Visual angle

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 diagram on the right shows an observer's eye looking at a frontal extent (the vertical arrow) that has a linear size formula_1, located in the distance formula_2 from point formula_3.
For present purposes, point formula_3 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 formula_5 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 formula_5 on the retina at point formula_7.
The central line of the bundle represents the chief ray.
The same holds for object point formula_8 and its retinal image at formula_9.
The visual angle formula_10 is the angle between the chief rays of formula_5 and formula_8.
Measuring and computing.
The visual angle formula_10 can be measured directly using a theodolite placed at point formula_3.
Or, it can be calculated (in radians) using the formula, formula_15.
However, for visual angles smaller than about 10 degrees, this simpler formula provides very close approximations:
The retinal image and visual angle.
As the above sketch shows, a real image of the object is formed on the retina between points formula_7 and formula_9. (See visual system). For small angles, the size of this retinal image formula_19 is
where formula_21 is the distance from the nodal points to the retina, about 17 mm.
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 about 0.01 rad or 0.57°. Thus they have the same retinal image size formula_22.
That is just a bit larger than the retinal image size for the moon, which is about formula_23, because, with moon's mean diameter formula_24 formula_25 , and earth to moon mean distance formula_2 averaging formula_27 (formula_28), formula_29formula_30.
Also, for some easy observations, if one holds one's index finger at arm's length, the width of the index fingernail subtends approximately one degree, and the width of the thumb at the first joint subtends approximately two degrees.
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 formula_1). What matters is the visual angle formula_10 which determines the size of the retinal image.
Terminological confusions.
In astronomy the term "apparent size" refers to the physical angle formula_10 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. One is the perceived visual angle formula_34 (or apparent visual angle) which is the subjective correlate of formula_10, also called the object's perceived or apparent angular size.
The perceived visual angle is best defined as the difference between the perceived directions of the object's endpoints from oneself.
The other "size" experience is the object's perceived linear size formula_36 (or apparent linear size) which is the subjective correlate of formula_1, 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.
Representation of visual angle in 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 formula_19 of a given retinal image determines the extent of the neural activity pattern eventually generated in area V1 by the associated retinal activity pattern. Murray, Boyaci, & Kersten (2006) recently used Functional magnetic resonance imaging (fMRI) to show that an increase in a viewed target's visual angle, which increases formula_19, also increases the extent of the corresponding neural activity pattern in area V1.
The observers in Murray 'et al.'s' experiment viewed a flat picture with two discs that subtended the same visual angle formula_10 and formed retinal images of the same size formula_19, but the perceived angular size formula_34 of one was about 17% larger than for the other, due to differences in the background patterns for the disks. It was shown that the areas of the activity in V1 related to the disks were of unequal size, despite the fact that the retinal images were the same size. This size difference in area V1 correlated with the 17% illusory difference between the perceived visual angles. This finding has implications for spatial illusions such as the visual angle illusion.