Representation of dioptric power in Euclidean 3-space

Every dioptric power of the usual form sphere/cylinder x axis may be represented by means of a point in three-dimensional space. Graphical representation of data in this manner is important for statistical analysis. In particular, graphical representation may be used to display confidence regions about a mean power, for example. A disadvantage of these representations, however, is that simply changing the reference meridian for cylinder axes changes distances in the space and, therefore, changes the shapes of the confidence regions. Because the shapes define the nature of the variation and give, in particular, the principal components of variation, a researcher who happens to measure the orientation of cylinder axes from the 20 degree meridian, for example, instead of the conventional horizontal meridian, could be led to different statistical conclusions. The implication is that such conclusions are unlikely to have much physical meaning. A representation is described here in which distances and shapes do not depend on the meridian that happens to be chosen as reference. Each dioptric power is represented by a point in Euclidean 3-space. Several examples of graphical representation are given. The spherical powers occupy a particular line in the space, the Jackson crossed cylinders occupy a plane, the cylindrical powers occupy a cone, and so on, for all types of conventional dioptric power. These lines and surfaces are illustrated. The statistical implications are discussed briefly. The representation satisfies the requirements of the statistics and is proposed as the standard one for future use.