an applied two dimensional b-spline model for interpolation of data

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –

6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME

322

AN APPLIED TWO-DIMENSIONAL B-SPLINE MODEL FOR

INTERPOLATION OF DATA

Mehdi Zamani

Civil Engineering Department, Faculty of Technology and Engineering, Yasouj University,

Yasouj, IRAN, [email protected]

ABSTRACT

The literature about the interpolation of data is less than the one's for the

approximation of data especially for three dimension data. The three methods of

interpolation, two-dimensional Lagrange, two-dimensional cubic spline and two-dimensional

explicit cubic spline are investigated. In the present study a new model with two-dimensional

B-spline approach has been developed. The presented model has the advantage of simplicity

and applicability with respect to the volume of operation calculations and using non-uniform

and non-symmetric data. Three problems were selected for the testing and verifying the

model with the square arrangement of the data and its non-uniform distribution. The results

indicate that this model is simple, efficient and applicable for the two-dimensional

interpolation of data. The model can be generalized for non-regular and non-uniform

distribution of data, which can result in a bounded sparse matrix for the governing linear

system of equations.

Keywords: approximation, B-spline, cubic spline, interpolation, Lagrange method

I. INTRODUCTION

The interpolation models are applied considerably in all branches of engineering

activities. Examples are the evaluation of surface area and volume of non-uniform and

irregular objects, determination of volumes of cutting, filling and embankment in earth work,

and engineering surveying. Volume of surface topography and morphology forms such as

hills and valleys. Here interpolation means to find a model function such as )(xp which

satisfies n data points fi therefore; ii fxp =)( for ni ,,3,2,1 L= . In approximation approach, the

model curve does not necessarily cross the data points. The classic methods of interpolation

are explained in details by (Fomel, 1997b), (Collins, 2003), (Dubin, 2003), (Kiusalaas, 2005),

(Nocedal and Wright, 2006) and (Zamani, 2009a). The methods for two-dimensional

INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN

ENGINEERING AND TECHNOLOGY (IJARET)

ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 3, Issue 2, July-December (2012), pp. 322-336 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2012): 2.7078 (Calculated by GISI) www.jifactor.com

IJARET

© I A E M E



323

943

73432313

7421

2

72423212

74311

71413121

7432

)()())()()((

))()()((

))()()((

))()()((

))()()((

))()()((),(

zzzyyyyxxxx

yyyyxxxx

zyyyyxxxx

yyyyxxxxz

yyyyxxxx

yyyyxxxxyxz

+++−−−−

−−−−

+−−−−

−−−−+

−−−−

−−−−=

L

x

y

interpolation in engineering literature are not frequent. The most important models in this

area are Lagrange method, B-spline method, two-dimensional cubic spline method, and

explicit two-dimensional cubic spline method. The introduction of the paper should explain

the nature of the problem, previous work, purpose, and the contribution of the paper. The

contents of each section may be provided to understand easily about the paper.

1.1 Lagrange Method The Lagrange method for this case is the generation of one-dimensional Lagrange

formulation for two-dimensional interpolation problems (Dierckx, 1993) and (Burden and

Fairs, 2010). For a problem with a set of 9 triple data A ε {(xi, yi, zi) i=1,2,…,9)} Figure 1.

The two-dimensional Lagrange equation is as follows,

(1)

Fig: 1 Simple data distribution

The two-dimensional Lagrange model is limited in terms of applicability. The degree and

order of the Lagrange polynomial increases as the number of data point increases. Therefore,

it results in the oscillation and sinus behavior of the governing curve. Hence, the Lagrange

curves give high inclination and deviations from the real curves for the points between the

data or nodes. Therefore, the model is not applicable and efficient for huge engineering data.

1.2 B-spline Approximation Method B-spline methods have been extended since 1970s. The designated B stands for Basis,

so the full name of this approach is the basic spline. B-spline methods are mostly used for

curves and surfaces in computer graphics. The 2nd

and 3rd

degree B-splines which are used

extensively for approximation of data are less applicable for the interpolation of data. With

some modifications and corrections the B-spline models specially the second and third

degrees can be applied for interpolation of huge data, particularly for one-dimensional

interpolation problems. The 2nd

and 3rd

degrees B-spline polynomials can be used for

approximation of n control points )1,,2,1,0,( −= nipi L (Saxena and Sahay, 2005) and (Salomon,

2006) as,



324

2,,2,1

011

022

121

2

11)(

1

1

2 −=

−

−

=

+

−

ni

p

p

p

tttP

i

i

i

i L

[ ]{ } 3,,2,1

0141

0303

0363

1331

6

11)(

2

1

1

23 −==

−

−

−−

=

+

+

−

nipBt

p

p

p

p

ttttP

i

i

i

i

i L

]4[6

1

]4[6

1

212

111

++

+−

++=

++=

iii

iii

pppq

pppq

[ ][ ][ ] { }

−

−

−−

−

−

−−

==

++++++−

++++++−

++−

−+−+−−−

10001

1333

4063

1331

6

1

0141

0303

0363

1331

6

11

),(

2

3

2,22,12,2,1

1,21,11,1,1

,2,1,,1

1,21,11,1,1

23

,

u

u

u

zzzz

zzzz

zzzz

zzzz

ttt

uBzBtutp

jijijiji

jijijiji

jijijiji

jijijiji

T

jii

[ ][ ][ ][ ] [ ] { }yABZBAxyxpTT

i '),( =

12

1

12

12

1

12

22

2

2

',1

'

,,1

,

100

''0

'''2'

',

1

02

00

yy

yb

yya

xx

xb

xxaba

bbaa

A

bb

aab

a

AT

−−=

−=

−−=

−=

=

=

(2)

(3)

The above B-spline curves will not cross through the control points but pass near them. The initial

and terminal points (q1 and q2) of the cubic B-spline curve which are joint points can be obtained

from Eq. (4).

(4)

The two-dimensional cubic B-spline can be obtained from the product of Eq. (3) in two dimensions

as,

[ ]{ } [ ]{ }[ ]Tjii pBupBtutp =),(

(5)

(6)

Where [ ]1,0, ∈ut . Using the linear transformation from t and u to x and y; respectively for the

uniform B-spline surface patch in Fig. (2), the graph equation ),( yxpiresults as follows,

(7)

where 12xxx = , 12

yyy = and

(8)



325

x

y

[ ])4()4(4)4(36

11,11,1,1,1,,11,11,1,1 ++++−+−−+−−− ++++++++= jijijijijijijijijiij zzzzzzzzzq

3333

322322

3322

22

)()()()(

)()()()()()(

))(()()())(()()(

)())(()()()(),(

jijiijjiij

jiijjiijjiij

jiijjiijjiijjiij

jijjiijiijjijiijijij

yypxxoyyxxn

yyxxmyyxxlyyxxk

yyxxjyyxxiyyxxhyyxxg

yyfyyxxexxdyycxxbayxs

−+−+−−+

−−+−−+−−+

−−+−−+−−+−−+

−+−−+−+−+−+=

Fig: 2 Data distribution for B-spline surface patch

A network of data with (m+1)×(n+1) control points P00 through Pmn has (m+1) rows and

(n+1) columns with uniform x and y increments ),( yx ∆∆ . From the above network about (m-

2)×(n-2) local cubic B-spline graphs or patches can be obtained which have continuity at least

c1 on each element side. The final graph which consists of local graphs passes (m-1)×(n-1)

internal points ijq . If the data points transfer to those internal joint points

ijq the final graph

would be a two-dimensional cubic B-spline which is obtained by the interpolation method.

Hence, the final graph crosses all the data points. For the details of this method, the situation

of internal joint points, and their coordinates refer to (Salomon, 2006). The ijq can be

determined with respect to z value as Eq. (8).

(8)

The interpolation by two-dimensional cubic B-spline of a network data with mesh m×n and

uniform x∆ and y∆ requires 2(m+n+2) more equations in order to satisfy the uniqueness of

the governing system of equations. However, it relates to the degree of polynomial B-spline

considered. The above extra equations should be obtained by satisfying the boundary

conditions. The governed linear system of equations is a kind of nonadigonal system. It is

clear that this model is not practical for problems having huge data points.

1.3 Two-dimensional Cubic Spline Method This method is obtained by the product of two cubic spline equations in directions x and y as

Eq. (9).

(9)

Where ijs is bicubic spline which defines for each rectangular element and has 16 parameters.

Those parameters are obtained by satisfying the following 16 continuity equations on the

corners of each element.



326

3),(3),(

3),(3),(

3),(1),(

2

2

2

2

1,111

⇒∂

∂⇒

∂

∂

⇒∂

∂⇒

∂

∂

⇒=⇒= ++±±

ji

ij

ji

ij

ji

ij

ji

ij

jijiijijjiij

yxy

syx

y

s

yxx

syx

x

s

fyxsfyxs

3

9

2

8

2

7

3

6

2

2

54

2

3210

)(

))(()()()()()(

)())(()()()(),(

j

jijiijiij

jjiijiij

yya

yyxxayyxxaxxayyxxg

yyayyxxaxxayyaxxaayxs

−+

−−+−−+−+−−+

−+−−+−+−+−+=

=

=

=

=

++++

++

++

1,111

1,1

,11

),(

),(

),(

),(

jijiij

jijiij

jijiij

ijjiij

fyxs

fyxs

fyxs

fyxs

=∂

+∂

=∂

+∂

=∂

+∂

=∂

+∂

=∂

∂

=∂

∂

+

+

+

+'

1,,

'

,1,

'

,1,

'

1,,

'

,,

'

,,

)1,(

),1(

),1(

)1,(

),(

),(

jiy

ij

jiy

ij

jix

ij

jix

ij

jiy

ij

jix

ij

fy

jis

fy

jis

fx

jis

fx

jis

fy

jis

fx

jis

(10)

The above 16 equations of continuity for each element are written and combined together to

form a nonadiagonal linear system of equations. By adding the governing boundary

conditions to the above linear system it is complemented to a linear system of equations

which has a unique solution. This method of solution gives the c2 continuity along each node

and at least c1 continuity at the element sides.

1.4 Explicit Two-dimensional Cubic Spline Method An explicit two-dimensional interpolation model has been developed by (Zamani, 2010) that

is the simplification of the above two-dimensional cubic spline method. He applied uniform

network of data set (rectangular distribution). The interpolation equation which is

implemented in his model is as follows,

(11)

The coefficients 0a to

9a are obtained by using c1 continuity on each element sides. They

consist of 4 equations of continuity for function and 6 equations of continuity for the first

partial derivatives along the x and y axis at 4 nodes of each element, Eqs. (12) and (13).

(12)

(13)

With having the above partial derivatives the problem is unique and it consists of 10

equations and 10 unknowns which can be solved explicitly (it’s not necessary to form any

linear system of equations). In most cases the partial derivatives are not available at each



327

,,, '

,2

'

,10 ijyijxij fafafa ===

'

,1,,121023

1323jix

i

ji

iii

fh

fh

ah

ah

a ++ −+−−=

'

1,,2

'

,1,1,11,,1104

111)(

111++++++ +−+−++−−= jix

ji

jiy

i

jijiji

jijji

fk

ah

fh

fffkh

ak

akh

a

'

1,,1,22025

1323++ −+−−= jiy

j

ji

jjj

fk

fk

ak

ak

a

'

,1,2,1312036

1212jix

i

ji

iii

fh

fh

ah

ah

a ++ +−+=

'

1,,1,11,,121027

1)(

111+++++ −−+−+= jix

ji

jijiji

jijiji

fkh

fffkh

akh

akh

a

'

,1,1,11,,122028

1)(

111jiy

ji

jijiji

jijiji

fkh

fffkh

akh

akh

a +++++ −−+−+=

'

1,,21,322039

1212++ +−+= jiy

j

ji

jjj

fk

fk

ak

ak

a

[ ] )()()()()( 1100

1

ucucucxucxs nnii

n

i

i ββββ +++==∑=

L

node but can be removed or eliminated by using numerical derivatives approaches. The

coefficients 0a to

9a of Eq. (11) can be determined from the following equations.

(14)

Where ih and

ik are length and width of element ij ; respectively. For a huge set of data

which usually exists in engineering problems the above model is simple and requires less

calculation operations with respect to the other methods of two-dimensional interpolation.

The only limitation for this model is determining the partial derivatives with respect to x and

y axis at each node and they should be approximated by the governing methods.

II. MODEL FORMULATION

The model is the generation of B-spline method in two dimensional, therefore; it is

better to explain a brief review of the one-dimensional B-spline.

2.1 One-dimensional B-spline

This model is explained in more detail by (Salamon, 2006) but in this study it is briefly

discussed. The B-spline curve consists of a series of B-spline basic functions that are linearly

independent in domain [a, b] as,

(15)

where ]2,2[−∈u , hxxxu ii /)()( −= , nabh /)( −= and n is the number of elements. Each B-

spline basic function consists of four cubic polynomial functions as Eqs. (16).

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976

6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July

u

uu

uu

u

ui

0

)2(

1(3)1(3)1(31

1(3)1(3)1(31

)2(

)(

3

2

2

3

−

−−−+−+

+−++++

+

=β

aecaeu h

xxb

ub

i

i

−=−=

−−

−

2

2

)(β

)()(4)( 110011 xcxcxc =++−− βββ

)()(4)( 0

'

110

'

000

'

11 xcxcxc ++−− βββ

The following basic equation is applied for B

applicability and continuity order (Zamani, 2009b).

Where 016478.4=a , 3740615.1=b

function for Eq. (17).

Fig: 3

In order to promote the accuracy and efficiency of B

)(11 uc −− β and )(11 uc nn ++ β are added to Eq. (15) for adding the effects of boundary conditions.

The coefficients 1−c and 1+nc can be calculated with respect to the B

points according to the following equations.

After making derivatives from the B

(19) it can be written as,


6499(Online) Volume 3, Number 2, July-December (2012), © IAEME

328

otherwise

u

u

u

u

u

u

,

21,

10,

01,

12,

)

)3

3

≤≤

≤≤

≤≤−

−≤≤−

−

+

c

)(xf

)() 0

'xf=

The following basic equation is applied for B-spline basic function for increasing its

applicability and continuity order (Zamani, 2009b).

3740615 and 016478.0=c . Fig. (3) shows the B

Fig: 3 B-spline basic function

In order to promote the accuracy and efficiency of B-spline method two basic functions

are added to Eq. (15) for adding the effects of boundary conditions.

can be calculated with respect to the B-spline curve at boundary

points according to the following equations.

After making derivatives from the B-spline basic functions )(1 x−β , )(0 xβ and


December (2012), © IAEME

(16)

spline basic function for increasing its

(17)

. Fig. (3) shows the B-spline basic

spline method two basic functions

are added to Eq. (15) for adding the effects of boundary conditions.

spline curve at boundary

(18)

(19)

and )(1 xβ in Eq.



329

'

00

'

11 )( fxfcc ==+− − αα

ab

ehfc

fcc

b

2

'

01

'

011 −=−=−

α

ab

ehfc

fcc

b

nn

nnn

2

'

1

'

11 +=+= −−+α

ceayx hb

k −=−

22

2

),( l

ρ

β

[ ]ceayx

ji yyxxh

b

k −=−+−− 22

22)()(

),( lβ

[ ]ceayx

ji yyxxh

b

ij −=−+−− 22

2)()(

2

),(β

(20)

(21)

where h

abe

h−

=2

α . In the same way the coefficient 1+nc for the last boundary point can be

obtained.

(22)

With regarding to '

0f and '

nf the derivatives at the first and the last points are not available

and should be approximated. This model gives a tridiagonal linear system of equations with

dimensions (n+1)×(n+1) with 4 on the main diagonal entries and 1 on the sub diagonal and

sup diagonal components. The obtained linear system of equations is easy to solve. This can

be done by Thomas algorithm. The above model results in smooth curves for data of having

nonlinear distributions and uniform elements. The model can also be extended for data of

non-uniform element length.

2.2 Two-dimensional B-spline The two-dimensional B-spline model is the generation of one-dimensional model in two

directions and in cylindrical coordinates is as follows,

(23)

(24)

where k is number of node having (xi, yi) coordinates. Eq. (24) for 2/2=l is,

(25)

Where h is element length. The two-dimensional B-spline Eq. (25) for h=1 and 2=ρ ,

2/2 and 0 equal 0, 1 and 4; respectively. The effective radius for B-spline function of Eq.

(25) is 2 . Figs. (4) and (5) show the graph of Eq. (25) and its contour line.

Fig: 4 Two-dimensional B-spline basic function graph



m

n

m

m ccyxcyxs ββ +==∑=

11

1

),(),(

mkmkmmmmm ccfyxs −−− +== ββ 1,),(

=

===== ++−−

4

11

m

kmmmkm a

β

αββββ

c

c

c

c

4

4

4

40

40

40

4

00004

OOOOO

O

OO

OO

OO

OOOO

O

αα

ααα

ααα

αα

αα

ααα

ααα

αα

Fig: 5 Two-dimensional BSuppose there exist n data points with uniform rectangular or square mesh. The element dimensions

are constant all over the mesh with length

dimension B-spline function for this set of data can be written as,

where n is the total number nodes and m is the node number. For simplification of the

let’s consider square elements with h

The values of B-spline basic functions from Eq. (27) equal,

If Eq. (27) is applied on each node the following linear system of equations Eq. (29) will be

obtained where it is pentadiagonal system with 4 on the main diagonal entries and

diagonal components. Since the values of diagonal entries are constant,

to save the values of 4, α and their addresses to save matrix A. This saves computer memory

especially for problems with huge number of data.

Three problems are chosen for investigation of checking and efficiency of the a

which are explained here.



330

nnccc βββ L++ 3333

kmkmmmmmm ccc ++++− +++ ββββ 111

−2bea

[ ]{ } { }fcA

f

f

f

f

c

c

c

c

n

n

n

n

==

=

−− 1

2

1

1

2

1

M

M

M

M

dimensional B-spline basic function contour Suppose there exist n data points with uniform rectangular or square mesh. The element dimensions

are constant all over the mesh with length iii xxhh −== +1 and

jj yykk −== +1

spline function for this set of data can be written as,

where n is the total number nodes and m is the node number. For simplification of the

1== kh . The B-spline function ),( yxs for node m is,

functions from Eq. (27) equal,


obtained where it is pentadiagonal system with 4 on the main diagonal entries and α on the four others

diagonal components. Since the values of diagonal entries are constant, therefore; it is only necessary

and their addresses to save matrix A. This saves computer memory

especially for problems with huge number of data.

Three problems are chosen for investigation of checking and efficiency of the above formulation



Suppose there exist n data points with uniform rectangular or square mesh. The element dimensions

jy . The two-

(26)

where n is the total number nodes and m is the node number. For simplification of the formulation

for node m is,

(27)

(28)


(29)

on the four others

therefore; it is only necessary

and their addresses to save matrix A. This saves computer memory

bove formulation



331

[ ] 5.0)2(13.1)1(71.0 22

),( −+−== yxeyxfz

),(),(),(20

1

25

1

yxcyxcyxs ib

i

ibi

i

i ββ ∑∑==

+=

xxxxxx

xxxxxx

yyyyyy

yyyyyy

i

ibib

ccc

ccc

ccc

cccyxc

25256655

21212211

252522222121

552211

20

1

),(

βββ

βββ

βββ

ββββ

+++

++++

++++

++++=

−−−−−−

−−−−−−=

∑

L

L

L

L

2.2.1 Problem 1

There are 25 data points { }25,,2,1;,, L=izyx iii in this problem. They form a mesh point with

5 rows and 5 columns and square elements as Fig. (6). The element side is 1== kh . The

domain of function ),( yxf is ]4,0[∈x , ]4,0[∈y . The data are obtained from Eq. (30).

(30)

As indicated in Fig. (6) 20 auxiliary points or nodes around the data are considered for

including the effects of boundary conditions. This action causes an increase in accuracy and

efficiency of presented model for solving these kinds of problems.

Fig: 6 Data distribution in problem 1

Partial derivatives are required with respect to x axis and y axis at boundary nodes for this

purpose. For the most engineering data the above partial derivatives for boundary points are

not available and should be somehow approximated. The B-spline model which is used for

this problem is,

(31)

In Eq. (31) the second set of equation is related to the effects of boundary conditions which

can be expanded as Eq. (32).

(32)

For calculation ),( yxs or z value for internal points in each element it is necessary to consider

only a few terms of Eq. (31). Most of its terms are zero for each point inside the elements



332

because of the nature of B-spline basic function. Fig. (7) shows the amount of the

effectiveness with regard to the number of B-spline basic functions which contributes to

determining ),( yxs . For example in elements with nodes 8, 9, 13, and 14 the number of

required B-spline basic functions for calculation ),( yxs are from minimum 4 to maximum 7

Fig. (7). This problem is solved by the presented model. Figs. (8), (8a), (9) and (9a) show

the comparative behavior between the model and the real values of ),( yxf function for the

central points of elements. As it can be seen from the figures there are a suitable relationship

and agreements between the model output and the real data.

Fig: 7 Required number of B-spline basic function

Fig: 8 Graph of real data Fig: 8a Contour lines of real data

0 0. 1 1. 2

0 1 2 3 4

x

0

1

2

3

4

y



817.610[),( 5.2(2.1)5.4( 2

+= −−−− eeyxf xx

Fig: 9 Graph of B-spline model Fig: 9a Contour lines of B

2.2.2 Problem 2 This problem in relation to the previous one has more complexity with respect to the

variations of data. In this example there are 121 data points

form a uniform mesh having 11 rows and 11 columns with the domain of

]6,1[∈x , ]6,1[∈y . The data are generated by the following equation.

About 44 auxiliary boundary nodes for

considered. The partial derivatives with respect to x and y axis at boundary points are

calculated by the forward and backward divided difference formulas. The graph and contour

lines for Eq. (33) are in Figs. (10) and (10a). They are shown from the B

Figs. (11) and (11a) as well. Comparison of the figures shows a good agreement and suitable

relationship between the real data and the interpolated data.

Fig: 10 Graph of real data

0 1 2 3

x

0

1

2

3

4

y



333

8

)4(] 53.2)5 2 π−yCos

spline model Fig: 9a Contour lines of B-spline model

This problem in relation to the previous one has more complexity with respect to the

variations of data. In this example there are 121 data points{ 121,,2,1;,, L=izyx iii

form a uniform mesh having 11 rows and 11 columns with the domain of function

. The data are generated by the following equation.

About 44 auxiliary boundary nodes for applying the effects of boundary conditions are



s. (10) and (10a). They are shown from the B-spline model in


relationship between the real data and the interpolated data.

Fig: 10 Graph of real data Fig: 10a Contour lines of real data

4

middle points f(x,y)



spline model

This problem in relation to the previous one has more complexity with respect to the

}121 . The data

function ),( yxf

(33)

applying the effects of boundary conditions are



spline model in


Fig: 10a Contour lines of real data



[ ])/(8)( 22yxArctgSinyxz +=

Fig: 11 Graph of B-spline model Fig: 11a Contour lines of B

2.2.3 Problem 3

This example consists of 49 triple data points

they form a uniform square mesh with 7 rows and

element size 0.1== kh . The data which are highly nonlinear are obtained from Eq. (34).

About 28 auxiliary points are considered for including the effects of boundary conditions.

Fig: 12 Network of data and boundary points

The domain of data for this problem is

the partial derivatives with respect to x axis and y axis at the boundary points are calculated

by the numerical method differentiation. The parameters governing to the boundary points

can be obtained from the above deri

determined from Eqs. (35) and (36).

43y

-43x 43

-36 36

-29 29

-22 22

-15 15

-8 8

-1x 1

-1y



334

spline model Fig: 11a Contour lines of B-spline model

triple data points{ }49,,2,1;,, L=izyx iii. According to Fig. (12),

form a uniform square mesh with 7 rows and 7 columns of data and 36 elements with

. The data which are highly nonlinear are obtained from Eq. (34).


Fig: 12 Network of data and boundary points

The domain of data for this problem is ]7,1[∈x , ]7,1[∈y . Similar to the previous problem



can be obtained from the above derivatives. The parameters xc7and

yc 7−for example can be

determined from Eqs. (35) and (36).

43y +44 +45 +46 +47 +48 +49y

43 44 45 46 47 48 49 +49x

36 37 38 39 40 41 42 +42

29 30 31 32 33 34 35 +35

22 23 24 25 26 27 28 +28

15 16 17 18 19 20 21 +21

9 10 11 12 13 14 +14

2 3 4 5 6 7 +7y

1y -2 -3 -4 -5 -6 -7y



spline model

. According to Fig. (12),

7 columns of data and 36 elements with

. The data which are highly nonlinear are obtained from Eq. (34).

(34)


. Similar to the previous problem



for example can be



335

)7(1 '

67 xx fA

cc +=

)7(1 '

147 yy fA

cc −=−

[ ]ceayx ji yyxxb

k −=−+−− 22 )()(2

),(β

(35)

(36)

where babeA −= 8 . The two dimensional B-spline basic function which is applied for this problem

for 0.1=h and 2/2=l is,

(37)

The graphs and contour lines obtained by the model and from the function ),( yxf are in Figs. (13),

(13a), (14) and (14a). A comparison between these figures presents a good and suitable relationship

for real data and the two-dimensional B-spline model.

Fig: 13 Graph of real data Fig: 13a. Contour lines of real data

Fig: 14 Graph of B-spline model Fig: 14a Contour lines of B-spline model

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

x

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

y

f(x,y)

f(x,y)

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

x

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

y

B-spline model B-spline model



336

III. CONCLUSIONS AND DISCUSSION

The interpolation methods especially the two-dimensional ones have many

applications in all branches of engineering science. Therefore; the development, extension

and improvement of these methods are necessary. The presented model is easy to implement

and results in a linear pentadiagonal systems of equations which is diagonally dominant and

simply solvable. The presented model for interpolation of data with rectangular and square

elements is powerful. Also it can be expanded for non-uniform mesh or triangular elements

of data. The final linear system of equations for those cases is sparse and bounded while

about more than 90% of matrix entries are zero. The graphs obtained by this method have no

limitations of continuity properties inside and across the elements because of the continuity

conditions of the applied B-spline basic function. These conditions explain the advantage

and robustness of the presented model to the available methods of two-dimensional

interpolation.

REFERENCES

Burden, R. and Fairs, J. (2010). Numerical Analysis, 9

th Ed. Brooks-Cole Pub.

Comp.Boston.

Collins, G. W. (2003). II Fundamental Numerical Methods and Data Analysis, (internet pdf

file).

Dierckx, P. (1993.) Curve and Surface Fitting with splines, Clarendon Press, NY.

Dubin, D. (2003). Numerical and Analytical Methods for Scientists and Engineers Using

Mathematica, John Wiley & Sons, Inc., Pub., Hokoben, New Jersy.

Fomel, S. (1997b). On the General Theory of Data Interpolation, SEP. 94, 165-179.

Kiusalaas, J. (2005). Numerical Methods In Engineering with MATLAB, Cambridge Univ.

Press,UK.

Lobo, N. (1995). Curve fitting using spline sections of different orders, Proceedings of the

First International Mathematics Symposium, P. 267-274.

Nocedal, J. and Wright, S. J. (2006). Numerical Optimization, 2nd

Ed., Springer, Science

+Business Media, LLC, USA.

Salomon, D. (2006). Curves and Surfaces for computer Graphics, Springer Science

+Business Media, Inc. USA.

Saxena, A. and Sahay B. (2005). Computer Aided Engineering Design, Springer, Anamaya

Publishers, New Delhi, India.

Zamani, M. (2009 a). Three simple spline methods for approximation and interpolation of

data, Contemporary Engineering Science, Vol.2, No.8, P. 373-381.

Zamani, M. (2009 b). An investigation of B-spline and Bezier methods for interpolation of

data, Contemporary Engineering Science, Vol.2, No.8, P. 361-371.

Zamani, M. (2010). A simple 2-D interpolation model for analysis of nonlinear data, Natural

Science, Vol. 2, No. 6, June 2010, P. 641-646.

M. Nagaraju Naik and P. Rajesh Kumar, “Spectral Approach To Image Projection With

Cubic B-Spline Interpolation” International journal of Electronics and Communication

Engineering &Technology (IJECET), Volume3, Issue3, 2012, pp. 153 - 161, Published by

IAEME

an applied two dimensional b-spline model for interpolation of data

Documents

approximation of data

bspline method

twodimensional lagrange

nonsymmetric data

dimension data

volume of nonuniform

engineering literature

interpolation models