(Song of the day:
Remember Me Lover from The Incident by Porcupine Tree)
You probably don't understand this as well as you think you do. And this topic isn't as mundane as you probably imagine. Let's get the usual explanations, which aren't wrong only incomplete, out of the way. With an...
Introduction
Our eyes focus light coming from objects onto the retina to form (
approximate) images on the retina. These images are communicated to the brain via electrical impulses, which sees the original objects. Objects appearing out of focus is directly correlated with the image formed on the retina being out of focus.
Focusing the light from objects onto the retina sharply requires an appropriate optical power of the eye-lens depending on the distance of the object from the eye. This is why sometimes we have to choose between focusing on one or another object in our field of view. We can
focus on objects at different distances by varying the curvature of the lens, by contracting and relaxing the ciliary muscle in the eye.
Ideally the human eye can focus on objects placed between around 25 cm away from the eye (near point) to infinitely far away (far point). The two common defects of the eye, nearsightedness (myopia) and farsightedness (hypermetropia/hyperopia), result from the far-point of the eye reducing and the near-point of the eye increasing, respectively. In other words, the eye can only focus on objects in this reduced zone of comfortable viewing. There are many reasons for this to happen, including eye-ball defects and deterioration of the ciliary body.
In terms of bending light rays, a myopic eye bends light from objects beyond it's far-point too much (cannot not bend too much), that the image forms in front of the retina. A hypermetropic eye bends light from objects in front of its (abnormal) near-point too little, that the image forms behind the retina. A picture depicting this can be found
here.
A small digression here, before we get to the non-trivial stuff. A fairly
common mistake is the suggestion that a hypermetropic eye cannot bend parallel rays of light (from infinity) sufficiently to form a image on the retina - that is a rather extreme case of hypermetropia where the near-point itself is at infinity. Many people see the issue with this and phrase it with a catch as follows:
In hypermetropia, parallel rays get focused onto a point behind the retina when there is no accommodation from the ciliary muscle ("when staring off into the distance, for example"). But they're only narrowing down the definition of the ideal eye, making (potentially completely wrong) claims they can't justify about the relaxed state of the eye, all while leaving open the possibility that a person who's hypermetropic by their definition could very well have a near point of under 25 cm.
Back to the shitty eyes not being able to bend light appropriately. If the eye cannot not converge light too much, we can help by putting a diverging corrective lens in front of it. And if the convergence isn't sufficient, we can help by putting a converging lens in front it. A sample image depicting this can be found
here. This is a common explanation you find, even in some textbooks. Although not incorrect, it isn't the best way of looking at the functioning of corrective lenses - it doesn't automatically raise questions that the other (also common) way of looking at it does.
The other way of looking at it is as follows (we'll restrict our discussion to the nearsighted eye and it's correction, although the arguments extend to farsighted eyes as well): The eye can only see clearly up to a certain far-point. Any object beyond that cannot be seen clearly. When we place a corrective lens in front of the eye, the lens forms virtual images of all objects in front of it. These virtual images act as real objects for the eye. In other words, the eye doesn't
see the world anymore, it sees the image of the world formed by the corrective lens. And if we choose the corrective lens appropriately, we can bring the entire world within the far-point of the eye, thereby letting the eye see everything clearly. Check out figure 19.12
here.
This way of looking at things is really neat - it divorces the working of corrective lenses from that of the eye.
- Eye cannot see everywhere clearly.
- Lens brings objects to where the eye can see clearly.
And it also raises some rather interesting questions.
The good stuff
The question
To fully appreciate what follows, let's see what the far-points are like for typical myopic eyes. One of the lowest powers of corrective lenses you can buy, the one I started with as a 6th grader, is -0.25. That corresponds to a focal length of -4 m. What this means is that, the virtual image of the stars formed by a lens of power -0.25 will be at a distance of 4 m in front it. And since the eye is practically right behind the lens, the lens brings the stars to within 4 m of the eye!
A digression. It's not like the person cannot see clearly beyond 4 m, but chances are if the person can see clearly beyond, say, 10 m, they wouldn't have gotten the prescription in the first place. I don't have exact statistics, but I imagine most people in the world will have their far point not at infinity but at something more modest like under 15 m. Yes, the slightest deviation from perfect eyesight (you don't have it) reduces the far point that much - This has to do with the behavior of the function 1/x, for those familiar with the lens and/or lens maker's formulae. Not to worry, you can still see far, well enough without glasses!
Where were we? Power = -0.25, stars at 4 m. It gets better/worse really fast. If the power's -1, the stars are at 1 m, and if like me your (lenses') power is -4, your far-point is a depressing 25 cm. That's supposed to be the ideal eye's near-point! Nearsightedness? More like cantseeshitedness *smh* Anyway, onto the big question that usual explanations of corrective lenses neither raise, nor resolve.
Let's take the example of a person wearing corrective lenses with power of -1. The focal length is 1 m. This lens forms a virtual image of the stars at 1 m from the lens, and squishes the entire universe from right in-front of the lens all the way to infinity within that 1 m distance! The image of an object at infinity will be at 1 m, the image of an object 1 km away will be at 0.999 m. If the object's at 2 m, the image will be at 67 cm; and if the object's at 1 m, the image will be at 50 cm. Which raises the question: Why do those wearing glasses not freak out the first time, or need time to get adjusted to this trippy shit? More formally,
Why do people seeing through glasses, not perceive the objects to be located at the corresponding images formed by the glasses?
At this point you may be questioning the neat and divorced way of looking at corrective lenses. Maybe the whole 'image formed by lens acts as the object the person sees' is alright to do calculations with, but perception doesn't really work like that. Allow me to convince you.
If you have glasses of modest power say under (over, technically) -3, remove them and hold them at a distance of about 20 cm from you (doesn't have to be exact). Now look at a distant object, say a tree, through the spectacles. You'll see a much closer and smaller image of the object through the lens. This means that you do see the image formed by the lens, when you look through it. Try moving the glasses back and forth. Now, slowly bring the glasses up to your eyes and wear them. You'll find the distant object magically go to where it's supposed to be.
If you don't wear glasses, you can do this with a friend's pair of glasses. Make sure they are nearsighted. You're not supposed to have farsighted friends. And no, you will not 'catch' their power by looking through their glasses. But if you want to do the part wear you gradually bring it close to your eyes and wear the glasses, make sure the power is under -1.5, you may feel uncomfortable otherwise. Also, be wary of eye infections. You don't know where their eyes have been. And those with power over -3, you too may have to use a friend's pair all while wearing your own :^(
You may want to pause here for a bit and ponder about the mindfuck I just unleashed on you. No? Okay.
A possible explanation
When I was first playing with the whole moving my glasses back and forth thingy, I felt that the explanation had to do with the brain. When the glasses are held at a distance, the glasses and the images it forms occupy a small part of our visual field. But as we bring the glasses close to the eyes, the glasses occupy the (almost) entire field of vision. Maybe, at this point, the brain perceives the farthest thing visible to be infinity, and somehow places the rest of the objects at their right places from experience. Maybe it's all the training my brain's had from before my days with glasses, that helps my brain perceive the right positions of the objects - by guessing. I knew the explanation was bonkers, but I settled for it. Even marveled at it. In retrospect, it was my God excuse, in some sense - don't understand something, must be how the brain works.
The brain is capable of a great many things perception wise, but this simply can't be because of that. It's hard to imagine yourself walking into an empty room with just one tiny light bulb in the center and going 'Oh, what a beautiful star!' You're not going to place it at infinity in your head just because it's all you see.
The explanation
There's a closely related question, answering which will lead to the explanation for the original question.
Why do people seeing through glasses not see funhouse mirror/lens versions of the world?
Think about it. If you look though an arbitrary piece of glass, you half expect to see distorted images. So, why does it not happen with corrective lenses? What about corrective lenses makes sure that objects don't appear fatter/thinner/funnier? (Does it perhaps have to do with the fact that our eyes themselves have spherical lenses, so using spherical corrective lenses to compliment them works? To keep this short, no.)
To answer this question consider looking through one lens with one eye (taking out the whole depth perception issue). What does it mean for the eye to see the same with or without glasses? It means that the angular positions of different objects is the same with and without the lens.
Lets say looking through the lens, you point with your two hands towards the top and bottom of a tree. Now after removing the lens, if you find that you're still pointing towards the top and bottom of the tree, then it means that the image of the tree you saw wasn't distorted vertically. If it holds for any point in your field of view (the direction of the point is the same with and without the lens), then you see an undistorted image of the world. Note that the distances needn't be the same - one eye only sees a 2D photograph and doesn't perceive depth. You can bring a tree closer while simultaneously making it smaller, and have one eye not register it while the other is closed.
And what about spherical lenses ensures this? In locating the image formed the lens, one of the rays we use is the ray passing through the optic center. And the ray passing through the optic center is undeflected. So, if you put your eyes right next to the optic center (which is what you do when you bring the lens real close to the eyes), the angular positions of different objects and their corresponding images will be the same. Check out the second image on this page with five objects in red labeled from 1 to 5 and their images images in blue. When you place one eye right behind the lens, looking at the images in blue is equivalent to looking at the objects in red, because, again the angular positions of the images are the same as that of the corresponding objects.
So it doesn't matter what the shape of the lens in our eyes is (probably not spherical), the corrective lens can be spherical.
But how does this help us answer the original question of correct depth perception? We're already kind of there. As we stated just now, depth perception is a result of binocular vision. Our brain pieces together the two different images seen by our two eyes to perceive depth. And if each eye is seeing what it would've without the glasses, the depth perception will work itself out! But but...
But then why did it not work itself out when the glasses were held at a distance?
- Your eyes weren't right behind the lens. And more importantly...
- Both of your eyes were seeing the images formed by both the lenses in the glasses.
So with both your eyes you could see that the image formed by the left lens was closer than the actual tree. And the same with the image formed by the right lens. But when you wear the glasses, your two eyes see two different images both within 1 m, the combination of which tells your brain that the star you see is far away. That right there is VR. The illusion of an infinite world, in a one meter radius. Yes, I am that cheesy. In fact, let's cheese it up more. The gift of corrective spherical lenses is something we neither appreciate nor deserve.
And that's how corrective lenses work. The end.
PS: Luckily for us, the brain doesn't use the information of how contracted or relaxed the ciliary muscle is in it's perception of depth. Or, at the very least, doesn't use it enough to get us all confused.