Aperture referral in heterocentric astigmatic systems William F Harris Department of Optometry, University of Johannesburg, Johannesburg, South Africa Citation information: Harris WF. Aperture referral in heterocentric astigmatic systems. Ophthalmic Physiol Opt 2011, 31, 603–614. doi: 10.1111/j.1475-1313.2011.00860.x Keywords: aperture referral, astigmatism, blur, effective corneal patch, elliptical pupil, heterocentricity, restricted pencil, transference Correspondence: William F Harris E-mail address: wharris@uj.ac.za Received: 1 March 2011; Accepted: 17 June 2011 Abstract 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 general- ization 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 effec- tive 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. Introduction Retinal blur patch, 1–4 effective corneal patch, 5–10 projec- tive field, 11 field of view 12–15 and other concepts, usually regarded as distinct, can all be unified under the general heading of referred apertures. The purpose of this paper is to develop a complete and general theory of aperture referral in linear optics for dioptric systems that may be heterocentric and astigmatic. Apertures may be ellipti- cal and decentred. Referral transforms the geometry of an aperture. General equations are derived for the transformation. Expressions for retinal blur patch, effec- tive corneal patch and other ostensibly-distinct aperture- dependent concepts then become particular applications of the general theory. As an optical device the typical eye differs in two impor- tant respects from a camera: its refracting surfaces are nei- ther centred on nor invariant under rotation about a common axis. Consequently many concepts suitable for the camera loose their sharpness when applied to the eye. For example, because of heterocentricity, the eye has no optical axis as traditionally defined; at best one has to be Ophthalmic & Physiological Optics ISSN 0275-5408 Ophthalmic & Physiological Optics 31 (2011) 603–614 ª 2011 The College of Optometrists 603