cornea. This reduces the angle of incidence for light entering it, and the angle of

refraction as a result. Concept 2 shows the eye after laser eye surgery for

nearsightedness. The dashed line shows the original curvature of the eye.

With farsightedness, the muscles of the eye cannot contract the lens enough. Close

objects are focused behind the retina, again causing blurring. This problem is shown in

Concept 3. To address it, the surgeon increases the steepness of the cornea. Since the

angle of incidence of incoming light rays is increased, they refract more. The bottom

illustration shows a farsighted eye after laser eye surgery. Again, the dashed line

represents the initial shape of the eye.

The problem: farsightedness

Image forms behind retina

The solution

Cornea steepened to correct vision

After surgery í image forms on retina

33.13 - Angular size

Angular size: How much of

the field of vision is filled by

an object or image.

Imagine yourself in Seattle, standing two kilometers

from the Space Needle. You look at the structure, and

then raise your hand so that your thumb is a half-

meter from your eyes. Your thumb now appears to be

about the same size in your field of vision as the

Space Needle, although you know that the Space

Needle is much bigger.

This effect occurs because of the relative angular sizes of your thumb and the Space

Needle. The angular size refers to the angle, measured in radians, subtended by an

object when viewed from a particular distance. The subtended angle is perhaps as well

explained with a diagram as in text. The subtended angle is labeled ș in Concept 1 to

the right. It measures the angle of your vision “blocked out” by the object.

Trigonometry can be used to determine angular size when the height h of the object

and the distance d to the object are known. The angular size can be approximated as

the height divided by the distance to the object. This approximation uses a small-angle

approximation, ș§ tanș for small angles, where the angle must be measured in

radians. This approximation for angular size is very good (within one percent) for

angular sizes less than 0.17 rad (about 10°). Approximations like this can be useful in

astronomy to approximate distance or size. For instance, it can be useful to know that

the Moon subtends about 0.5° (0.009 radians). If you know its distance, you can quickly

approximate its diameter using this fact and some trigonometry.

To leave space and return to Seattle: The thumb in the photograph above is 5 cm or so

tall and 60 cm away. It has an angular size of 5 cm divided by 60 cm, which equals 0.08 rad. At a distance of 2 km, the Space Needle, which is

184 meters tall, would subtend an angle of about 184 m/ 2000 m, or 0.09 rad. These values confirm that the thumb is about the same angular

size as the Space Needle.

Sizing up the Needle.

Angular size

Amount of visual field filled by object

Measured as angle

(^618) Copyright 2007 Kinetic Books Co. Chapter 33