I asked this one of Cecil a while ago, but it didn’t seem to take his fancy.
Can anyone explain to me WHY perspective IS? I don’t understand how objects can appear smaller with distance… I once asked this of a nuclear physicist of my aquaintance, and he furnished me with an answer I didn’t understand at all, though I nodded sagely as he was talking.
Can anyone help this simple antipodean to whom all you propodeans are just tiny, prespective-shrunk specks?
Think of it this way. Normally, you have about 180 degrees of vision, with maybe 90 degrees of clear vision - the stuff that’s directly in front of you. If there is a four foot wide object directly in front of you, it blocks nearly all of your 90 degrees of clear vision. If there is a four inch wide object directly in front of you, it blocks maybe 5 degrees of your vision (I’m pulling these numbers out of my ass, but you get the idea). If you take that four-foot wide object and put it 100 feet away, it will only block 5 degrees of your vision, just like the smaller object did when it was closer, so it looks “smaller”.
Obviously, if you know how far away an object is, you can compensate for this effect, or you can compare it to something of known size that is at the same distance.
Does that make sense to you?
mischievous
In a perfect world (I’ll get back to this), the size that you see something to be is directly related to its retinal image size. The larger the retinal image, the larger the object. The retinal image size depends on the actual size of the object you’re looking at and the distance of that object.
Say R is the retinal image size.
If S = size and D = distance, then R = S / D.
If an object is twice as far away as a different object, then its retinal image is half as big. Therefore, smaller.
However, the world is not perfect. Your brain can be fooled quite easily. I’m sure you’ve seen numerous optical illusions of size. The most widely observed is the optical illusion that a full moon appears bigger at sunrise and sunset.
Getting back to the formula. Since R = S / D, S = R * D. In other words, the perceived size of the object is its retinal image size times it’s distance. That is why when you see an adult human at a distance you realize that it’s not some three inch tall adult (based on its retinal image size), but rather a full sized adult at a distance.
Now, when the distance cues are messed up, the perceived size of an object can be misleading. The full moon illusion is due to the fact that people perceive the horizon as being further away than the sky directly overhead (this perception has been verified in numerous studies). Therefore, the rising moon seems to be further away, yet its retinal image is identical to when the moon is directly overhead, so since perceived size (s) = R * D, the size appears large.
I’ve gone a ways astray from your question in the OP, but it all ties together.
Oh. And welcome to the SDMB.
Are you sure about this, Algernon? I thought the larger appearance of the moon near the horizon was an optical effect of the atmosphere. When the moon is near the horizon, the rays it reflects are entering the atmosphere at an angle (relative to you) and they must pass through a much greater distance of atmosphere. I thought the refraction and/or scattering effect of the atmosphere was responsible for the apparently larger size, and for the yellow/orange color visible when the moon is near the horizon.
Not that I know much about optics, though.
mischievous
Of course, if I had search the Straight Dope archives, I wouldn’t have just made an ass of myself. Cecil says exactly what you do, Algernon.
http://www.straightdope.com/classics/a2_110.html
mischievous
Thank you very much, both of you, for your lucid and generous answers.
Nevertheless… Maybe I’m dumb, but I must admit, I think I’ve still got a problem.
It’s this: you mention we have an 180 degree range of vision (approx), and go on to use that to explain why objects appear smaller with distance, i.e. because they block fewer degrees of our vision as they recede… But *why * is it they block fewer degrees of our cone of vision? Am I a dunderhead, or are you just saying ‘things appear smaller with distance’, but using degrees rather than feet and inches to say it?
Sorry if this is totally obtuse…
Take a piece of paper. Make a dot somewhere on it. Now draw a 1 inch long straight line 2 inches away from your dot. Draw lines from the dot to the end points of the line. The angle you made is what exists between your “eye” and the object.
Now draw the same line 2 inches beyond the first (keep them parallel). Draw new lines from your dot (“eye”) to the new lines end points. Notice anything?
Yep. The angle create by the new lines is smaller than the first one.
You seem to be saying that you understand that if an object occupies a smaller part of your field of view, it will appear smaller. Your question is why, a more distant object does this.
One simple way to see this would be to use an example in two dimensions. Take a sheet of paper and draw a horizontal line 20 cm long. From its center, draw a vertical line that rises 4 cm, and do the same from its right end. Connect the top ends of these two 4-cm lines to the left end of the horizontal line.
You thus have two right triangles, each 4 cm high. It’s easy to see that although they are the same height, they are not the same shape: The longer one is “thinner.” The angle at its left end is noticeable smaller. Stretch the base of this triangle out to, say, 10 meters while keeping the 4 cm height, and the angle would get really small. So we see that the “angular size” of a 4-cm line depends on how far away from it the angle is being measured.
Vision works the same way. Think of a triangle that consists of a visible object and lines that join your eye with its top and its bottom. As you increase your distance from the object, either its height has to increase or its angular size will decrease.
Imagine for a moment a world in which objects block the same number of degrees, regardless of their distance from you:
Within that world, encircle yourself with a set one inch wooden blocks spaced 1/2 inch apart.
Turn around in a circle, and verify that each block takes up the same amount of space in your visual field. Count the blocks. Let’s say you end up with 100 blocks in the circle.
Now take another set of 100 blocks and hike half a kilometer from your starting point.
Set the blocks down 1/2 inch apart, in a circle around your starting position.
Since each block takes up a fixed amount of visual field regardless of it’s distance from you, you’ll find that your set of 100 blocks is sufficient to create a circle of blocks around your starting position with a spacing between the blocks of 1/2".
This will be true regardless of the distance you walk from your staring position, how big the circle is.
Now, let’s calculate the circumferences of the two circles, and try to derive a rule relating a circles circumference to its diameter. In regular old space the rule is of course circumference = pi x diameter.
To get the rule in your world, we need only count the blocks and spaces between them to get the diameter. For the small circle, that would be 100 x 1 + 101 x 0.5 = 150.5. For the 1 km diameter circle that would be 100 x 1 + 101 x 0.5 = 150.5. The same number. So the rule relating a circles circumference to its diameter in this imaginary world is that there isn’t a relationship. ALL circles have a diameter of 150.5 inches. -This of course implies that your walk in setting up the second circle of blocks wasn’t nearly as long as you thought it would be. In fact you only traveled 1/2k out 150.5 inches around, and 1/2k back.
It’s a very weird world indeed, and probably collapses from internal contradictions.
Algernon said:
I think the image will be one quarter as big, because of the inverse square law. The intensity (and amount) of light (or other forms of radiation, or gravity, for that matter) falls off as the square of the distance. I think this also explains why objects appear smaller as they recede from us. We are seeing less light from them, because of the inverse square law.
I’ve never read this as an explanation for the effect, just figured it out myself, so if I’m wrong, maybe some physicist can explain why.
A correct statement is that the retinal image will be half the distance from one edge to the other, and will thus occupy one-quarter the area.
But this has nothing to do with the inverse square law. This law governs how many photons reach the retina each second, but has nothing to say about where on the retina they fall. Were it otherwise, an object’s apparent size would change with the illumination - a man would shrink to the size of a child as he stepped out of sunlight into shadow.
Xema is correct. While the retinal image is half as tall (or wide) when the object is twice as far away, the total image area is one quarter as big. Simple math of course.
As far as objects in the distance being dimmer due to fewer photons hitting the retina, it has no direct effect on the perception of size except to the extent it provides distance cues. Far away mountains are hazy and bluish, giving us distance cues. It helps us determine that these are far away mountains rather than nearby foothills. So it does indirectly assist in size perception.