Vol.2, No.6, 641-645 (2010) Natural Science
http://dx.doi.org/10.4236/ns.2010.26080
A simple 2-D interpolation model for analysis of
nonlinear data
Mehdi Zamani
Department of Civil Engineering, Faculty of Technology and Engineering, Yasouj University, Yasouj, Iran;
Received 1 March 2010; revised 24 April 2010; accepted 13 May 2010.
ABSTRACT
To determination the volume and weight of non-
uniform bodies, such as in ore deposits evalua-
tion for mining and rock cutting for construction,
the methods of interpolation are usually used.
The classic curves, which are frequently used to
interpolate one-dimensional data are cubic spl-
ine, Bspline and Bezier curves. These methods
have good efficiency for determination of geom-
etric characteristics of nonregular masses. They
have some limitations and problems with two-
dimensional interpolation analysis such as for-
ming large linear systems of equations with a
lot of entries and difficulty encounter with their
solutions. In this research the two-dimensional
splines are used, which have the advantages of
simplicity and less computational operations ef-
fort. The spline functions that are applied have
the continuity of C1 at elements boundaries. The
presented model has suitable efficiency for vo-
lumes of large extents governing to lots of data.
Keywords: Simulation; Approximation; Least
square; Cubic Spline; Optimization; Curve Fitting;
Prediction
1. INTRODUCTION
The most popular methods for interpolation of data are
Lagrange method, Neville iteration approach, Newton
divided difference methods, Cubic spline, Hermit spline,
Bspline and Bezier methods. For the Neville and New-
ton divided difference method, the higher degree of pol-
ynomial generation is straightforward. In conic spline
approach each curve consists of a number of about n
segment curves. The general way is to divide the interval
to collect subintervals or segments and construct differ-
ent approximating polynomial on each interval. Appr-
oximation by functions of this type is called piecewise
polynomial approximation. This method has the advan-
tage of removing the oscillatory nature of high degree
polynomials. In this approach there exists the continuity
of C2 on the segments boundaries. However; the conti-
nuity order is optional to users willing. The formulation
for creating cubic spline curve results in the solution of a
three-diagonal system of equations. Hermit interpolation
are based on two points p1 and p2 and two tangent values
'
1
p and '
2
p at those points. It computes a curve seg-
ment that starts at p1, going in direction to '
1
p and ends
at p2 moving toward '
2
p. Hermit interpolation has an
important advantage. The Hermit curve can be modified
by changing the tangent values.
The Bezier curve is a parametric curve p(t) that is a
polynomial function of the parameter t. The degree of
the polynomial depends on the number of points used to
define the curve. The method applies control points and
presents an approximating curve. The Bezier curve does
not pass through internal points but the first and last
points. Internal points influence the direction and posi-
tion of the curve by pulling it toward themselves [1-3].
B-spline methods were first proposed in the 1940 for
curves and surfaces. The B-spline curves can be ap-
proximating or interpolating curves. The advantage of
B-spline curves to Bezier curve is the obtaining the
higher continuity for the individual spline segments
[2-5]. If there are a set of triple data ),,,{( '
iii yyxc 
},,2,1,0ni 

The cubic spline functions Si(x) can be
obtained on each interval [xi, x
i+1]. With this model the
continuity of C1 exists on the boundary of each segment.
The spline function of degree 3 for each interval is:














i
i
i
i
iiiii
d
c
b
a
xxxxdxcxbaxg 3232 1)( (1)
If the Eq.1 is written based on the first point of each
segment (xi , xi+1) then,

M. Zamani / Natural Science 2 (2010) 641-645
642































i
i
i
i
i
ii
iii
iiiiiiii
d
c
b
a
x
xx
xxx
xxx
xxdxxcxxbaxs
1000
3100
3210
1
1
)()()()(
2
2
32
32
32
(2)
The first derivatives of the spline function (2) is as
follows,
























i
i
i
i
ii
iiiiii
d
c
b
x
xx
xx
xxdxxcbxs
300
620
321
1
)(3)(2)(
2
2
2'
(3)
The coefficients iii cba ,, and i dfor each interval
can be obtained explicitly from the Eq.5.















)(
1
)(
2
)2(
1
)(
3
)(
)(
''
1
2
1
3
''
11
2
''
ii
i
ii
i
i
ii
i
ii
i
i
iii
iii
ff
h
ff
h
d
ff
h
ff
h
c
fxsb
fxsa
, (4)
where hi is the element size or the element interval (hi =
xi+1 – xi). The problem with this approach is that we
generally don’t have the tangent values at n + 1 point.
This can be provided by applying the following ap-
proximation equations (forward difference, central dif-
ference and backward difference approximation, respec-
tively) for the first derivatives at the points and for uni-
form intervals hi.















]3,4,1[
2
1
]1,0,1[
2
1
]1,4,3[
2
1
'
'
'
i
i
i
i
i
i
h
f
h
f
h
f
, (5)
The following two examples show the interpolation of
two sets of data by the above approach.
2. EXAMPLES
2.1. Example 1
This example is in the field of groundwater engineering.
When a production well in unconfined aquifer is pumped,
water is continuously withdrawn from storage within the
aquifer as the cone of depression progresses radially
outward from the well. Because of the absence of a re-
charge source in the form of vertical leakage, there can
be no stabilization of water levels and the head (h) in the
aquifer will continue to decline supposing the aquifer is
infinite in areal extent. However, the rate of decline of
head (h) continuously decreases as the cone of depres-
sion spreads. The partial differential equation governing
the unsteady-state radial flow in the nonleaky confined
aquifer in polar coordinate system is,
t
h
T
S
r
h
r r
h







1
2
2
. (6)
The Theis solution for the above partial differential
equation is,
du
u
e
T
Q
hhs
u
u



 4
0. (7)
Theis approximated the above equation by the fol-
lowing series,









 
!33!22
log5772.0
4
32
0
uu
uu
T
Q
hh e

(8)
where,
Tt
Sr
u
2
87.1

s = drawdown, in feet,
Q = discharge, in gpm,
t = time after pumping started, in days,
T = coefficient of transmisivity, in gallons per day per
foot (gpd/ft) and
S = coefficient of storage, dimensionless.
The value of brackets equals W(u). The data for in-
terpolation have logarithm values of W(u) respect to u
[6]. For the interpolation of data about ten points are
considered. Figure 1 shows the comparison between the
W(u) vs u
W(u)
u
mo d el
W(u)
Figure 1. The comparison between the data and the model.

M. Zamani / Natural Science 2 (2010) 641-645
643
643
data and the model for the nine spline curves. Based on
the figure there is a suitable and good relationship be-
tween those.
2.2. Example 2
In this problem about five pairs of data are obtained
from the parametric Eq.9 with non-uniform segments.
]24,0[,
12
cos2sin2
12
sin
12
sin2sin2
12
cos 







t
tty
ttx


. (9)
About four cubic spline curves are generated accord-
ing to the formulations of Eqs.3, 4 and 5. Figure 2
shows the relationship between the real values and in-
terpolated ones. As it can be seen, there is a relative ex-
pectable fitness between them.
3. FORMULATION FOR 2-D
INTERPOLATION
The bicubic spline formation that applies hear for two
dimensional interpolation and four nodes patch is,
223
01 23456
223
789
(, )
ij
g
xyaax ay axaxy ayax
axy axyay
 

(10)
or in matrix notation,
 

























3
2
6
73
841
9520
32
1
000
00
0
1),(
y
y
y
a
aa
aaa
aaaa
xxxyAxyxgij
. (11)
If the above equation is written based on (i x,i y) or the
coordinates of the lower left point on each segment, Eq.12
y
x
Figure 2. The comparison between the data and the model.
obtains. Then, the formulation for determination of the
coefficient will be simpler.
2
01 23
2
45
32
67
23
89
(,)( )()( )
()()()
()()()
()()()
ijij i
ij j
iij
jj j
s
xya axxayyaxx
axxy yay y
ax xax xy y
axxy yay y
 
 
 
 
(12)
or in matrix notation it is,















































3
2
23
2
2
32
32
1
133
012
001
0001
1000
3100
3210
1
1),(
y
y
y
yyy
yy
y
A
x
xx
xxx
xxxyxs
iii
ii
i
i
ii
iii
ij
.
(13)
The above bispline function has ten parameters (un-
knowns) for each segment or patch. Therefore; ten equa-
tions are required for defining them. Four equations sat-
isfy the function values at each corner and six equations
satisfying partial derivative in x and y directions for
three nodes. Thus there exists the continuity of C1 on
each of four sides or boundaries of every element. The
first derivatives in x and y direction for the Eq.12 are:
















































22
6
73
841
2
2
1
12
01
001
00
0
100
310
321
321
),(
y
y
yy
y
a
aa
aaa
x
xx
xx
x
yxs
jj
j
i
ii
ij
.
(14)
















































22
7
84
952
2
2
3
2
1
133
012
001
00
0
100
210
1
1
),(
y
y
yy
y
a
aa
aaa
x
xx
xx
y
yxs
jj
j
i
ii
ij
.
(15)
The coefficients 910 ,,, aaa can be derived explicitly
from satisfying the following conditions,

M. Zamani / Natural Science 2 (2010) 641-645
644
'
,2
'
,10
),(
,
),(
,),(
ijy
ij
ijx
ij
ijij
fa
y
jis
fa
x
ji
s
fajis







(16)
'
,1,6
2
31 32),1( jixii
ij fahahaji
x
s




, (17)
'
1,,9
2
52 32)1,( 



jiyjj
ij fakakaji
y
s
, (18)
'
,1,7
2
42
),1( jiyii
ij fahahaji
y
s




, (19)
'
1,,8
2
41
)1,(  


jixjj
ij fakakaji
x
s
, (20)
jiiii
jiij
fahahaha
jisjis
,16
3
3
2
10
,1 ),1(),1(




, (21)
1,9
3
5
2
20
1,)1,()1,(




jijjj
jiij
fakakaka
ji
s
ji
s
, (22)
1,19
3
8
2
7
2
6
3
5
2
43
3
2
101,1 )1,1()1,1(





jijji
jiijjiij
ijiij
fakakh
akhahakakhahak
ahaji
s
ji
s
(23)
The above ten equations give a linear system of equa-
tions. If it is solved analytically the governing parame-
ters are obtained as follows,
,
1323'
1,,1
2
10
2
3jix
i
ji
i i i
f
h
f
h
a
h
a
h
a  (24)
'
1,1,11,,1
2104
1
)(
1
111
 

jix
j
jijiji
ji
ijji
f
k
fff
kh
a
h
a
k
a
kh
a
, (25)
,
1323 '
1,1,
2
20
2
5 ijy
j
ji
j j j
f
k
f
k
a
k
a
k
a (26)
,
1212 '
1,
2
,1
3
1
2
0
3
6jix
i
ji
iii
f
h
f
h
a
h
a
h
a  (27)
'
1,1,11,,1
2
10
2
7
1
)(
111


ijx
ji
jijiji
ji
ji
ji
f
kh
fff
kh
a
kh
a
kh
a
, (28)
'
1,1,11,
,1
2
20
2
8
1
)
(
111
jiy
ji
jiji
ji
ji
ji
ji
f
kh
ff
f
kh
a
kh
a
kh
a




, (29)
 
'
1,
2
1,
3
2
2
0
3
9
1212
ijy
j
ji
jjj
f
k
f
k
a
k
a
k
a (30)
where jjjiii yykxxh 


 11 ,,
),1(,)1,( '
1,
'
1, ji
y
f
fji
x
f
fjiyijx





 . (31)
4. PROBLEM 3
For testing the above formulation several problems were
considered. Here only one of them is presented. The data
for this problem are obtained from the Eq.32.
]3,0[]6,0[,,4),( )5.00867.0(22  yxeyxz yx (32)
The domain of problem consists of 9 elements and 16
nodes. The elements are uniform where 2

i
h and
1

i
k. The above formulations were applied to to the
governing data. Figure 3 shows the interpolation curves
along x = 0, x = 2, y = 0 and y = 2. As it can be seen the
2-D model presents a close relationship to the relating
set of data.
5. THE RESULTS AND CONCLUSIONS
In this research two models for 1-D and 2-D interpola-
tions were presented. The models are easy and straight-
forward to handle. Therefore; it can be applied for prob-
lems having huge data. It has the advantage of less
computational efforts respect to the cubic spline, Bezier
and B-spline methods especially for 2-D problems. The
continuity for each segment is C1 therefore; this is suffi-
cient for many problems such as calculation of surfaces
and volumes of nonuniform bodies. It is recommended
to compare the 2-D model with Bezier and B-spine
methods respect to the fitness and accuracy.
comparison
x
, y
0 1 2 3 4 5 6
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
f
(x, y)
f(x, y)
model
f(x, 0)
f(x, 2)
f(2, y)f(02, y)
Figure 3. The comparison between the data and 2-D model.

M. Zamani / Natural Science 2 (2010) 641-645
645
645
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