W. F. Harris

Curvature of ellipsoids and other surfaces.

From differential geometry one obtains an expression for the curvature in any direction at a point on a surface. The general theory is outlined. The theory is then specialised for surfaces that are represented parametrically as height over a transverse plane. The general ellipsoid is treated in detail as a special case. A quadratic equation gives the principal directions at the point and, hence, the principal curvatures associated with them. Equations are obtained for ellipsoids in general that are generalisations of Bennetts equations for sagittal and tangential curvature of ellipsoids of revolution. Equations are also presented for the locations of umbilic points on the ellipsoid.

Nodes and nodal points and lines in eyes and other optical systems.

The typical stigmatic optical system has two nodal points: an incident nodal point and an emergent nodal point. A ray through the incident nodal point emerges from the system through the emergent nodal point with its direction unchanged. In the presence of astigmatism nodal points are not possible in most cases. Instead there are structures, called nodes in this paper, of which nodal points are special cases. Because of astigmatism most eyes do not have nodal points a fact with obvious implications for concepts, such as the visual axis, which are based on nodal points. In order to gain insight into the issues this paper develops a general theory of nodes which holds for optical systems in general, including eyes, and makes particular allowance for astigmatism and relative decentration of refracting elements in the system. Key concepts are the incident and emergent nodal characteristics of the optical system. They are represented by 2 × 2 matrices whose eigenstructures define the nature and longitudinal position of the nodes. If a system’s nodal characteristic is a scalar matrix then the node is a nodal point. Otherwise there are several possibilities: Firstly, a node may take the form of a single nodal line. Second, a node may consist of two separated nodal lines reminiscent of the familiar interval of Sturm although the nodal lines are not necessarily orthogonal. Third, a node may have no obvious nodal line or point. In the second and third of these classes one can define mid‐nodal ellipses. Astigmatic systems exist with nodal points and stigmatic systems exist with no nodal points. The nodal centre may serve as an approximation for a nodal point if the node is not a point. Examples in the Appendix , including a model eye, illustrate the several possibilities.

The exponential-mean-log-transference as a possible representation of the optical character of an average eye.

Considering a set of eyes, how does one define an average whose optical character represents an average of the optical characters of the eyes in the set? The recent proposal of the exponential‐mean‐log‐transference was based on a conjecture. The purpose of this note is to provide justification by proving the conjecture and defining the conditions under which it fails.

Effective corneal refractive zone in terms of Gaussian optics

&NA; The portion of the cornea that transmits light for vision is clinically important in several contexts, including corneal ablation in refractive surgery. In contrast to geometric optics, Gaussian optics allows one to obtain simple, explicit formulas for the geometry of the effective corneal refractive zone for distant object points that are on or off the line of sight. In this article, Gaussian optics was used to derive the formula for the diameter of the zone and, when the zone is annular, the inner and outer diameter, as a function of corneal power, anterior chamber depth, pupil diameter, and angular position of the object point.

Visual axes in eyes that may be astigmatic and have decentred elements

The visual axis of the eye has been defined in terms of nodal points. However, astigmatic systems usually do not have nodal points. The purpose of this note is to offer a modified definition of visual axes that is in terms of nodal rays instead of nodal points and to show how to locate them from knowledge of the structure of the eye. A pair of visual axes (internal and external) is defined for each eye. The visual axes then become well defined in linear optics for eyes whether or not they are astigmatic or have decentred elements. The vectorial angle between the visual axes and the optical axis defines the visuo‐optical angle of the eye.

Cardinal points and generalizations

In the presence of astigmatism a focal point typically becomes the well‐known interval of Sturm with its pair of axially‐separated orthogonal line singularities. The same is true of nodal points except that the issues are more complicated: a nodal point may become a nodal interval with a pair of nodal line singularities, but they are not generally orthogonal, and it is possible for there to be only one line singularity or even none at all. The effect of astigmatism on principal points is the motivation behind this paper. The three classes of cardinal points are defined in the literature in a disjointed fashion. Here a unified approach is adopted, phrased in terms of rays and linear optics, in which focal, nodal and principal points are defined as particular cases of a large class of special structures. The special structures arising in the presence of astigmatism turn out to be described by mathematical expressions of the same form as those that describe nodal structures. As a consequence everything that holds for nodal points, lines and other structures now extends to all other special points as well, including principal points and the lesser‐known anti‐principal and anti‐nodal points. Thus the paper unifies Gauss’s and Listing’s concepts of cardinal points within a large class of special structures and generalizes them to allow for refracting elements which may be astigmatic and relatively decentred. A numerical example illustrates the calculation of cardinal structures in a model eye with astigmatic and heterocentric refracting surfaces.

Subjective refraction: the mechanism underlying the routine.

The routine of subjective refraction is usually understood, explained and taught in terms of the relative positions of line or point foci and the retina. This paper argues that such an approach makes unnecessary and sometimes invalid assumptions about what is actually happening inside the eye. The only assumption necessary in fact is that the subject is able to guide the refractionist to (or close to) the optimum power for refractive compensation. The routine works even in eyes in which the interval of Sturm does not behave as supposed; it would work, in fact, regardless of the structure of the eye. The idealized subjective refraction routine consists of two steps: the first finds the best sphere (the stigmatic component) and the second finds the remaining Jackson cross‐cylinder (the antistigmatic component). The model makes use of the concept of symmetric dioptric power space. The second part of the refraction routine can be performed with Jackson cross‐cylinders alone. However, it is usually taught and practiced using spheres, cylinders and Jackson cross‐cylinders in a procedure that is not easy to understand and learn. Recognizing that this part of the routine is equivalent to one involving Jackson cross‐cylinders only allows one to teach and understand the procedure more naturally and easily.

Refractive variation under accommodative demand: Curvital and scaled torsional variances and covariance across the meridians of the eye

Autorefractor measurements were taken on the right eye of 10 students with an external target at vergences -1.00 and -3.00 D. The refractive errors in the form of sphere, cylinder, and axis were converted to vectors h and variance-covariance matrices calculated for different reference meridians. Scatter plots are drawn in symmetric dioptric power space. The profiles of curvital and scaled torsional variances, the scaled torsional fraction, and the scaled torsional-curvital correlation are shown using a polar representation. This form of representation provides a meridional pattern of variation under accommodative demand. The profile for scaled torsional variance is characteristically in the form of a pair of rabbit ears. At both target vergences curvital variance is larger than scaled torsional variance in all the meridians of the eye: the relative magnitudes are quantified by the scaled torsional fraction. An increase in accommodative demand generally results in an increase in variance. The rabbit ears usually become larger but less well divided. The correlation between curvital and torsional powers is usually positive in the first quadrant and negative in the second quadrant. Typical, atypical, and mean typical responses are discussed.

Reduction of artefact in scatter plots of spherocylindrical data.

Round‐off of spherocylindrical powers, to multiples of 0.25 D (for example) in the case of sphere and cylinder, and 1 or 5° in the case of axis, represents a type of distortion of the data. The result can be artefacts in graphical representations, which can mislead the researcher. Lines and clusters can appear, some caused by moiré effects, which have no deeper significance. Furthermore artefacts can obscure meaningful information in the data including bimodality and other forms of departure from normality. A process called unrounding is described which largely eliminates these artefacts; each rounded power is replaced by a power chosen randomly from the powers that make up what is called the error cell of the rounded power.

Achromatic axes and their linear optics

If a polychromatic ray segment enters an optical system, is dispersed into many slightly different paths through the system, and finally emerges at a single point, then the incident segment defines what Le Grand and Ivanoff called an achromatic axis of the system. Although their ideas of some 65 years ago have inspired important work on the optics of the eye there has been no analysis of such axes for their own sake. The purpose of this paper is to supply such an analysis. Strictly speaking optical systems, with some exceptions, do not have achromatic axes of the Le Grand-Ivanoff type. However, achromatic axes based on a weaker definition do exist and may for practical purposes, perhaps, be equivalent to strict Le Grand-Ivanoff axes. They are based on a dichromatic incident ray segment instead. The linear optics of such achromatic axes is developed for systems, like the visual optical system of the eye, that may be heterocentric and astigmatic. Equations are obtained that determine existence and uniqueness of the axes and their locations. They apply to optical systems like the eye and the eye in combination with an optical instrument in front of it. Numerical examples involving a four-refracting surface eye are treated in Appendix A. It has a unique achromatic axis for each retinal point including the center of the fovea in particular. The expectation is that the same is true of most eyes. It is natural to regard the Le Grand-Ivanoff achromatic axis as one of a class of six types of achromatic axes. A table lists formulae for locating them.