# Why don't spectacles form these weird images?

It's an established fact that:

1. Convex lenses produce inverted images of objects beyond the focus, on the other side of the lens.

2. Any object placed at a finite distance from a concave lens appears to be somewhere between the focus and the optical centre when viewed from the other side.

Both these facts come from the lens formula, $$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$$

But I wonder:

• why a person wearing convex lenses doesn't see inverted images of objects beyond the focus, and

• why a person wearing concave lenses doesn't feel that the furthest objects are at a distance of $f$ from their eyes.

Hope my question is clear.

## An example

Consider a person wearing spectacles with concave lenses of power $-1$ $\mathbf{D}$. The focal length would be $-1$ $\mathrm{m}$. If an object is at an object-distance $u=-3$ $\mathrm{m}$, simple calculations show that the image would be at an image-distance $v=-0.75$ $\mathrm{m}$ away.

But obviously the object doesn't appear to be so close to the person wearing the spectacles.

More confusingly still, the image would be magnified by a factor of $0.25$, and hence would appear to be rather small.

But again, as anyone wearing concave lenses would tell you, that isn't what they see!

Why is this so?

Objects through spectacles appear both at a different distance and magnified. It turns out these effects cancel out, and the object subtends the same solid angle in your vision as without the spectacles. Your eye cannot directly perceive distance- your brain estimates it in several ways.

One way is by comparing the angular size in your vision: objects thought to be the same physical size with larger angular size appear closer. So since the spectacles do not change the angular size of objects, they appear to be at the same distance.

Using your numbers, the image is $\frac{1}{4}$ as big, but also at $\frac{1}{4}$ the distance away. Some simple geometry shows that the angular size has not changed.

Another way your brain can determine the distance of objects is by the direction your eyes are pointing when both are focused on a single point. Spectacles don't change that direction either (at least not much), and so don't mess with depth perception.

• Thanks, @Chris! Does that work even for inverted images? – Harry Weasley Nov 7 '17 at 12:49
• @HarryWeasley No. It only works in the first place because spectacles are placed very close to your eyes. If you wear a pair of reading glasses, you'll notice that nothing is inverted- you have to hold them away from your face to see an inverted image. – Chris Nov 7 '17 at 18:04
• And what if someone wearing reading glasses looks at something beyond F? Would they still 'see' the object as though it were erect, as opposed to inverted? – Harry Weasley Nov 8 '17 at 17:06
• @HarryWeasley You typically see a blur because your lens fails to make an image on your retina at that point. Even if you see an image, it is not inverted, because the reading glasses invert it, but with where the image is, the lens of your eye doesn't invert it. Since your eye normally inverts everything, this has the same effect as no lens- the image appears upright. – Chris Nov 8 '17 at 19:36
• Great, that clears up a lot of confusion! Thanks, I've accepted your answer! – Harry Weasley Nov 9 '17 at 10:46

The eye itself has its own lens that is intended to create an image at the retina. IF this lens is faulty for one reason or another, the image does not form at the retina - rather, it forms in front of or "behind" the retina. The additional lenses (glasses) shift the path of light rays by a small amount before they enter the eye, moving the location of the focal point of the glasses-eyes system closer to the retina.