Besides the obvious cases where I'm behind a "one-way" mirror or have goggles/glasses on: is there one where I can see someone's eyes, and they can't see mine?
[Physics] If I can see someone’s eyes, can they see mine
everyday-lifevisible-lightvision
Related Solutions
You can't see clearly underwater for a couple of reasons. One is the thickness of your lens, but the main one is the index of refraction of your cornea.
For reference, here's the Wikipedia picture of a human eye.
According to Wikipedia, two-thirds of the refractive power of your eye is in your cornea, and the cornea's refractive index is about 1.376. The refractive index of water (according to Google) is 1.33. In water, your cornea bends light as much as a lens in air whose refractive index is
$$\frac{1.376-1.33}{1.33} + 1 = 1.034$$
That means you're losing about 90% of your cornea's refractive power, or 60% of your total refractive power, when you enter the water.
The question becomes whether your lens can compensate for that.
I didn't find a direct quote on how much you can change the focal distance of your lens, but we can estimate that your cornea is doing essentially nothing, and ask whether your lens ought to be able to do all the focusing itself.
For a spherical lens with index of refraction $n$ sitting in a medium with index of refraction $n_0$, the effective focal length is
$$f = \frac{nD}{4(n-n_0)}$$
The refractive index of your vitreous humor is about 1.33 (like water), and the refractive index of your lens, according to Wikipedia, varies between 1.386 and 1.406. Let's take 1.40 as an average. Then, plugging in the numbers, the effective focal distance of a spherical eye lens would be five times its diameter.
The Wikipedia picture of a human eye makes this look reasonable - a spherical lens might be able to do all the focusing a human eye needs, even without the cornea.
The problem is that your eye's lens isn't spherical. From the same Wikipedia article
In many aquatic vertebrates, the lens is considerably thicker, almost spherical, to increase the refraction of light. This difference compensates for the smaller angle of refraction between the eye's cornea and the watery medium, as they have similar refractive indices. [2] Even among terrestrial animals, however, the lens of primates such as humans is unusually flat.[3]
So, the reason you can't see well underwater is that your eye lens is too flat.
If you wear goggles, the light is refracted much more as it enters the cornea - the same amount as normal. If you want to wear some sort of corrective lenses directly on your eye like contact lenses, they should have a refractive index as low as possible.
Googling for "underwater contact lens", I found an article about contact lenses made with a layer of air, allowing divers to see sharply underwater.
Dear Rootosaurus, when you're looking at an image of a chair behind you in a flat mirror, then you're observing the so-called virtual image of the chair. If the mirror's surface is located in the $x=0$ plane and the coordinate of the real chair is at $(x,y,z)$, then the virtual image of the chair is at $(-x,y,z)$.
However, the light rays coming from the real chair that are reflected by the mirror and that ultimately end up in your eyes have exactly the same directions as the light rays of a hypothetical chair that would be actually located at the point $(-x,y,z)$ behind the mirror. So geometrically, you can't really distinguish a reflection of a chair that seems to be located at $(-x,y,z)$ from a real chair that is located at $(-x,y,z)$ and that you're observing through a window without a mirror. The geometry of the light rays is identical. That's why the concept of images is so useful.
In particular, myopia means that one has some trouble to observe distant objects. Distant objects - imagine a distant point - have the property that they emit light rays that are nearly parallel. The further an object is, the more parallel its light rays look when they arrive to your eye. However, the lenses in your eyes have to convert these parallel mirrors into converging mirrors - so that all the light rays coming from the distant star end up at one point of the retina.
Myopic eyes are good in converting "steeply divergent" light rays from nearby objects to converging ones, but they're doing too much of a good thing. When you get too parallel light rays, myopic eyes make them converge too much, too early - the intersection will be inside the liquid in your eye. Hyperopia is the opposite disorder in which eyes make the light converge less than is needed. But what's relevant for your question is that the virtual image of the chair at $(-x,y,z)$ "emits" the same excessively parallel rays as a real chair at the same point, so a myopic eye will have the same trouble seeing it. After all, it shouldn't be paradoxical: the total distance that the light rays have traveled includes the distance of you from the mirror as well as the distance of the mirror from the real chair - because the colorful photons ultimately came from the chair and they were just reflected, not created, at the plane of the mirror.
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Best Answer
Fermat's principle says that the direction of travel for any light ray can be reversed. Therefore there is always a line of sight between a pair of eyes in both ways.
If one person is in the dark, then only one person can see the eyes of the other. So there needs to be enough light reflected from both person's eyes for this to work.