Visual angle

The visual angle is the angle a viewed object subtends at the eye, usually stated in degrees of arc. It also is called the object's angular size.

The sketch helps to define it 1,2

It shows an observer's eye looking at a frontal extent (the vertical arrow) that has a  linear size, S meters, located D meters from point O.

For present purposes, point O can represent the eye's nodal points at about the center of the lens, and also represent the center of the eye's entrance pupil that is only a few millimeters in front of the lens. The three lines from object endpoint A heading toward the eye indicate the bundle of light rays that pass through the cornea, pupil and lens to form an optical image of  endpoint A on the retina at point a. The central line of the bundle represents the chief ray

Likewise for object point B and its retinal image at b.

The visual angle, V deg, is the angle between the chief rays for A and B.

The relationship between S, D and V.
V deg can be measured directly using a theodolite placed at point O.

Or, it can be calculated 3 using the formula, V = 2 arctan(S/2D).

However, for visual angles smaller than about 10 degrees, the simpler formula, Equation 1, provides very close approximations.

tanV = S/D   (Equation  1).

Also for small angles, V = S/D radians. That is, V = 57.3 (S/D) degrees.

The retinal image and V deg
As the sketch shows, a real image of the object is formed on the retina between points a and b. (See Visual system,  physiological optics).

For small angles, the size of this retinal image, R mm, is given by Equation 2

R/n = tanV     (Equation 2). in which n is the distance  from the nodal points to the retina, about 17 mm.

Some Examples
If one looks at a one-centimeter object at a distance of one meter and a two-centimeter object at a distance of two meters, both subtend the same visual angle of 0.01 radian or 0.57 deg. Thus they have the same retinal image size, about R = 0.17 mm. That is just a bit larger than R for the moon, which is about  0.15 mm, because, with S = 2160 miles, and D averaging 238,000 miles, V =  0.009 rad, or 0.52 deg.

Also, for some easy observations, if one holds one's index finger at arm's length, the width of the index finger subtends approximately one degree, and the thumb subtends approximately two degrees (O'Shea, 1991).

Therefore, if one is interested in the performance of the eye or the first processing steps in the visual cortex, it does not make sense to refer to the absolute size of a viewed object (its linear size, S meters). What matters is the visual angle, V deg, which determines the size of the retinal image.

Some terminological confusions
In astronomy the term apparent size refers to the physical angle, V deg, or angular diameter.

But in psychophysics and experimental psychology the adjective "apparent" refers to a person's subjective experience. So, "apparent size" has referred to how large an object looks, also often called its "perceived size".

Additional confusion has occurred because there are two qualitatively different "size" experiences for a viewed object (Joynson, 1949, McCready, 1965, 1985, Baird, 1970).

One is the perceived visual angle, V' deg, (or apparent visual angle) which is the subjective correlate of V deg, also called the object's perceived or apparent angular size. Expressed in degrees of arc, V' is best defined as the difference between the perceived directions of the object's endpoints from oneself (Joynson, 1949, McCready, 1965, 1985).

The other "size" experience is the object's perceived linear size, S'  meters, (or apparent linear size) which is the subjective correlate of S meters, the object's physical width or height or diameter.

Widespread use of the ambiguous  terms "apparent size" and "perceived size" without specifying the units of  measure has caused confusion (degrees are not meters).

Visual Angle and the Visual Cortex
The brain's primary visual cortex, area V1 or Brodmann area 17 contains a spatially isomorphic representation of the retina see Retinotopy).  Loosely speaking,  it is a distorted "map" of the retina. Accordingly, the size ( R mm) of a given retinal image determines the extent of the  neural activity pattern eventually generated in Area V1 by the associated retinal activity pattern. Indeed. Murray, Boyaci, & Kersten (2006) recently used Functional magnetic resonance imaging  fMRI to show, convincingly, that an increase in a viewed target's visual angle,  which increases R mm, increases the extent of the corresponding neural activity pattern in Area V1.

Their most important finding, however, relates to the perceived visual angle, V' deg, and to a  visual angle illusion.

Angular size illusion and Area V1
The Murray, et al. observers viewed a flat picture with two disks that subtended the same visual angle (V deg) and formed retinal images of the same size (R mm), but the perceived angular size (V') of one was about 17% larger than V' for the other, due to differences in the background patterns for the disks.

The major discovery was that the sizes of the Area V1 activity patterns related to the disks were unequal, despite the fact that the retinal images were the same size. This size difference in Area V1 correlated almost perfectly with the 17% illusory difference between the perceived visual angles. These new findings are crucial for theories of  visual space perception and especially any Visual Angle Illusion.4