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 S {\displaystyle S} , located in the distance D {\displaystyle D} from point O {\displaystyle O} . For present purposes, point O {\displaystyle 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 {\displaystyle 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 {\displaystyle A} on the retina at point a {\displaystyle a} . The central line of the bundle represents the chief ray. The same holds for object point B {\displaystyle B} and its retinal image at b {\displaystyle b} . The visual angle V {\displaystyle V} is the angle between the chief rays of A {\displaystyle A} and B {\displaystyle B} .
Measuring and computing The visual angle V {\displaystyle V} can be measured directly using a theodolite placed at point O {\displaystyle O} . Or, it can be calculated (in radians) using the formula, V = 2 arctan ( S 2 D ) {\displaystyle V=2\arctan \left({\frac {S}{2D}}\right)} . However, for visual angles smaller than about 10 degrees, this simpler formula provides very close approximations:
tan ( V ) = S D . {\displaystyle \tan \left(V\right)={\frac {S}{D}}.}
The retinal image and visual angle As the above sketch shows, a real image of the object is formed on the retina between points a {\displaystyle a} and b {\displaystyle b} . (See visual system). For small angles, the size of this retinal image R {\displaystyle R} is
R n = tan V , {\displaystyle {\frac {R}{n}}=\tan V,}
where n {\displaystyle n} 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 R ≈ 0.17 mm {\displaystyle R\approx 0.17{\text{ mm}}} . That is just a bit larger than the retinal image size for the moon, which is about 0.15 mm {\displaystyle 0.15{\text{ mm}}} , because, with moon's mean diameter S = 3474 kilometers {\displaystyle S=3474{\text{ kilometers}}} ( 2159 miles ) {\displaystyle (2159{\text{ miles}})} , and earth to moon mean distance D {\displaystyle D} averaging 383 , 000 kilometers {\displaystyle 383,000{\text{ kilometers}}} ( 238 , 000 miles {\displaystyle 238,000{\text{ miles}}} ), V ≈ 0.009 rad {\displaystyle V\approx 0.009{\text{ rad}}}
≈ 0.52 deg {\displaystyle \approx 0.52{\text{ deg}}} . 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 S {\displaystyle S} ). What matters is the visual angle V {\displaystyle V} which determines the size of the retinal image.
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