This is one of my favorite demos. I like it because anyone can do it at home and people usually find the results surprising. Here is the situation:
How much of yourself could you see in a small mirror? What if you move farther away from the mirror, can you see more of yourself? If you want to do the demo yourself, now would be a good time. All you need is a small mirror (I used a platter from a 10 GB hard drive - they make awesome mirrors). It will help out a lot if you can mount the mirror on a surface that is very near to vertical.
Here is my version of this demo (in case you can't find a mirror).
To answer this, I will first answer a bonus question. How big of a mirror do you need to see your whole face. (I am going with the face because that is similar to the above video demo) The only things you really need to know are:
- In order to see the top of your head, light has to reflect off your head and to your eye
- Light reflects off the mirror at the same angle it hits it
I am not going to show where the light comes from, just to make it look less-crazy. But here is a diagram for a person looking at a small mirror.
I didn't really give away the question to "how big of a mirror do you need?". Anyway, what if you move farther away from the mirror? Here is another diagram.
The person can still see all of his/her face with that size mirror. It doesn't matter how far away the view is from the mirror.
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OK, when I read the first part of this, I was thinking you were talking about how far around from the front can you see yourself. So, if you were a cube head on you can see only one sixth, never more, never less. If you are a circle, it's probably just under half because you can't see the absolute horizon. And so on.
Assuming a planar mirror you should be able to see twice the angular width of the mirror at the distance of your eye. So, a mirror half the dimensions of your head would allow you to see your entire head if it was perfectly flat. I think there are two ranges where this wouldn't be the case, extremely close up, features would start masking as the rays become tangential. Slightly further back, depending on the shape of the object, you could see something larger then twice the size of the mirror. For instance, (if I have my geometry right) you could see an infinitely large cone if the mirror was larger then the conic section the same distance (but opposite direction) from the imaging point as the mirror. I know things are slightly different when you factor in the finite secondary aperture of the pupil, or the synthetic aperture formed by the distance between the pupils but I don't want to try and figure out how to calculate how much of a difference it makes. Any chance you could tell us how much more you can see over having a single viewpoint? I think its the same as the angular distance between your pupils that you would see in the mirror, but I'm unsure.
I just re-read that first sentence I wrote. Ick. What I meant to say was that with a planar mirror, the angle stays the same as the distance doubles, so you can see something twice as large.
I give similar problems to my intro physics students, and many students get it wrong (thinking they can see more of themselves as they move backwards). Before thinking about it, I also had a similar "gut" reaction.
The ray diagram makes the answer clear, but I'm curious as to why many folks have backwards intuition on this one.
Here's my favorite theory: Most folks look at themselves in the bathroom mirror. In a typical bathroom mirror - where your lower body is obscured by the sink, which juts a foot or two out from the mirror - you CAN see more of yourself by moving backwards. Maybe people are accustomed to this behavior in the bathroom, and assume it's a property of the mirror itself rather than the a property of the geometry of the bathroom mirror/sink combination?
A person can see more and more as they move away from a mirror. Why? Because they can move farther from side to side. Where did it say they had to be stationary? You ought to mention that or this doesn't make much sense.
Also, in the mirror, if you stay at the same spot, you can see an object twice the size if it is twice as far away. Most people see various sized objects at varying distances while staying at a fixed distance from the mirror they are using.