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Possibly several centuries ago, people began to use positive lenses to aid in reading. Before the astigmatic error of human eyes was discovered and corrected, only one number was used to represent the human vision. It is the sphere power of a correcting lens. For example, if you need a +2 D lens to see clearly at distance, you are said to be hyperopic of 2 diopters. If you need a -3.5 D lens to see clearly at distance, you are then a myope of -3.5 diopters. Once astigmatic error was discovered, people realized that eyes needed different powers of lenses at different meridians. A new way of representing the human vision with three numbers was invented. Using a lens of pure sphere and a lens of pure cylinder with a certain angle, one can denote the so-called spherocylindrical error of ocular aberrations with these three numbers: sphere power, cylinder power, and cylinder axis. However, when we consider the irregular error of ocular aberrations, we need to use more numbers to represent them.
There are two ways to represent ocular aberrations in a two-dimensional surface. One way is to use discrete values to represent the optical path differences at different locations within the pupil. The other way is to continue with the idea of our ancestors by extending three numbers to more numbers, i.e., to decompose the ocular aberrations into a series of basic shapes. For example, when we use one number, we can only say that one has -2.5 diopters of sphere. When we use three numbers, we may give a more accurate account, such as -3.0 D of sphere and 1.0 D of cylinder at 45°. If we want to be even more accurate, then we may add that the eye also contains 0.08 μm of spherical aberration, -0.12 ¹mm of vertical coma, and 0.09 ¹mm of horizontal trefoil, over a 6 mm pupil. The basic shapes of sphere, cylinder, spherical aberration, vertical coma, and horizontal trefoil are all defined shapes and can be represented mathematically.
The analytical representation of ocular aberrations is preferred over the discrete representation for three reasons. First, it gives the most important components of the ocular aberrations - sphere, cylinder, spherical aberration, coma, and trefoil - which are all familiar. The discrete representation gives only a topographic view, and thus is less intuitive. Second, the discrete representation has limited resolution once represented. To obtain the value between two neighboring points, an interpolation must be used. This can introduce error. On the contrary, the analytical representation has infinite resolution. Once represented, the value at any point in the two-dimensional surface can be accurately given. Finally, the manipulation and computation of ocular aberrations by means of the analytical representation are much more efficient than the discrete representation.
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