October 1972
On the Concept of the “Visual Angle” in an Optic Array and its History
J. J. Gibson, Cornell University
The World Wide Web distribution of James Gibson’s “Purple Perils” is for scholarly use with the understanding that Gibson did not intend them for publication. References to these essays must cite them explicitly as unpublished manuscripts. Copies may be circulated if this statement is included on each copy.
Ecological optics is concerned with the ambient light at a point of observation as distinguished from the radiant light, emitted from a point-source (Boynton). It assumes a steady-state of illumination in a medium, with omnidirectional light at all points in the medium (because of multiple reflection in a semi-enclosed space). And ecological optics is particularly concerned with the structure of the ambient light at any point of observation, this being what is meant by the ambient optic array. An array entails an arrangement.
One way of describing this array (if not the best way) is to say that it consists of visual solid angles all having a common apex at the point of observation. Each solid angle is separated from its topological neighbors by a transition or “contrast” of luminous intensity. The set of solid angles is “closed,” i.e., returns upon itself, and the ambient array is unbounded.
The component solid angles of the array at a fixed point of observation correspond to components of the environment, more exactly to all components that are not hidden (occluded) at the point of observation. The sky and the earth, for example, are components of the terrestrial environment. The sky provides one solid angle and the earth provides its complementary solid angle, the skyline providing the contrast between the two. Each, particularly the earth-angle, is then subdivided into smaller solid angles, and this subdivision has no limit. The faces and facets of environmental surfaces correspond to visual solid angles in an optic array. Note that all faces and facets are projected to some point of observation.
These visual solid angles each have a cross-sectional shape and an angular size. Each is a unit in the sense of a “form” but this is not to be confused with a metric unit in the sense of degrees, minutes, and seconds of arc. The units of the ambient array are nested, that is, there are forms within forms. We can speak of optical texture when the units are small and optical patterns (vaguely) when the units are large.
Is the concept of a visual solid angle new? Actually it has a long, if obscure, intellectual history. Euclid had the notion of what he called the “visual pyramid,” and Ptolemy spoke of the “visual cone,” the apex of the pyramid or cone being ” at the eye,” and the base being “at the object.” Neither of these ancient geometers could have meant that the base was always a rectangle or always a circle; they probably understood that it was actually what I have called the face of an object, the unhidden face, and not the object itself. But Ptolemy was still thinking in terms of Euclidean solid geometry and objects in empty space, not in terms of adjacent solid angles and the corresponding components of a surrounding environment. He probably did not understand that a solid angle might correspond to a hole or a patch of sky.
Nevertheless these ideas were the basis of what was called “natural perspective” in medieval times as distinguished from the “artificial perspective” which was discovered by the Renaissance painters. Visual angles had to do with the laws of vision for the world around us, not with optics as we now think of it. In a sense, however, the laws of “natural perspective” define a branch of optics, geometrical optics, and physiological optics.
Note that this early concern with visual angles is quite separate from the problem of how the eye gets an image of the object, or forms an image of it. There were all kinds of early speculation about images or copies being sent off from objects and transmitted to an eye. There were deep perplexities about light. Not until the time of Kepler was it suggested that an image was formed at the back of the eye by the refracting power of its lens, such that diverging rays form each point of the object converged to each focus-point in the image. This is now the unchallenged theory of image-formation for a chambered eye, or a camera. Up until that time it had more or less been assumed that an image was something that could be transmitted to an eye. But the study of visual angles, of natural perspective, or of ecological optics can be carried out independently of the study of eyes, rays, and images.
The “law of the visual angle” as dependent on the distance of the object is often treated as self-evident and unworthy of elaboration. But the laws of visual solid angles considered as the components of an ambient optic array that are specific to certain components of an environment are not at all self-evident and are in need of elaboration. The “laws of perspective” are often treated as cues for depth perception by psychologists and as conventions for the making of illusionistic pictures by painters. But when they are freed from the conceptual confusion that permeates the theory of pictures, including retinal pictures, the laws of perspective prove to be very interesting indeed.