B. WANG ET AL.
Copyright © 2013 SciRes. ENG
ter clockwise direction around the
-axis. Thus, the
plane can be coincided with the corneal meridian
section in the new coordinate system (
). The equa-
tions of the corneal meridian section in the new coordi-
nate system (
).are as follows:
222
12 0
0
2(1 )
x
yaz azrzQz
=
= +=−+
(7)
where
are the coordinates of the new coordinate
system (
).
Then by substituting
into
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
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
coordinate. The
is a parameter and the
values of a semi-meridian are selected from 0 mm to 3.5
mm at 0.1 mm intervals. The
coordinate values of
every point are calculated by substituting the corres-
ponding
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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