Engineering, 2013, 5, 477-481
http://dx.doi.org/10.4236/eng.2013.510B098 Published Online October 2013 (http://www.scirp.org/journal/eng)
The 3D Computer Image of the Anterior Corneal Surface
Bo Wang1, Xueping Huang2, Jinglu Ying3, Mingguang Shi3
1School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, China
2Wenzhou Medical College, Wenzhou, China
3Department of Ophthalmology, The Second Affiliated Hospital of Wenzhou Medical College, Wenzhou, China
Received 2013
ABSTRACT
In this paper, we derive a nonlinear equation of corneal asphericity (Q) using the tangential radius of curvature (rt) on
every semi-meridian. We transform the nonlinear equation into the linear equation and then obtain the Q-value of cor-
neal semi-meridian by the linear regression method. We find the 360 semi-meridional v ariation rule of the Q-value us-
ing polynomial function. Furthermore, we construct a new 3D corneal model and present a more realistic model of
shape of the anterior corneal surface.
Keywords: Cornea; Computer; Image
1. Introduction
It is well established that the anterior surface of the cor-
nea is the major refractive element of the human eye,
being responsible for approximately 75% of the eye’s
total unaccommodated refractive power [1]. Guillon [2]
and Bennett [3] assumed the human cornea to have a
conic section which can be described by the Baker’s eq-
uation:
22
0
2yr xpx= −
[4]. Here, p describes the as-
phericity of the corneal section. Cheung [5] calculated
the corneal asphericity (p) using the sagittal radius of
curvature (
s
r
) from corneal axial power map according
to Bennett’s equation:
22 2
0
(1 ) s r rpy= +−
[6]. Schwie-
gerling [7] used corneal height data from corneal height
map and Zernike polynomials to describe the shape of
the cornea. Corneal topography is commonly presented
as axial power map, tangential power map and height
map. To our knowledge, there is no report of investigat-
ing the corneal asphericity (Q,
1Qp= −
) calculation by
the tangential radius of curvature (rt) from tangential
power map.
In most previous studies, it has been reported the Q-
values which is representative of all corneal meridians or
the Q-values of two principal meridians. Dubbelman [8]
measured k-values (where
1kQ= +
) of six semimeri-
dians (0˚, 30˚, 60˚, 90˚, 120˚, 150˚) using Scheimpflug
photograph y and modeled the meridional variatio n of the
k
-value using the
2
cos
function. However, it indicated
that the
2
cos
function is not an adequate model to de-
scribe t he va riati on.
Sagittal radius of curvature (rs) is spherically biased
and is not a true radius of curvature [9-11] and it will
lead to erroneous result for an asymmetric corneal sur-
face. Tangential radius of curvature (rt) is a true radius of
curvature which can better represent corneal shape and
local curvature changes especially in the periphery [12].
In this paper we derive a nonlinear equation of corneal
asphericity (Q) using the tangential radius of curvature (rt)
on every semi-meridian for the first time. We obtain the
Q-value of corneal semi-meridian by the linear regres-
sion method and find the 360 semi-meridional variation
rule of the Q-value using polynomial function. Further-
more, we construct a new 3D model of shape of the ante-
rior corneal surface.
2. Derivation of the Corneal Model
The Bausch & Lomb Orbscan II corneal topographer is
used to acquire images of the topography of the right eye
of 66 normal young subj ects. All subj ec ts have no history
of ocular disease and ocular surgery with emmetropic
eyes. A series of data point on a semi-meridian are ar-
ranged at 0.1 mm intervals. The interval between two
semi-meridians is 1˚. The tangential radius of curvature
(rt) and perpendicular distance from the point to optical
axis (
y
) of all data point on a se mi-meridian and vertex
radius of curvature (
0
r
-value) can be obtained from the
raw data of tangential power map of anterior corneal
surface.
A three dimensional Car tesian coordinate system is set
with its origin at vertex nor mal to the corn eal interse ction
of the optic axis of the corneal topographer [13]). The
Z
-axis,
Y
-axis,
X
-axis of the coordinate represent the
optical axis direction, the vertical direction and the hori-

B. WANG ET AL.
478
zontal direction, respectively.
θ
is the angle between
the corneal meridian section and the
XOZ
plane. The
corneal meridian section is located on the
YOZ
plane
when
θ
= 90˚. At this time, we assume that the equa-
tion of corneal meridian can be correspondingly de-
scribed by the conic equation:
22
12 12
,,yazaza aR=+∈
.
This conic equation is an improvement of the Baker’s
equation [4]. While located on the
XOZ
plane when
θ
= 0˚ and described by the conic equation:
22
12
xaz az= +
. For any other angle
θ
except for 0˚,
90˚, 180˚, 270˚, the
YOZ
plane can be coincided with
the corneal meridian section by rotating the coordinate
system. Thus the corneal meridian section of any other
angle
θ
can also be described by the conic equation
22
12
yaz az= +
in the new coordinate system (see Sec-
tion 5.2) .
Here let us take corneal meridian section with
θ
=
90˚ for example, the formula of the curvature of a point
on the section can be expressed as [14,15]:
3
2
"
1
1( ')
t
y
Kry
= =

+

(1)
where
K
is curvature,
'y
and
''y
are the first and se-
cond derivatives with respect to
z
which is
z
-axis
coordinate value of the point. Differentiating both sides
of the conic equation
22
12
yaz az= +
with respect to
z
,
we get
12
2
2
a az
yy
+
′=
,
2
1
3
4
a
yy
−
′′ =
Then by substituting
y′
and
y′′
into Equation (1), we
obtain:
3
22
12
2
2
1
4((1) )
4
t
a
r ay
a
= ++
(2)
The conic equation
22
12
yaz az= +
can be rewritten
as:
2
12
2
22
11
22
2
()
21
4
4
a
zay
aa
a
a
+
+=
−
(3)
Since
2
Qe= −
, then by Equation (3) we have
2
(1 )aQ=−+
.
Finally by substituting
2
(1 )aQ=−+
into Equation
(2), we obtain
3
22
2
0
2
0
1[]
t
rr Qy
r
= −
(4)
3. Solution to Q-Value Calculation Problem
Since rt is a nonlinear function of
y
in Equation (4), it
is difficult to calculate
Q
-value. To transform the non-
linear problem to the linear one, the Equation (4) is con-
verted to another form which can be written as:
2
23
t
yb cr= +
(5)
where
b
and
c
are constants, a straight line graph of
2
y
(on the ordinate) vs
2
3
t
r
(on the abscissa) is plotted.
By the linear regression method, we get
2
0
r
bQ =
and
4
3
0
r
cQ = −
, that is,
2
3
b
Qc = −
. The straight line gives a
coefficient of determination (2
R
). The Q-value of the
given semi-meridian is calculated includ ing from the first
point at 0.1 mm to 3.5 mm. Figure 1 illustrates a func-
tion scatterplot of perpendicular distance squared versus
tangential radius of curvature to the two-thirds power on
the nasal horizontal principal semi-meridian of the right
eye for subject number 1.
4. 360 Semi-Meridional Rule of the Q-Value
The corneal zone analyzed is up to diameter 7.0 mm
which is large enough to cover the pupillary area. The
near vertical meridians will have a diameter limit im-
posed by the eyelids and eyelashes. In our earlier study,
we found that the semi-meridians which the peripheral
points wer e up to 3.5 mm were mainly distributed within
50˚ of the horizontal including 0˚ - 50˚, 130˚ - 180˚, 181˚-
230˚, 310˚ - 359˚. The Q-value of each semi-meridian in
these near horizontal regions was calculated. According
to the Q-values of the near horizontal semi-meridians, we
use regression analysis to model the 360 semi-meridional
variation of the Q-value and fit the Q-value of
Figure. 1. Scatterplot of perpendicular distance squared
versus tangential radius of curvature to the two-thirds
power.

B. WANG ET AL.
479
each semi-meridian in the near vertical regions including
51˚ - 129˚, 231˚ - 309˚. The form of a polynomial func-
tion is:
234
01 234
( )...f xppxpxpxpx=+++++
where x is semi-meridian angle
θ
(degree) and f(x) is
corresponding Q-value. Here, the degree must be con-
verted to the radian when calculating the polynomial
fitting.
To determine which degree polynomial will provide an
optimal fit to the 360 semi-meridional variation of the
Q-value, we calculate the RMS fit error of the polynomi-
al function from 5th degree to 9th degree. We find that the
RMS fit error become relatively stable at approximately
0.02 for fits higher than 6th degree.
The 360 semi-meridional variation of the Q-value is
well fitted using the 7th degree polynomial function for
all subjects. Figure 2 shows an example of the variation
of the Q-value as a function of semi-meridian for subject
number 22 with the following 7th degree polynomial
function:
Red: Fitted curve of 360 semi-meridional variation of
the Q-value.
Figure 3 shows that the majority of right eyes display
the goodness of fit (r2) of polynomial function for all
subjects for the asphericity above 0.9 and the median
value is 0.94. The mean RMS fit error of polynomial fit
is 0.02 ± 0.008.
Figure 4 shows the variation in asphericity with semi-
meridian region of anterior corneal surface for all sub-
jects. It can be seen that the Q-value distribution of ante-
rior corneal surface presents bimodal variation. These
two peak va l ues repres ent the l e a s t negative Q-values.
Figure 2. Typical example of the variation of the Q-value as
a function of semi-meridian.
Figure 3. Box and whisker plot for the goodness of fit (r2) of
the polynomial function for all subjects for the asphericity
(Q).
Figure 4. Variation in asphericity as a function of semi-
meridian region of anterio r corneal surface for all subjects.
5. Construction of a 3d Model of Corneal
Shape
5.1. Rotation of the Coordinate System
A new coordinate system (
''X OY
) is obtained by rotat-
ing the original coordinate system (
XOY
)
θ
degree in
the counter clockwise direction.
P
is an arbitrary point
in the coordinate system with
(, )Pxy
in the original
coordinate system and
( ',')Px y
in the new coordinate
system. We can obtain the following coordinate rotation
formula:
' sincos
' cossin
xy x
yy x
θθ
θθ
= +

= −

(6)
5.2. The Parametric Representation of the
Equations of the Corneal Meridian Section
We set the angle between a given corneal meridian sec-
tion and the
XOZ
plane is
θ
degree. A new coordi-
nate system (
XOY
) is obtained by rotating the original
coordinate system (
XOY
)
0
-(90 -) θ
degree in the coun-

B. WANG ET AL.
480
ter clockwise direction around the
Z
-axis. Thus, the
YOZ
plane can be coincided with the corneal meridian
section in the new coordinate system (
XOY
). The equa-
tions of the corneal meridian section in the new coordi-
nate system (
XOY
).are as follows:
222
12 0
0
2(1 )
x
yaz azrzQz
=

= +=−+


(7)
where
(, )xy
are the coordinates of the new coordinate
system (
XOY
).
Then by substituting
0
-(90 -) θ
into
θ
given in the
formula (6), we obtain the following coordinate rotation
equations of our corneal model:
sin cos
sin cos
xx y
yy x
θθ
θθ
= −

= +


(8)
We substitute the
,xy
given in Equation (7) into the
Equation (8). The equations of the corneal meridian sec-
tion on the original coordinate system (XOY) are as fol-
lows
22
0
sincos 0
( sincos)2(1)
xy
yxrz Qz
θθ
θθ
−=


+= −+


(9)
Finally, we transform the Equation (9) into the fol-
lowing f orm:
2
0
2
0
2(1)cos
2(1)sin
x rzQz
y rzQz
θ
θ
= −+


= −+

(10)
5.3. Generation of a 3D Corneal Model
360 semi-meridians are all chosen. Every point has an
(,,)xyz
coordinate. The
z
is a parameter and the
z
values of a semi-meridian are selected from 0 mm to 3.5
mm at 0.1 mm intervals. The
,xy
coordinate values of
every point are calculated by substituting the corres-
ponding
z
value into the Equation (11), 3D corneal sur-
face plot is generated with the Visual C++ 6.0 program-
ming [16]. Figure 5 shows a colorized 3D surface plo t of
anterior corneal surface from two different perspectives
for the same subject as in Figure 2. Variation of color
shows semi-meridional variation of the Q-value with
0.02 color steps. From the top to bottom of color scale,
the Q-value becomes more negative gradually. Figure 2
shows that the Q-value of each semi-meridian is negative
value (−1 < Q < 0) corresponding to the most common
corneal shape (prolate ellipse) ([17]). Thus, the 3D sur-
face plot of anterior corneal surface approximates a pro-
late ellipsoid shown in Figu re 5.
6. Conclusion
In contrast to the sagittal radius of curvature (rs), the
Figure 5. 3D surface plot of anterior corneal surface for the
same subject as in Figure 2.
tangential radius of curvature (rt) is a true radius of cur-
vature which can better represent corneal shape and local
curvature changes especially in the periphery.
In this paper, we proposed a nonlinear equation of
corneal asphericity (Q) using the tangential radius of
curvature (rt) on every semi-meridian. The 360 semi-
meridional variation of the Q-value was well fitted using
the 7th degree polynomial function for all subjects. We
constructed a new 3D corneal model and present a more
realistic model of shape of the anterior corneal surface.
Our mathematical model could be helpful in the contact
lens design and detection of corneal shape abnormalities,
such as kerat oconus or previous la s er surger y.
7. Acknowledgements
This study was supported by grant No. 30872816 from
the National Natural Scientific Found a tion of China.
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