Conclusions

Theory versus Reality
During this project, we made several simplifying assumptions that allowed us to derive a functional theoretical model. However, if we wish to extend this model to a real-life application, we would need to take several additional factors into consideration.

The largest of these factors is the inherent nature of the cornea. In this project, we analyzed the surface of the eye in one-dimension. In actuality, the eye is made up of infinitely many slices that combine to complicate our approach. When the model is extended to account for the two-dimensional eye, the difficulty of the situation increases tremendously. The equation must be transformed into a two-dimensional equation .

Another simplifying assumption that may cause a loss in the functionality of our model is the assumption that the cornea is a convex surface. There may be some badly misshapen eyes where parts of the surface are convex and parts are concave. If this situation were to occur, it is quite possible that incorrect incisions would be made, possibly permanently affecting a patient’s eyesight.

The final drawback of this method is the precise nature of the measurements. The film data is reported with an accuracy of 10^-16, which is probably more accuracy than is realistically possible. Inherent in this point is the concept of noise, or data that is not measured exactly due to observation error. Our model does not take into account noise. When using an iterative process where the value of the next result depends on the previous result (as we are in this approach), noise could be a factor which, although small at first, may cause serious problems as the error is propagated throughout the system.

Accuracy and Implementation
Accuracy is a key factor involved in a manner as delicate as surgery involving the eye. Modern technology allows for surgical procedures using pulses of ultraviolet laser light that may sculpt the eye with an accuracy of 0.0004 millimeters of tissue. (The width of a human hair is approximately 0.125 millimeters.) Ideally, we would like our method of approximating the surface of the cornea to have accuracy greater than 0.0004 millimeters in order to ensure that error in approximation is not adversely affecting the eye.

Using 21 points, Forward Euler’s method has an average error of 0.0015205 millimeters, while Backward Euler’s method has an average error of 0.0013263 millimeters. Both values are between 3 and 4 times our minimum standard for approximation accuracy. However, recalling that both Euler’s methods are of order 1, while Runge-Kutta is of order 4, we are hopeful that the Runge-Kutta method will provide us our desired level of accuracy.

In fact, using 21 points, Runge-Kutta has an average error of 0.00010728 millimeters, which is well below our threshold for accuracy. However, this is an average measure of error, and as we move farther away from the center of the eye we experience more and more error. At the far corner of the eye, the error is as high as 0.00090590 millimeters, which is double our desired level.

Remembering that we may use more points in order to approximate a curve, we can employ the Runge-Kutta method with more points so that a better approximation is calculated. Using 51 points, we are able to reduce our average error to 0.000091948 millimeters, while the maximum error is 0.00041197 millimeters. These values are close enough to the desired level of error for comfort.

Additional concerns to address in the use of these mathematical models are the stability of the system and the computational cost involved in implementing the model. Backward Euler’s method is generally more stable than either Forward Euler’s method or the Runge-Kutta method. However, when we are concerned with the relatively smooth curve of the human eye, stability is not as great of an issue as it may be other times. Concerning the computational cost involved with the three methods, we observe that all methods are relatively quick to execute on a personal computer. The computational cost of all methods considered is in the order of O(n).

As a result, I recommend that the Runge-Kutta method be implemented. It gives the greatest accuracy with minimal, straightforward, calculations. Additionally, the method may be easily modified based upon different conditions, such as a different light source.

Additional Considerations
As with any real-life problem, there is always room for improvement. Given sufficient time and resources, I would make the following extensions:

1. Explore the two-dimensional geometry of the eyeball. In this project, we analyzed a single cross-section of the eyeball. In real life, doctors wish to know information about the entire eye.
2. Develop an integrated graphing module written in C++. Although Microsoft Excel provides a nice graph, the program would be more functional and robust if the entire program were self-contained.
3. Combining concepts from extensions one and two, the modified program would include the ability to produce a contour map of the eye.
4. Develop generalized equations governing the positions of the various observed points on the eye, allowing doctors to analyze the curvature of the eye at any point.