Descriptors of elliptical shape

The different terminology used to define ellipse shape can seem confusing at first. ‘Eccentricity’ is the preferred mathematical term, but the terms ‘Shape Factor’ and ‘Asphericity’ are also in widespread use. So which should you use? When it comes down to it this will usually depend on what is specified by the design you are using so from your topography software you just need to find the required descriptor. Fortunately,  it is a pretty simple process to convert between them if the descriptor you require is not available.

Eccentricity (ecc)

I’m not going to dive into the full mathematical derivation of eccentricity, an internet or Wikipedia search which provide this if you are really interested, but essentially it describes the ratio between the short and long axes of an ellipse. Eccentricity values range 0 < 1, where 0 describes a circle (short and long axes are the same) and positive values indicating a prolate surface. Values from 1 and beyond are possible but then technically the name for the surface changes from being a parabola at a value of 1 and hyperbola at values higher than 1. These different technical names are irrelevant for OrthoK fitting purposes, however, so you can think of positive values of eccentricity existing on the same scale where 0 describes a spherical surface and positive values indicate a prolate surface that flattens towards the periphery with higher values indicating a greater amount of peripheral flattening.

The major problem with eccentricity is that it cannot be used to describe the opposite scenario of an oblate surface that steepens towards the periphery because this would require the square root of a negative number, which is nonsensical, and where Shape Factor comes into play.

Shape Factor (p)

Shape Factor overcomes the troublesome problem of the square root of a negative number by removing the need for performing the square root. The result is a value of 1 describing a circle (spherical surface), values between 0 and 1 describing a prolate surface and values greater than 1 describing a prolate surface. While this is mathematically sound it introduces further problems, namely that it doesn’t neatly define a circle as 0, instead this is now 1. A further point of difference is that it reverses the scale compared to eccentricity because increasing positive values beyond 1 now indicate greater amounts of surface oblateness. Enter ‘Asphericity’ from stage left.

Asphericity (Q)

Asphericity essentially takes Shape Factor and shifts it by 1 so that 0 again represents a circle (spherical surface), which makes things easier to visualize with negative values indicating a prolate (peripheral flattening), and positive values an oblate (peripheral steepening) surface.

Relationship between Eccentricity, Shape Factor and Asphericity

The relationship between these different terms is as follow:

Ellipse descriptors

The terminology most often cited in OrthoK lens fitting is eccentricity and asphericity, because they are simpler to understand with 0 referring to a circular (spherical) surface in both cases. Even though it is mathematically nonsensical, to allow the more familiar eccentricity term to describe a full range of corneal shapes, the use of negative eccentricity values to describe oblate surfaces has become widely adopted in the industry. To reach a negative eccentricity value requires some mathematical fudging to avoid the negative square root that arises when describing oblate surfaces, however, the resulting value can still be used to recreate the surface and more importantly calculate sag height to assist OrthoK lens fitting that will be covered in the next section on computer modeling.

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About Paul

Dr Paul Gifford is a co-founder of Eyefit, an information resource to assist contact lens practitioners in all modes of practice. Learn more about him here.

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