.

Example 2. The regular positive surface data are generated

from the square function hðx; yÞ ¼ 4

ðx2þy2Þ2þ0:0001. Although this

function is positive over every domain but for the ease of

Figure 8 Positive C1 bivariate rational quadratic trigonometric

function.

computation the domain is restricted to S1 · S2, where

S1 = S2 = {�3,�2,�1,0,1,2,3}. The tabular form of thispositive data set is (see Table 2)

The positive shape of the data is produced in Fig. 4 by itslinear interpolation. C1 bivariate rational quadratic trigono-metric function for arbitrary values of shape design parame-

ters, is unable to preserve the positive shape of the data. It isexposed in Fig. 5 (li = 0.6, gi = 2, lj ¼ 0:8 and gj ¼ 2:5),Fig. 6(xz-view) and Fig. 7(yz-view). Fig. 8 is the test of Theo-rem 7 on this data. The surface produced in Fig. 8 is positive

over the whole domain.

6. Conclusion

In this study, a Bezier like rational quadratic trigonometricspline is developed to assure C1-continuity in rational qua-dratic structure. The shape preserving schemes proposed in

[13] was C1 for uniform knot, [17] was C0. In [14,15] orderof continuity was dependent on multiplicity of knot and shapeparameters, in [19] C1-continuity was dependent on knot vec-

tors and choice of derivatives, [20] restricted the derivativesto Di for tangent continuity. The data arising in most of theapplications are non-uniform and does not restrict the deriva-

tives; hence, these schemes are not applicable to a wide rangeof functions where derivative preservation is also mandatory.The order of continuity of rational quadratic trigonometricspline of this paper is independent of knot spacing, slope of se-

cant line and shape parameters.The developed curve and regular surface data interpolants

are likely to preserve the shape of data, unlike

[1,4,5,11,12,18,19,21], without constraining the interval andderivatives.

The order of approximation of the developed scheme is

O h3i� �

in quadratic structure, whereas the order of approxima-tion of rational interpolant used in [17] was O h2i

� �.

References

[1] Asim MR, Brodlie KW. Curve drawing subject to positivity and

more general constraints. Comput Graph 2003;27:469–85.

[2] Bao F, Sun Q, Pan J, Duan Q. A blending interpolator with value

control and minimal strain energy. Comput Graph

2010;34:119–24.

[3] Brodlie K, Mashwama P, Butt S. Visualization of surface data to

preserve positivity and other simple constraints. Comput Graph

1995;19:585–95.

[4] Butt S, Brodlie KW. Preserving positivity using piecewise cubic

interpolation. Comput Graph 1993;17(1):55–64.

[5] Casciola G, Romani L. Rational interpolants with tension

parameters. In: Lyche T, Mazure ML, Schumaker LL, editors.

Curves and surfaces design, Saint Malo. Brentwood: Nashboro

press; 2002, 2003.

[6] Delgado J, Pena JM. Are rational Bezier surfaces monotonicity

preserving? Comput Aided Geomet Des 2007;24(5):303–6.

[7] Duan Q, Bao F, Du S, Twizell EH. Local control of interpolating

rational cubic spline curves. Comput Aided Des 2009;41:825–9.

[8] Duan Q, Djidjeli K, Price WG, Twizell EH. A rational cubic

spline based on function values. Comput Graph

1998;22(4):479–86.

[9] Duan Q, Zhang Y, Twizell EH. A bivariate rational interpolation

and the properties. Appl Math Comput 2006;179:190–9.

[10] Duan Q, Zhang H, Zhang Y, Twizell EH. Error estimation of a

kind of rational spline. J Comput Appl Math 2007;20(1):1–11.

220 M. Hussain, S. Saleem

[11] Farouki RT, Manni C, Sestini A. Shape-preserving interpolation

by G1 and G2 PH quintic splines. IMA J Numer Anal

2003;23:175–95.

[12] Floater MS. A weak condition for the convexity of tensor-product

Bezier and B-spline surfaces. Adv Comput Math 1994;2(1):67–80.

[13] Goodman TNT, Ong BH. Shape-preserving interpolation by

splines using vector subdivision. Adv Comput Math

2005;22(1):49–77.

[14] Han X. Cubic trigonometric polynomial curves with a shape

parameter. Comput Aided Geomet Des 2004;21:535–48.

[15] Han X. C2 quadratic trigonometric curves with local bias. J

Comput Appl Math 2005;180:161–72.

[16] Han Xi-An, Ma Y, Haung X. The cubic trigonometric Bezier

curve with two parameters. Appl Math Lett 2009;22:226–31.

[17] Hussain MZ, Ayub N, Irshad M. Visualization of 2D data by

rational quadratic function. J Inform Comput Sci

2007;2(1):17–26.

[18] Lamberti P, Manni C. Shape-preserving C2 functional interpola-

tion via parametric cubics. Numer Algorithms 2001;28:229–54.

[19] Manni C, Sablonniere P. C1comonotone Hemite interpolation via

parametric surfaces. In: Drehlen M, Lyche T, Schumaker LL,

editors. Mathematical methods for curves and surfaces. Vender-

bilt University Press; 1995.

[20] Schultz MH. Spline analysis. Englewood Cliffs, New Jersey:

Prentice-Hall; 1973.

[21] Zhang Y, Duan Q, Twizell EH. Convexity control of bivariate

rational interpolating spline surfaces. Comput Graph

2007;31:679–87.