Keating's asymmetric dioptric power matrices expressed in terms of sphere, cylinder, axis, and asymmetry

Twelve years ago Keating pointed out that dioptric powers existed which could not be represented by the familiar three parameters sphere, cylinder, and axis. They are the equivalent powers of optical systems (including many eyes) with separated obliquely crossing astigmatic elements. Four parameters are required to represent such powers, and all four are unfamiliar to most clinicians and researchers. This note shows that it is, in fact, possible to transform the four parameters so that the three familiar parameters are retained and only one (called asymmetry) remains unfamiliar. The consequence is that it is always possible to represent a power by means of sphere, cylinder, axis, and asymmetry. Powers commonly used in practice all have asymmetry equal to zero which is why only the first three are usually necessary. Powers, however, do exist, and are of potential interest in optometry, for which asymmetry is not zero and cannot be omitted from the representation. Two numerical examples are given, including Keating's model eye.