Objects in rigid motion do not look to change their shape, despite the effects of perspective. Yet sometimes under unusual conditions of foreshortening, an object’s shape may not be judged very accurately, or else an object in rigid motion may be mistaken for an object in non-rigid motion. Foreshortening and other geometric characteristics of light – i.e., the geometry of the medium by which we see – can affect the estimates of shape which we make with the help of vision. Sometimes observers have difficulty in separating the medium from the message in that respect. There are some properties of shape which are unaffected by the geometrical characteristics of the medium, that is, they are unaffected by the uniform propagation of light. The measure of those properties can be used to gauge how well observers judge the changing or unchanging shape of an object despite the effects of perspective. Such measures can be said to abstract from any transformation in shape that is imposed by the propagation of light. The properties in question are called projective invariants. The present article extends the use of these measures, to include the projective invariants of conics. Though descriptive tools for the measurement of projective invariants are well understood – not only for conics, but also for cubics, quadric surfaces and more general classes of shapes – these mathematical tools have not been applied in experiment to problems of visual shape constancy.

The experiments that follow will concentrate on the projective invariants of conics, but why study the projective invariants of conics? They may seem to represent an obscure detail of geometry – a detail not easily generalized or not readily applied to practical constructions. Yet beyond their applications in computer vision, the projective properties of conics find application in the industrial modelling of complex shape. They find interpretation in the geometry standards of curve and surface design, as in the design of aircraft fuselage and automobile bodies. “Conics may be pieced together to form a more complicated curve, whose shape is too complex to be captured by a single conic. Such composite curves are called conic splines”. The projective invariants of conics have immediate application to the business of measuring and describing complex stimuli for purposes of the study of vision, when the geometry of the viewing situation is unspecified. “Projective geometry in computer vision is primarily applied to the study of uncalibrated cameras, for which the linear intrinsic parameters are unknown and the non-linear effects are assumed negligible”. This is only one way to proceed, though; another is to apply a still more subtle mathematics. What follows is a first demonstration or in any case, an effective demonstration of shape constancy for the projective properties of conics under change in perspective.

On first consideration, the notion that a geometric quantity unchanged under all perspective transformations can be used to gauge the accuracy of visual form perception may seem implausible. There has been little to demonstrate that observers’ estimates are stable and reliable in terms of these measures (though a recent effort began with Cutting, 1986; see Niall, 1987, for a critical review of that effort). Although perceptual psychologists have continued to outline their intuitions on the topic of invariance, little has been done to measure projective invariants in the context of experiment. Perhaps a classical approach to geometry has been thought too rigorous. Or else it may have seemed implausible that observers compute the measures of quantitative invariants, which consideration may have sparked study of ‘qualitative invariants’ in the study of human form perception. A classical and quantitative approach to the application of projective invariants to human vision has not yet been exploited to its potential. In this study we begin to assess accuracy of shape perception by measuring the projective invariants of conics.

Seven students and employees of DCIEM (the Defence and Civil Institute of Environmental Medicine, North York, Ont., Canada) were tested, of whom four were women and three were men. All had normal uncorrected acuity. None were aware of the specific purpose of the study, apart from the experimental instructions given. Each observer signed a document indicating informed consent in the study, as did all observers in the other experiments that will be reported. Observers were given a few dollars per diem allowance for their participation.

The basic stimuli for the first two experiments consist of 18 pairs of ellipses. These differ in the eccentricity of the ellipses, and in the relative orientation of the major axis of one ellipse to the major axis of the other. The pairs of ellipses differ in the values of their projective invariants, as a consequence. The ellipses of each pair have a common centre, that is, a common midpoint between their foci. The 18 pairs of ellipses are formed in this way: three ellipses are chosen that differ in the ratio of the length of their minor axis b to the length of their major axis a. That ratio is 5:1 for the first ellipse, 9:1 for the second ellipse, and 9:5 for the third ellipse.