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Yes, I know all about perspective (I'm an artist). I even have some basic knowledge of descriptive geometry. I know how it works. My question is more about why it works.

I have a sneaking suspicion it has something to do with the curvature of space. However, I'm not certain whether the two concepts are related.

Another thing that's been bothering me for a while (this may or may not be related to the question: please enlighten me) is the relation of the vanishing point to the horizon:

Strictly speaking, the geometrical horizon is where the vanishing point is. The actual horizon is a bit lower, but the difference is negligible.

Thus, if you sketch a perfectly straight street lined with buildings of equal size stretching all the way to the horizon, you shouldn't even be able to see anything beyond 2.9 miles (the distance to the horizon): all parallel lines will merge - and vanish. And yet, if the buildings are large enough, they will be visible beyond that point.

enter image description here

Which would suggest that smaller objects become a point source before they reach the true vanishing point, and that the true vanishing point is a lot further than 2.9 miles (which is why we can see the actual shape of the moon, as well as some of the planets: they're not point sources; they're actual disks). I would even go so far as to posit that the reason we see stars as point sources is they aren't large enough; and the Andromeda Galaxy, which appears to the naked eye as a cloud, is not a point source; which leads me to believe that the absolute vanishing point is at an infinite distance from the observer.

With all of the above in mind, my question still is - WHY do objects appear smaller in the distance?

P.S. "That's how perspective works" is not a good answer in this case unless physics has nothing more to contribute at this juncture.

P.P.S. Yes, about those angles (the light hitting the eye, etc). Why should the angle get narrower as the distance increases? Well, for one thing, that's how our brain interprets the signal it receives from the eye. I daresay if it were the other way around (i.e. if the angle got broader as the distance increased), our brain would find a way to adjust, and we'd all be saying "It's bigger because it's further away, that's how it works").

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marked as duplicate by Chris White, ACuriousMind, Kyle Kanos, John Rennie, gigacyan Mar 11 at 13:05

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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This is not about physics. However, the answer simple: The available "viewing surface" at distance $r$ grows as $r^2$ (in flat space!), while the objects stay the same size. So the farther they are, the smaller their ratio to the available surface, and so they appear smaller, because they make up a smaller solid angle. – ACuriousMind Mar 9 at 21:41
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Why is "that's how perspective works" not a good answer? You say that you know "all about perspective." So what is it about perspective that dissatisfies you? – james large Mar 9 at 21:42
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The surface grows because the surface of a sphere of radius $r$ is $4\pi r^2$, and this isn't physics because, well, it is not about any natural phenomenon, it is about an abstract geometrical fact. – ACuriousMind Mar 9 at 22:17
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But this has nothing to do with how the brain works. A camera would see the same thing. Concerning the curvature of space, this doesn't influence the vanishing point on such small scales as Earth, and even the Solar system, the Galaxy, and the local Universe. However, on very large scales (billions of lightyears), the curvature of space (if there were any) actually can affect the geometry of the Universe, and hence the vanishing point. – pela Mar 9 at 22:36
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@Ricky, The laws of perspective don't "work by faith." Perspective is a simple application of geometry---pure math. The diagrams in cobaltduck's answer should be pretty close to self-explanatory. – james large Mar 9 at 22:49

It's all about the angles made by the object when light from it enters the eye.

enter image description here

Consider this crude doodle of an eye looking at two identically sized trees. The light entering the eye from the nearer tree makes a broader angle at the eye, and the further tree makes a sharper angle. The brain interprets this as the further tree seeming to be smaller.

Try this- Go outside during a full moon. Take a quarter (or an equivalent sized coin if you are not in the U.S.) and hold it out at arm's length. Move the quarter over the moon. Does the quarter just about cover the moon? You can also use smaller coins and hold them closer.

enter image description here

Above is another crude doodle, and here is a photo. The coin and moon seem to be the same size because the angles made by them at the eye are equal.

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If the answer of "Because isosceles triangles with taller heights have more acute apex angles and acute angles generate smaller images" goes to "But why does the angle become more acute?", then this question belongs on the Math stack exchange. Physics has done its work here. – Ryan Cavanaugh Mar 10 at 17:13
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This skips the fact that lenses and how we see light is key to this problem. Without a pinhole/lens, the idea that the arc is how big we see things doesn't happen: an eye that was just a bunch of photon detectors with no lens/hole wouldn't behave that way. – Yakk Mar 10 at 18:44
    
@Yakk " an eye that was just a bunch of photon detectors with no lens/hole wouldn't behave that way. " -- Think about how such an "eye" would see the sun. – Jim Balter Mar 10 at 21:04
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@JimBalter It would either see "oh, lots of light" or "nope, no light" if there was something occluding the sun. Now imagine placing various diffraction gratings over different sensors -- this creature now "sees" a Fourier transform of what we see (and again, the image is different). It is our physical apparatus that generates the image. – Yakk Mar 10 at 21:33
    
what this and other answers is missing that a picture on the retina works the same. This is what OP asks for – joojaa Mar 11 at 5:07

It is because light travels in more-or-less straight rays.

Let's assume for simplicity that your eye is like a pinhole camera; it has a pinhole in front and a screen at the back. Then an image forms by the light rays that pass through the pinhole.

(from https://commons.wikimedia.org/wiki/File:Pinhole-camera.png)

Consider two points on an object, such as the top and bottom of a tree. If you move an object further, the two points remain at the same distance from one another but are further from the pinhole, and hence the angle that it makes with the pinhole is smaller. But the screen is still the same distance from the pinhole, and hence the image is smaller. (See also this related BBC article.)

As for your question about the vanishing point, it's not clear at all what you are asking. In perspective drawing we have the projection of the scene onto a flat screen through an origin (the eye). Under the projection, consider all the lines that do not pass through the origin. Each of them maps to a line, and any two of them that intersect map to intersecting lines. If we stipulate that any two of them that are parallel actually intersect at a point at infinity, then it is even nicer because then any two lines intersect at a unique point. Also, parallel lines on a horizontal surface would all intersect at the same point at infinity, which after projection maps to a point on the screen, through which the images of all those parallel lines pass through.

Points at infinity do not exist in the Euclidean model of the world, just as parallel lines do not intersect. It is just that if we conceptually add them to the Euclidean model we obtain the projective space that has nice properties, including a meaningful notion of the horizon as the image of some line of points at infinity. This enables us to draw perspective drawings, which is basically to draw the image on the screen given what we know about the lines in the scene.

The surface of the Earth is not flat but slightly curved, and so we cannot see the whole surface of the Earth, and we see a slightly curved horizon. This has nothing to do with the line at infinity in perspective drawings, since that line is still there, just not relevant to the horizon that we see between the earth's surface and the sky. If this is where you got your "2.9 miles" from, then it is simply how far on the surface of the earth you can see, which is of course unrelated to the fact that you can see all the way to the stars.

enter image description here

(from https://en.wikipedia.org/wiki/Horizon#/media/File:Horizons.svg)

In the above diagram, the astronomical horizon corresponds to the line at infinity if you were standing on a really flat surface. The true horizon is what you perceive to be the divide between the sky and ocean since the Earth isn't flat. The visible horizon is what you perceive to be the divide between the sky and land since there are usually lots of things like trees on the land blocking the view of the actual ground surface.

In short, perspective drawings don't work for things that are too far away on the Earth's surface because they assume a perfectly flat ground surface.

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@Ricky: If you want to justify rigorously that the angle decreases when the image is brought further away, then you have no choice but to set up a mathematical model, such as using Euclidean geometry, and using the axioms we can indeed prove the fact. – user21820 Mar 10 at 5:06

An angle is just a measure of the relationship between the arc length of a segment of a circle and the radius of said circle. An object (let's say a pencil) of length $30~\text{cm}$ will always be that length, but the distance the pencil is from your eye can change.

The definition of an angle is the arc length, $s$, over the radius, $r$

Even though the pencil doesn't change its size, the radius increases as you move it away from your eye, thus the denominator increases while the numerator stays the same, so the fraction results in a smaller value (i.e. a smaller angle). The brain perceives things for the angle at which light comes from (as you know and others have pointed out). This is due to the fact that the brain estimates sizes by the percentage of the field of view that the object takes up. That is, how big its angle is compared to the entire angle we can see (approx. $180º$, since we can see almost everyhing in front of us).

There is no limit, as you say, to the actual reach of eyesight, but there is a limit to the angle at which light enters to our eyes. A dust particle can be literally 1 cm away from your eye, but since the its length is so small, the angle stays small, so light coming from the top of the particle and the bottom is indistiguishable from one another. On the other hand, the Moon, despite being thousands of kilometers away, is also hundreds of kilometres wide.

These two examples can be assigned values:

Dust particle of length is $2.5~\mu\text{m} = 2.5 \cdot 10^{-6}~\text{m}$ (looked this size up in Google). So, if the particle is $1~\text{cm} = 10^{-2}~\text{m}$ away from your eye, the angle at which paths of light come from above and below it form an angle of $\frac{s}{r} = \frac{2.5 \cdot 10^{-6}}{10^{-2}} = 2.5 \cdot 10^{-4}~\text{rad}$. This is, to put it simply, less than the resolution of our eyes.

Whereas, for the Moon example:

Moon distance to Earth $384 000~\text{km}$, we'll say $3.8 \cdot 10^{8}~\text{m}$ to simplify. Moon diameter $3,474~\text{km}$, so $3.5 \cdot 10^{6}~\text{m}$ again to simplify. The angle here is about $0.01~\text{rad}$, which is about $0.5~\text{degrees}$. Since our whole perspective is almost $180º$, this is about $0.2\%$ of our field of view, small, but considerable nontheless.

Hope this helps!

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I think this is the right answer: describe objects' sizes in terms of what percentage of our field of vision they consume. – Snowman Mar 10 at 5:02
    
Angular diameter of the Moon is 0,5 degrees. You missed one zero while converting 384.000 km to m, the correct distance to the Moon is $3.8 \cdot 10^{8}~\text{m}$ – Iván Pérez Mar 10 at 9:57
    
Whoops, my bad. Though the numbers may be wrong, the concept is right, which is what matters. I'll edit right away! Thanks for the feedback! – Yottaphy Mar 10 at 15:27

TLDR

Look up Euclidian and Minkowski space.

Perspective and the shape of space

Geometrical perspective works because we happen to live in a very nearly Euclidian 3-dimensional space. In such a space, by definition, the familiar rules of 3D geometry apply and perspective follows from the rules of geometry.

We can imagine and describe other spaces in which the 'familiar rules of geometry' do not apply and in these spaces our rules of perspective would not work.

At present the best mathematical description of the space we live in is that of Minkowski space, according to Special Relativity. Minkowski space is warped from Euclidian space by density of mass and energy. But, around here, Minkowski space is very nearly Euclidian. Gravitational Lensing is a characteristic of Special Relativity where normal perspective does not apply.

Vanishing points in your drawing

I think it easiest to think of a vanishing point as a point on your drawing surface where the 2D projections of a set of parallel (3D) lines in the scene converge. Parallel lines in Euclidian 2D and 3D space do not converge, but the projected lines on your drawing plane will radiate from a point in that plane.

Consequently the position of any vanishing point in your drawing is entirely dependent on: your point of view, the direction of the centre of your scene, the field of view and the 'up' direction in your picture.

For example if you are looking down a bit, as at a lily-pond, then the horizon vanishing point in your picture will be above the centre of your picture. If you are looking up a bit then the opposite is true.

Points at infinity

When we project a pair of parallel lines to an imaginary sphere at infinity the lines will intersect with the sphere at distinct points. If project those points to the drawing plane from your point of view, the projected points will have effectively zero separation. This is because while something divided by infinity is undefined, we can take it as indistinguishable from zero, here.

Optical effects

The rules of perspective apply true lines from the scene, through the drawing plane, to your point of view. But light only travels in a straight line in a uniform medium. That is why lenses work: the speed of light is slower in glass than in air. That is why we can get heat distortion and mirages in air. We could argue that space appears to be non-Euclidian under these conditions but this is due to the light ray not being straight.

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Yes, about those angles (the light hitting the eye, etc). Why should the angle get narrower as the distance increases? Well, for one thing, that's how our brain interprets the signal it receives from the eye. I daresay if it were the other way around (i.e. if the angle got broader as the distance increased), our brain would find a way to adjust, and we'd all be saying "It's bigger because it's further away, that's how it works").

I think you've got it a little backwards. The fact that the angle gets narrower when an object gets farther away is a consequence of the GEOMETRY we live in. We live in a geometrical space where angles get smaller when things move further apart. This is both something that can be proven using the postulates of Euclidean geometry, AND an experimental fact.

Now, why does our brain interpret small angles to mean far away? Because we've evolved to have our senses be USEFUL. The rodents who interpreted big lions to be far away are all dead. There's a strong evolutionary pressure to have our brain interpret our senses to reflect reality.

Like you said, if we lived in a geometrical space where angles got larger when things were further away, our brains would have evolved to interpret distance appropriately. But we DON'T live in that kind of space; we live in Euclidean space. You can tell because in our space, parallel lines don't intersect and the angles of triangles add up to 180 degrees. One of the mathematical consequences of this is that things that are farther away look smaller.

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I think what it boils down to is: Because that's how the math works out.

That is how Perspective projection works. And it does so because you can approximate a camera (or your eye) as a single point.

I feel like the last "why" to accept one of the other answers is this: I does work that way because we imagine the focal point of the projection to be behind the plane of projection.

So maybe it will clear things up for you if we did the opposite.

Imagine your retina as a huge canvas. You look at things through a window of considerable smaller size (the plane of projection - or the lens in the following mock-up).

enter image description here

Then indeed, the closer (orange) object would appear smaller then the farther (purple) object. But simply put: This is not how an eye works! In fact the consequences of a system like that are rather hard to imagine.

Fun and obvious fact: In reality this focal point really is in front of the plane of projection. With your eyes as well as cameras. However it is inside the whole apparature and thus no light is emitted from within there and we can ignore that space for our purposes. Thus an approximation as a point is sufficient.

To a degree the following is true: Objects inside your eye appear larger the further from the retina they are.

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First one that covers the fact that this effect is because of "quirks" (useful ones!) in how we see light -- in the construction of the eye. – Yakk Mar 10 at 18:46
    
The comment above is nonsense that does not pertain to the answer, which correctly says it is how the math works out. Perspective projection is not dependent on "quirks" of eye construction, but rather is how to project a 3D world onto a 2D patch. Eyes, cameras, and perspective drawing all work this way of necessity. The drawing of an "eye" here makes no sense ... the "lens" does no lensing, and there is a point source of light which is casting shadows on the retina. Light reflected off of surfaces would land all over the retina and vision would not be possible. – Jim Balter Mar 11 at 6:19
    
@JimBalter That is the entire point of this deliberately wrong construct: In order to see the desired effect of larger objects at a distance, you have to build up your projection (or at least the math) in a fundamentally different way. However, since this is not how any of this works, the argument arises that the math behind perspective projection is correct because that is an adequate approximation of the real process. This is what I was aiming at: You can construct it in a different way but it would not represent reality in a meaningful way. – Zuzel Mar 11 at 9:31
    
Yes, but my point was that the other commenter is just wrong about the construction of the eye being the answer to the OP's questions, and misconstrues your answer as supporting his mistaken notion (which he has repeated elsewhere on this page, including in a wrongheaded answer). – Jim Balter Mar 11 at 9:44

This is because of how our eyes are constructed.

Our eyes have a lens and a hole and a retina. These take the light heading towards us, and project an "image" in one set of directions of it on our retina. The retina then divides things up based off angles and sends the information to our brain. Our brain then interprets it as what we see.

A decent approximation of how the hole in our eye plus the lens work is to treat the pupil as a point that only allows light to go "strait through it" and project on the retina behind it.

Take this diagram:

A                 B                #
A                 B             *  #
A                 B                #

The * is the pupil. The #s are the retina. The As and Bs are two objects that are nearer/farther away.

Use a strait edge (pencil, or piece of paper) draw a line from the top of the Bs, through the center of the pupil *, onto the retina #. Do the same with the bottom. This is the image that B projects onto our retina.

Compare that to the height of the As are projected onto the retina. The closer one projects a larger image.

Now, the retina/pupil can also move. And it has a "high resolution" center of vision. Keeping the pupil stationary, imagine the retina rotates around it until the center of vision points at the top/bottom of each of the As and Bs.

A has a wider angle than the B does.

These two effects -- projects over a larger area on the retina, and requires your eye to rotate more to go from one side to the other -- is what "looks bigger" means.

It happens because the optics of a small hole with lens basically discard light that does not travel "strait" through the hole.

Light is like a swimming pool full of children, with waves all over the place. Our eye is a little device floating in the corner. It is a box with a hole in one side. Inside the box is a bunch of floats. Using the position of the floats caused by the waves moving, it constructs an image of where everyone in the pool is and what they are doing, so long as there is a strait line from the device to the thing making the waves. It is a ridiculously amazing device, but the image we see is only an interpretation (a useful one!) of what the waves (light) is doing.


How a pinhole+lens cause this to happen (only allow light at goes "strait through" to pass) is a problem of optics and/or quantum mechanics, depending on how deep you want to go, and beyond the scope of your problem.


The next question is the horizon. The horizon we see is caused by two things -- things getting farther away (and hence smaller), and the Earth getting in the way.

On an infinitely flat world, what you'd see pretty close to the artists vanishing point. All things "on the earth" would get shorter and shorter as they got further away, in proportion to how far away they are. Parallel lines also get closer and closer together. They would never reach 0 height or width -- instead, it would describe a slightly more complex curve, where parallel lines 2x as far away are 1/2 the distance between each other. If they do go forever, however, strait lines are a decent approximation. But the space between "distance milestones" would also shrink.

On the Earth, however, we are typically located just shy of 2 meters above the surface. And the surface curves away.

Seeing something 0 meters tall (ie, the actual Earth's surface) is about 5 km away. At that point, the Earth itself prevents you from seeing the Earth itself.

Taller things will indeed be visible further away. An infinitely tall thing sticking right out of the Earth would only be completely under the horizon if it was on the exact opposite side of the Earth.

For reasonably short things (like, buildings or mountains), you can see things about 5 + $3.6 \sqrt{h}$ km away, where $h$ is in meters above the ground (assuming you are human-height). Source.

More generally, $3.6 \sqrt{h_0}$ + $3.6 \sqrt{h_1}$, where $h_0$ and $h_1$ are the heights of the two objects, is how far away you can see something over the horizon.


None of these effects require "curvature of space" to work.

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What does it mean to make a drawing which is accurate with respect to perspective, such as the picture in your question? The 2-d sheet of paper is very clearly not the same thing as the 3-d scene, so we can't just say "they're the same."

If I may borrow the image from cobalt duck:

enter image description here

Perspective drawing is done in a way such that, if you hold the sheet of paper up in front of your eye, the lines and edges drawn in the drawing approach your eye in the same direction as the actual edges in the 3d scene.

Consider, in cobalt ducks' image, the 25 cent piece to be our paper that we are drawing on, and the moon to the right to be the actual object. In the real world case, photons from the moon emit in all directions, some of which go towards your eye. In order to make a perpsective drawing of the moon, one needs to make the paper emit photons along the same lines of sight. (I'm intentionally ignoring the fact that we tend to draw edges in black pen... pretend they were glowing neon edges instead!)

So if I want to represent a point in 3d space on my 2d sheet of paper, where should I put it? I should draw a line between the point in 3d space and the eye. Wherever that point intersects the paper is the correct point of representation on the paper.

This is a mapping process, and a key takeaway is that it only depends on the direction of the vector from the point in 3d space to the eye. The actual distance wont matter (in reality, there's some effects that make objects in a distance appear dim, but that's completely separate from the question you are asking).

Let's now consider an object with some extent. What does the moon look like when projected onto the paper? Let's consider just the topmost point and the bottommost point of the moon, which correspond to the two red lines in the diagram. These are the directions the photons are traveling to reach the eye. To properly represent these two points on the paper, we simply look at where those two vectors intersect the paper. Thanks to the clever choices of scales in cobalt duck's image, we see that the top and bottom of the 25 cent piece line up right along those red lines.

Now lets consider lengths of lines, because that's where your question of "smaller" comes from. We can draw a line between the top and bottom of the moon, forming a triangle with the 2 rays going to the eye. Likewise, we can draw a line from the top to bottom of the 25 cent piece, forming a triangle with the 2 rays going to the eye. Because the 25 cent piece (the representation of the moon on the paper) shares the edges of the triangle with the larger triangle from the moon, we can use Side-Angle-Side from geometry to show that the two triangles must be similar. These similar triangles are the "why" you seek.

Consider the ratio between the distance to the paper and the distance to the object. The two lines we drew from top to bottom must also share that ratio. If, for purposes of drawing in perspective, we presume the paper is at some fixed distance, this leads to the final relationship: if an object is further away, it should be drawn smaller because the ratio of distances between the eye-to-paper and eye-to-object is greater, which means the ratio between the size of the object and the size on the paper must also be greater by the same ratio. If we were to consider moving the object forward and backwards, we would see that, when it is closer, its perspective representation must be larger to maintain these ratios, and when it is further its perspective representation must be smaller.

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Consider a species with two eyes, set in a line parallel to the ground. They would judge distance by the perceived angle used by eyes to focus on an object. In the extreme case, you become cross-eyed looking at things very close to your face.

They can judge the angle between their eyes as an estimate of distance. Predators that are perceived as being at the maximum angle are judged to be distant, and safe. Predators rapidly decreasing the angle needed to see them, should be judged to be approaching rapidly, encouraging our specimen to scamper away.

Because you wish to create an illusion of distance, you create a smoothly changing scaled drawing where distant objects are small, crowded together and having a similar angle that mimics the sensation of a distant object. While the viewer doesn't have to go cross eyed to see a close object, they are reminded of the feeling of seeing a close object because they have to move their eye around to take in all of the details of the 'close' object.

Now, a one eyed man has to judge distance only by size. Which why he is amazed when a lion grows to be so big that his last thought is, "My, look how large those teeth..."

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The first part of this answer about the angle between two eyes is totally irrelevant to this question. Your final point that a one-eyed man can still judge distance from size only drives that home. – Matt Mar 10 at 10:48

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