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Ellipses are features of several structures in astigmatic eyes; they include retinal blur patches. How can one calculate change or averages or perform other quantitative analyses on such elliptical structures? The matrix A in the equation r T Ar=1, commonly used to represent an ellipse, is positive definite; such matrices do not define vector spaces. They are unsuitable, therefore, for quantitative analysis of ellipses. This paper defines a generalized radius R of an ellipse that is positive or negative definite for locally diverging or converging rays, respectively, indefinite between line foci, and singular at foci. Generalized radii of ellipses constitute a vector space and are suitable for quantitative analysis of elliptical ocular structures.

Purpose: To show that 14-dimensional spaces of augmented point P and angle Q characteristics, matrices obtained from the ray transference, are suitable for quantitative analysis although only the latter define an inner-product space and only on it can one define distances and angles. The paper examines the nature of the spaces and their relationships to other spaces including symmetric dioptric power space., Methods: The paper makes use of linear optics, a three-dimensional generalization of Gaussian optics. Symmetric 2 × 2 dioptric power matrices F define a three-dimensional inner-product space which provides a sound basis for quantitative analysis (calculation of changes, arithmetic means, etc.) of refractive errors and thin systems. For general systems the optical character is defined by the dimensionally-heterogeneous 4 × 4 symplectic matrix S, the transference, or if explicit allowance is made for heterocentricity, the 5 × 5 augmented symplectic matrix T. Ordinary quantitative analysis cannot be performed on them because matrices of neither of these types constitute vector spaces. Suitable transformations have been proposed but because the transforms are dimensionally heterogeneous the spaces are not naturally inner-product spaces., Results: The paper obtains 14-dimensional spaces of augmented point P and angle Q characteristics. The 14-dimensional space defined by the augmented angle characteristics Q is dimensionally homogenous and an inner-product space. A 10-dimensional subspace of the space of augmented point characteristics P is also an inner-product space., Conclusions: The spaces are suitable for quantitative analysis of the optical character of eyes and many other systems. Distances and angles can be defined in the inner-product spaces. The optical systems may have multiple separated astigmatic and decentred refracting elements., (© 2017 The Authors Ophthalmic & Physiological Optics © 2017 The College of Optometrists.)

Purpose: Cardinal points are used for ray tracing through Gaussian systems. Anti-principal and anti-nodal points (which we shall refer to as the anti-cardinal points), along with the six familiar cardinal points, belong to a much larger set of special points. The purpose of this paper is to obtain a set of relationships and resulting equalities among the cardinal and anti-cardinal points and to illustrate them using Pascal's ring., Methods: The methodology used relies on Gaussian optics and the transference T. We make use of two equations, obtained via the transference, which give the locations of the six cardinal and four anti-cardinal points with respect to the system. We obtain equalities among the cardinal and anti-cardinal points. We utilise Pascal's ring to illustrate which points depend on frequency and their displacement with change in frequency., Results: Pascal described a memory schema in the shape of a hexagon for remembering equalities among the points and illustrating shifts in these points when an aspect of the system changes. We modify and extend Pascal's ring to include the anti-cardinal points. We make use of Pascal's ring extended to illustrate which points are dependent on the frequency of light and the direction of shift of the equalities with change in frequency. For the reduced eye the principal and nodal points are independent of frequency, but the focal points and the anti-cardinal points depend on frequency. For Le Grand's four-surface model eye all six cardinal and four anti-cardinal points depend on frequency. This has implications for definitions, particularly of chromatic aberrations of the eye, that make use of cardinal points and that themselves depend on frequency., Conclusions: Pascal's ring and Pascal's ring extended are novel memory schema for remembering the equalities among the cardinal and anti-cardinal points. The rings are useful for illustrating changes among the equalities and direction of shift of points when an aspect of a system changes. Care should be taken when defining concepts that rely on cardinal points that depend on frequency., (© 2017 The Authors Ophthalmic & Physiological Optics © 2017 The College of Optometrists.)

Purpose: To determine the optimal distribution of refractions in monofocal, pseudophakic monovision., Methods: A previously reported mathematical method for describing defocus for a single eye (Acta Ophthalmol, 89, 2011, 111) is expanded to the binocular situation. The binocular distribution of refractions yielding the least defocus over the most extended fixation intervals is identified by mathematical optimization. The results are tested in a group of 22 pseudophakic patients., Results: For the fixation interval 0.25-6.0 m, the optimal refractions are pure spheres of -0.27D for the distance eye and -1.15D for near eye. The mathematically derived defocus structure is confirmed in the clinical series., Conclusions: The reported mathematical method enables identification of the optimal distribution of refractions over any fixation interval. Monovision with refractions of approximately -0.25 and -1.25D may lead to spectacle independence for distance and intermediate vision. Binocular problems--such as monovision suppression, reduced stereoacuity and binocular inhibition--are likely to be minimal with the suggested anisometropia of 1.0D. This moderate monovision is fully reversible with spectacle correction, as the induced aniseikonia is minimal and it therefore represents a safe alternative to multifocal intraocular lenses (IOLs)., (© 2013 Acta Ophthalmologica Scandinavica Foundation. Published by John Wiley & Sons Ltd.)

In 1945 Yves Le Grand published conditions, now largely forgotten, on the 4×4 matrix of an astigmatic eye for the eye to be emmetropic and an additional condition for retinal images to be undistorted. The conditions also applied to the combination of eye and the lens used to compensate for the refractive error. The conditions were presented with almost no justification. The purpose of this paper is to use linear optics to derive such conditions. It turns out that Le Grand's conditions are correct for sharp images but his condition such that the images are undistorted prove to be neither necessary nor sufficient in general although they are necessary but not sufficient in most situations of interest in optometry and vision science. A numerical example treats a model eye which satisfies Le Grand's condition of no distortion and yet forms elliptical and noncircular images of distant circles on the retina. The conditions for distant object are generalized to include the case of objects at finite distances, a case not examined by Le Grand., (Copyright © 2013 Elsevier B.V. All rights reserved.)

Background: The line of sight and the corneal sighting centre are important references for clinical work in optometry and ophthalmology. Their locations are not fixed but may vary with displacement of the pupil and other changes in the eye., Purpose: To derive equations for the dependence of the locations on properties of an eye which may be heterocentric and astigmatic., Methods: The optical model used is linear optics. It allows for the refracting surfaces of the eye to be astigmatic and tilted or decentred. Because the approach is general it applies not only to the natural eye but also to a pseudophakic eye and to the compound system of eye and any optical instrument in front of it. The analysis begins with the line of sight defined in terms of the foveal chief ray., Results: Equations are derived for the position and inclination of the line of sight at incidence onto the eye. They allow one to locate the line of sight and corneal sighting centre given the structure (curvatures, tilts, spacings of refracting surfaces) of the eye. The results can be generalized in several ways including application in the case of extra-foveal fixation and when there is a lens or other optical instrument in front of the eye. The calculation is illustrated in the Appendix for a model eye with four separated, astigmatic and tilted refracting surfaces., Conclusions: The equations allow routine calculation of the line of sight for an eye of known structure and of the eye combined with an optical device such as a spectacle lens. They also allow exploration of the dependence of the line of sight on the location of the centre of the pupil and on other properties in the eye. There is a dependence of the line of sight on the frequency (or vacuum wavelength) of light but this may not be of clinical significance., (Ophthalmic & Physiological Optics © 2012 The College of Optometrists.)

Two kinds of axes are described as achromatic in the vision science literature: those of Le Grand and Ivanoff, originally proposed in the 1940s, and those of Thibos and Bradley proposed in the 1990s. Thibos-Bradley axes are based on chief nodal rays, that is, nodal rays that intersect the pupil at its center. By contrast Le Grand-Ivanoff achromatic axes are pupil independent. The purpose of this paper is to develop the linear optics of Thibos-Bradley achromatic axes and to examine the sense in which such axes can be said to be achromatic. Linear optics is used to define the chief nodal ray of an arbitrary optical system whose refracting elements may be heterocentric and astigmatic and with nonaligned principal meridians. The incident segment of the ray then defines what is called here the incident chief nodal axis and the emergent segment the emergent chief nodal axis. When applied to an eye they become the external and retinal chief nodal axes of the eye. The axes are infinite straight lines. Equations are derived, in terms of the properties of the eye, for the inclination and transverse positions of both axes at incidence onto the eye. An equation is also derived for the position of the retinal chief nodal axis at the retina. The locations of the axes are calculated for a particular model eye in Appendix A. The equations are specialized for the system consisting of an eye and a pinhole in front of it. For a reduced eye the external and retinal chief nodal axes coincide and are independent of the frequency of light; and, hence, the Thibos-Bradley axes are strictly achromatic for that eye. However for more complicated eyes this is not usually the case; the external and retinal axes are usually distinct, dependent on frequency, and, hence, not strictly achromatic. It seems appropriate, therefore, to reserve the term achromatic axis for axes of the Le Grand-Ivanoff type and generalizations thereof, and to call Thibos-Bradley axes chief nodal axes., (Copyright © 2012 Elsevier Ltd. All rights reserved.)

Purpose: There is inconsistency in the literature in the definitions of longitudinal and transverse chromatic aberration, and there appear to be no definitions that make allowance for astigmatism and heterocentricity. The purpose is to propose definitions of longitudinal and transverse chromatic aberration that hold for systems which, like the typical eye, may be heterocentric and astigmatic and to develop the associated optics., Methods: Common definitions of longitudinal and transverse chromatic aberration based on Gaussian optics are generalized naturally in terms of linear optics to accommodate heterocentricity and astigmatism., Conclusions: The definitions offered here apply to systems in general, including the visual optical system of the eye, and hold for homocentric stigmatic systems in particular. Care is advocated in the use of the terms longitudinal and transverse chromatic aberration.

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., (Copyright © 2012 Elsevier Ltd. All rights reserved.)

Background: Retinal blur patch, effective corneal patch, projective field, field of view and other concepts are usually regarded as disjoint concepts to be treated separately. However they have in common the fact that an aperture, often the pupil of the eye, has its effect at some other longitudinal position. Here the effect is termed aperture referral., Purpose: To develop a complete and general theory of aperture referral under which many ostensibly-distinct aperture-dependent concepts become unified and of which these concepts become particular applications. The theory allows for apertures to be elliptical and decentred and refracting surfaces in an eye or any other optical system to be astigmatic, heterocentric and tilted., Methods: The optical model used is linear optics, a three-dimensional generalization of Gaussian optics. Positional and inclinational invariants are defined along a ray through an arbitrary optical system. A pencil of rays through a system is defined by an object or image point and an aperture defines a subset of the pencil called a restricted pencil., Results: Invariants are derived for four cases: an object and an image point at finite and at infinite distances. Formulae are obtained for the generalized magnification and transverse translation and for the geometry and location of an aperture referred to any other transverse plane., Conclusions: A restricted pencil is defined by an aperture and an object or image point. The intersection of the restricted pencil with a transverse plane is the aperture referred to that transverse plane. Many concepts, including effective corneal patch, retinal blur patch, projective field and visual field, can now be treated routinely as special cases of the general theory: having identified the aperture, the referred aperture and the referring point one applies the general formulae directly. The formulae are exact in linear optics, explicit and give insight into relationships., (Ophthalmic & Physiological Optics © 2011 The College of Optometrists.)

Pascal's ring is a hexagon each of whose corners represents one of the six cardinal points of an optical system and whose sides represent relationships of relative axial position of the cardinal points. Changes to the ring represent the axial displacements of the cardinal points of the visual optical system of an eye that are caused when a spectacle lens compensates for the ametropia. Pascal's schema was described some 70 years ago with little theoretical justification. The purpose of this paper is to derive expressions for the axial locations of the cardinal points of a compound system consisting of an optical instrument and a visual optical system and for the shift caused by the instrument, and to provide theoretical justification for Pascal's schema. The cardinal points are treated not as separate entities but in a unified manner as special cases of an infinite class of special points. Expressions are derived using Gaussian optics. The results are specialized for the case of the eye's ametropia compensated by optical instruments in general and by spectacle lenses in particular. Pascal's schema is shown to be broadly correct although some modification is necessary for the effects on the incident cardinal points especially for the myopic eye., (Copyright © 2011 Elsevier Ltd. All rights reserved.)

Background: The pupil admits to the back of the eye only some of the light arriving from a point in space. As a result only a portion of the cornea is involved when an eye views the point; it is the effective corneal patch for that point. The location, size and shape of the patch are of interest for corneal refractive surgery inter alia. Previous studies have used geometrical optics and a simple model eye (a naked eye with a spherical, single-surface, centred cornea and a concentric circular pupil). Even for the simplest situations geometrical optics provides only implicit formulae which give little insight into relationships and require numerical solution., Purpose: To show how linear optics leads to explicit formulae that estimate the geometry of the effective corneal patch in a very wide range of situations. The eye is not restricted to a single refracting surface; the surfaces may be astigmatic and decentred or tilted and the pupil may be decentred and elliptical. The eye may contain implants and it may be looking through a spectacle lens or other optical instrument which may also contain astigmatic and decentred surfaces., Methods: Linear optics is used to provide general formulae for the geometry of the corneal patch. An appendix illustrates application to some particular cases., Results: General formulae are obtained for the location and geometry of the effective corneal patch for object points that may be near or distant. Formulae are presented in particular for the special case of the naked eye and the case in which all surfaces are spherical and centred on a common axis. Numerical examples in the appendix allow comparison of results obtained via geometrical and linear optics., Conclusions: In using linear optics one sacrifices some accuracy at increasing angles away from the longitudinal axis but there is considerable gain in the complexity and range of problems that can be tackled, and the explicit formula one obtains clearly exhibit relationships among parameters of clinical relevance., (© 2010 The College of Optometrists.)

Purpose: To develop the linear optics of general catadioptric systems with allowance for both astigmatism and heterocentricity., Methods: Reflecting elements partition a catadioptric system into subsystems of four distinct types: (unreversed) dioptric subsystems, anterior catoptric subsystems, reversed dioptric subsystems, and posterior catoptric systems. Differential geometry of an arbitrary astigmatic and tilted or decentered surface is used to determine the anterior and posterior catoptric transferences of a surface., Results: The transference of a catadioptric system is obtained by multiplication of the transferences of unreversed and reversed dioptric subsystems and anterior and posterior catoptric transferences of reflecting elements. Formulae are obtained for the transferences of the visual system of an eye and of six nonvisual systems including the four Purkinje systems., Conclusions: The transference can be calculated for a catadioptric system, and from it, one can obtain other optical properties of the system including the dioptric power and the locations of the optical axis and cardinal structures.

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.

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.

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 x 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.

Back- and front-vertex powers are concepts of some importance in clinical practice. For example, the former is used for characterizing the typical spectacle lens and the latter for characterizing the addition of a bifocal lens. Typically, they are defined either in terms of vergence or the distance to a focal point. This note argues that current definitions are not as clear as they might be, that there is an unnecessary asymmetry between the definitions of front- and back-vertex powers, and that they are designed primarily for systems that are not astigmatic. The purpose of this note is to offer modified definitions that hold for optical systems in general, that is, for systems that may contain astigmatic and decentered refracting elements. The definitions are conceptually clear and provide a simpler derivation of Keating's elegant and general expressions for back- and front-vertex powers.

If a ray enters and leaves an optical system along the same straight line that line is an optical axis of the system. The number of optical axes that a system can have is none, one, or infinity. The purpose of the article is to show how to determine whether a system has an optical axis and to find the optical axis if it is unique and all the optical axes if there are an infinity of them. A simple system may have no optical axis or an infinity of them. A more complicated system is more likely to have a unique optical axis. The optical model is linear optics and the optical system may have refracting elements that are relatively decentered, separated, and astigmatic with non-aligned principal meridians. All the possible types of cases are treated in an appendix. In particular an example examines a simple eye that has an infinity of optical axes and a more realistic eye that has a unique optical axis.

The effective corneal refractive zone is that portion of the cornea traversed by the light that enters the pupil of the eye from object points at a specified angle from the line of sight. It is of relevance in corneal surgery and for understanding the effect of corneal opacities and lesions on vision. Gaussian optics is used in this paper to obtain explicit equations for the geometry of the effective corneal refractive zone for a simplified eye, when spectacle and contact lenses are worn. The theory shows that lenses of positive power increase the diameter of the effective corneal refractive zone and lenses of negative power decrease the diameter. For axial object points the diameter of the effective corneal refractive zone increases by about 0.015 mm per dioptre increase in the power of the spectacle or contact lens. For object points at 30 degrees from the longitudinal axis, the increase is about twice as much in the case of contact lenses and more than four times as much in the case of spectacle lenses.

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.

In the standard refraction routine, a Jackson cross-cylinder is reversed in front of the eye by the process of flipping it about its handle. Reverse telescopes have applications in low vision. In ophthalmoscopy, the practitioner views a patient's retina using light that traverses the eye in a direction opposite to that involved in vision. These represent examples of flipped systems that are involved in optometry. The purpose of this paper is to examine what happens to the optical character of an arbitrary optical system when it is flipped in this manner. The analysis is in terms of the ray transference of the system and, because the transference completely characterises the first-order optics of a system, is complete within the limitations of linear optics. It allows for elements in the system to be astigmatic and decentred. An expression is derived for the transference of the flipped system in terms of the transference of the system itself. Expressions are also obtained for the fundamental first-order optical properties and the dioptric power of the flipped system. Three numerical examples are given in the Appendix.

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.

Representation of astigmatic dioptric power as a power vector is satisfactory for basic operations such as summing and averaging powers. However, power vectors do not fully characterize the nature of dioptric power and are, therefore, unsatisfactory for representing power in general. The purpose of this note is to make the case that it is the power matrix instead that is proper for the representation of power in general. The mathematical nature of dioptric power is examined.

The ray transference matrix completely characterises the first-order optical nature of an optical system including the eye. It is in terms of the transference that quantitative analyses (for example, calculation of an average eye) can be performed. However, the fact that the entries of the transference do not have the same physical dimensions precludes the calculation of the usual scalar value (a Frobenius norm for example) for a change or difference between two optical systems. The purpose of this note is to show how to use the wavelength of the light as a natural unit of length to define a dimensionless transference and so make it possible to calculate a meaningful norm. In most practical applications, some components of the dimensionless transference may dominate unreasonably in the resulting norm in which case suitably weighted norms may be more appropriate. In an appendix, some of the issues are illustrated by application to a lens.

In quantitative analyses of the optical character of eyes (and related systems) it is sometimes necessary to deal with left and right eyes in the same context. In accounting for anatomical symmetry (mirror symmetry in the mid-sagittal plane) one treats a cylinder axis at 20 degrees , say, in a left eye as equivalent to an axis at 160 degrees in a right eye. But this is only one aspect of the linear optical character of an eye. The purpose of this note is to show how to account for anatomical symmetry in the linear optical character of eyes in general. In particular the note shows how to modify the optical properties of left (or right) eyes so that anatomical symmetry is accounted for in quantitative analyses in contexts in which both left and right eyes are involved.

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 Bennett's equations for sagittal and tangential curvature of ellipsoids of revolution. Equations are also presented for the locations of umbilic points on the ellipsoid.

Purpose: The purpose of this study is to derive the equation for the effect of tilt on the tilted power vector of a possibly astigmatic thin lens., Methods: The analysis makes use of the equation for tilted power in terms of the dioptric power matrix., Results: A simple equation is obtained for the tilted power vector of a thin lens in terms of the untilted power vector of the lens. A solution of the equation provides a means for compensating spectacle lenses for tilt. Numerical examples are presented in the (available online at www.optvissci.com)., Conclusions: The equation gives insight and is useful for researchers and clinicians working in terms of power expressed as power vectors.

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.

Purpose: The purposes of this article were, for any sequence of tilts applied to a thin lens, to calculate an equivalent turn and single tilt, to show how to use the equivalent turn and tilt to calculate the tilted power of the lens, and, given the desired tilted power, to calculate the power of the untilted lens necessary to compensate for the effects of tilt. The untilted lens may be stigmatic (spherical) or astigmatic (spherocylindrical)., Methods: The analysis makes use of rotation matrices to represent rotation in space and previous work in third-order optics on oblique central refraction., Results: Equations are presented for calculating the combination of turn and tilt that is equivalent to any sequence of tilts. They are specialized for the particular case of combinations of faceform and pantoscopic tilts and allow the decomposition of an arbitrary tilt into a combination of turn and pantoscopic and faceform tilts. The equations also lead to a procedure for calculating or compensating for the tilted power of a sequentially tilted thin lens., Conclusions: Previous work on the effect of tilt on thin lenses has been generalized to handle combinations of arbitrary tilts.

Purpose: : The purpose of this article is to analyze the geometry and examine the implications of the error cells of purely spherical powers in symmetric dioptric power space., Methods: : In the context of spherocylindrical data spherical data typically implies a cylindrical component that is less than some particular amount (often 0.125 D) in magnitude. This error or uncertainty in cylinder is over and above the error in sphere itself. The two components of error are used to define the error cells in symmetric dioptric power space., Results: : Error cells of spherical powers are constructed and presented as stereopairs. They are also shown in relation to error cells of powers in general., Conclusions: : An understanding of error cells can help the researcher avoid pitfalls in the analysis of spherocylindrical data. Perhaps surprisingly, the error cells of spherical powers are not invariant under spherocylindrical transposition.

Round-off of spherocylindrical powers, to multiples of 0.25 D (for example) in the case of sphere and cylinder, and 1 or 5 degrees 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.

There would appear to be little disagreement on what constitutes an astigmatic system in the case of a thin lens: the cylinder is not zero. A spherical thin lens is stigmatic or not astigmatic. The issue is less clear in the case of a thick system. For example, is an eye stigmatic merely because its refraction is stigmatic (spherical)? In this article, a system is defined to be stigmatic if and only if, through the system, every point object maps to a point image. Every other system is astigmatic. Thus, a system is astigmatic if and only if there exists a point object for which the image is not a point. This article is restricted to linear optics. The optical character of a system is completely determined by the ray transference of the system. The objective here is to find those conditions on the transference for which the system is stigmatic or astigmatic. The result is that, for a stigmatic system, all the 2 x 2 submatrices are scalar multiples of a common orthogonal matrix. For a system to be stigmatic, it is not sufficient that its power be stigmatic. An eye may be astigmatic despite having a stigmatic refraction.

Stigmatic optical systems are of two classes: proper and improper stigmatic systems. The purpose of the article is to explore the nature of the two classes. The image may be rotated in the case of proper stigmatic systems and is reflected in a line in the image plane in improper stigmatic systems. Chirality is preserved through proper stigmatic systems but reversed in improper systems. The image also undergoes magnification that is the same in all meridians and, in the case of decentered systems, a transverse translation. Negative magnification implies inversion in a point. The magnification depends on the distance of the object plane from the system, whereas the rotation and reflection do not. The article shows how to identify a system as astigmatic or proper or improper stigmatic from the transference and how to construct the transference of a system that will achieve a particular magnification and rotation or reflection.

For statistical and other purposes one needs to be able to determine an average eye. An average of refractive errors is readily calculated as an average of dioptric power matrices. A refractive error, however, is not so much a property of the eye as a property of the compensating lens in front of the eye. As such, it ignores other aspects of the optical character of the eye. This paper discusses the difficulties of finding a suitable average that fully accounts for the first-order optics of a set of optical systems. It proposes an average based on ray transferences and logarithms and exponentials of matrices. Application to eyes in particular is discussed.

The linear optical character of an optical system is represented by a particular type of 5 x 5 matrix. This article shows that the converse is also true, namely, that an optical system can be constructed in principle with linear optical character represented by any matrix of this type. In other words, every matrix of this type is realizable as an optical system. The system may have astigmatic and decentered elements.

A general expression is derived for the proximity factor in near image size magnification for an arbitrary instrument in front of an arbitrary eye. The proximity factor is a 2 x 2 matrix. The instrument and eye may be astigmatic and have decentred elements. The image on the retina may be blurred or not. The analysis is exact within the limitations of linear optics. The general results are specialized for the case of a stigmatic instrument and a stigmatic eye. The results are applied to the case of a thick, possibly bitoric, spectacle lens. The Appendix treats two numerical examples.

A previous article derives expressions for coefficients and magnifications for an arbitrary optical instrument in front of an arbitrary eye and for near object points. The purpose of this note is to correct an error in the interpretation of a parameter in that article, to show that the expression derived there for near image size magnification holds only under particular conditions, and to modify the expression so that it holds in general.

Traditional treatments of spectacle magnification for distant objects consider only stigmatic spectacle lenses and they compare the retinal image size in a refractively fully compensated eye with the image size in the uncompensated eye. Spectacle magnification is expressed as a product of two factors, the power and shape factors of the lens. The power factor depends on the position of the entrance pupil of the eye. For an eye with an astigmatic cornea, however, the position of the entrance pupil is not well defined. Thus, the traditional approach to spectacle magnification does not generalize properly to allow for astigmatism. Within the constraints of linear optics and subject to the restriction that the eye's iris remains the aperture stop, this paper provides a complete, unified and exact treatment for optical instruments in general. It compares retinal image size in a generalized sense (including image shape and orientation) for any instrument in front of an eye with that of the eye alone irrespective of whether the instrument compensates or not. The approach does not make use of the concept of the entrance pupil at all and it allows for astigmatism and for non-alignment of refracting elements in the instrument and in the eye. The concept of spectacle magnification generalizes to the concept of instrument size magnification. Instrument size magnification can be expressed as the product of two matrix factors one of which can be interpreted as a power factor (as back-vertex power) and the other factor for which the name dilation factor is more appropriate in general. The general treatment is then applied to a number of special cases including afocal instruments, spectacle lenses (including obliquely crossing thick bitoric lenses), contact lenses, stigmatic systems and stigmatic eyes. In the case of spectacle lenses, the dilation factor reduces to the usual shape factor.

Purpose: To provide a general approach to the analysis of refractive data that overcomes the shortcomings of traditional treatments and can be easily adapted to most spreadsheets., Setting: Corneal Service, Royal Liverpool Hospital, Liverpool United Kingdom, and Optometric Science Group, Rand Afrikaans University, Auckland Park, South Africa., Method: The basis of the analyses is the dioptric power matrix. Using a hypothetical sample of data on pre-event and post-event refractions, the calculation of the mean pre-event and post-event refractions and the effect of an event on the refractive outcome, for example the refractive surgical effect, are illustrated. The most important statistics, the mean and the variance-covariance of refractions, and the formal testing of hypotheses on the mean are provided., Results: The method of analysis demonstrated how an event such as cataract surgery, occlusion treatment of amblyopia, or anisometropia can be evaluated in terms of refractive outcome. Hypothesis testing showed how the significance of this effect may be demonstrated., Conclusions: This standardized method of analyzing and reporting refractive data enables a quantitative analysis and statistical hypothesis testing of the complete refractive data. This approach has generalized applicability in a range of commonly encountered contexts such as the refractive change after cataract and refractive surgery, corneal transplantation, and treatment for amblyopia or the significance of anisometropia. The method is relatively straightforward and can be adapted to most conventional spreadsheets.

Astigmatism poses a number of special difficulties when attempting to achieve a desirable refractive outcome during cataract surgery and intraocular lens implantation. One of the most pertinent difficulties is that traditional Gaussian theoretical formulas are simply unable to handle general astigmatic optical systems, that is, systems in which the principal meridians of the refracting interfaces do not lie in two mutually orthogonal planes. This means that the traditional approach falls short when the cornea is treated as bitoric or when other toric optical devices are already present in the eye. In this paper, the modified step-along procedure is utilized in a demonstration of the derivation of the traditional Gaussian theoretical formulas for the determination of intraocular lens power in a stigmatic eye. Matrices are then used to derive a general formula for the determination of the power of a thin toric intraocular lens in an astigmatic eye. This formula is then modified to allow the user to determine the power of the secondary intraocular lens required to compensate for any residual refraction in a phakic or pseudophakic astigmatic eye. The formulas hold under all conditions, including the case when the cornea is treated as thick and bitoric and when other toric optical devices are already present in the eye. Worked examples are provided.

Tilting changes the effective power of a thin lens. Blendowske has recently provided an elegant matrix expression for the tilted power in the case of tilt about a vertical axis (faceform tilt). He makes use of what can be called a tilt matrix. He shows that the same expression holds in the case of tilt about a horizontal axis (pantoscopic tilt) if one makes use of a different tilt matrix. The purpose of this note is to derive a general tilt matrix that makes Blendowske's equation valid for tilt about any axis. The general tilt matrix reduces to the particular tilt matrices in the case of rotation about the horizontal and vertical axes. The lens may be stigmatic or astigmatic and may be in air or in some other medium.

A corrected or uncorrected human eye may be astigmatic and noncoaxial. The first-order (i.e., paraxial) character of arbitrary astigmatic and noncoaxial optical systems can be compared quantitatively in terms of the ray transference of either an anterior or a posterior converter system. This article derives equations that give the relationships between the transferences of the anterior and posterior converter systems.