Longitudinal Magnification Drawing Mistake

Abstract

Lateral magnification in image formation by positive lenses, mirrors, and dioptrics is usually appropriately developed in most optics textbooks.1–9 However, the image of a three-dimensional object occupies a three-dimensional region of space. The optical system affects both the transverse and the longitudinal dimensions of the object and, in general, does it in different ways. The magnification in the direction of the optical axis (the longitudinal magnification) is seldom treated. In several texts, the concept of longitudinal magnification is not even considered. Symmetrical objects (such as arrows) are used and their images appear laterally inverted. It is not shown how a longitudinally nonsymmetric object is imaged. One of the few books where this subject is well treated is in the textbook by Hecht.10 We have repeatedly verified in our classes that there is some confusion related to this subject. Students tend to believe that the image is longitudinally symmetric with respect to the lens optic center. Some prestigious texts commit the same mistake. In addition, a very nice optics book,11 a catalogue of optical hardware,12 a worldwide scientific magazine,13 a paper in an optics journal,14 and a Spanish encyclopedia,15 for example, have also been found to contain this error in drawing the image of a three-dimensional object formed by a positive lens. In this paper we suggest that the teaching of longitudinal magnification should be done with some care and we include a figure showing a properly drawn image.