Link to a very brief article that makes some claims about the sizes of mirror images, to wit:
The size of my mirror image (as viewed by me) is half of my size.
The size of my mirror image (as viewed by me) stays the same as I move away from the mirror.
The size of someone else’s mirror image (as viewed by me, I assume?) shrinks as that person moves away from the mirror.
The size of someone else’s mirror image (as viewed by me?) grows as I move away from the mirror.
One overarching question I have is: How is the size of “a mirror image as viewed by X” measured?
I imagine that will explain 1 to me. 2 I understand already. 3 I think I understand, but I’m afraid that’s an illusion. 4 has me utterly flummoxed. Answering the overarching question might help. But it might not. Why is 4 true?
ETA: It occurs to me that the size of “a mirror image as viewed by X” might just be the size of the outline, on the mirror surface, that X would draw when outlining the image.
That explains 1. Wait, no it doesn’t. I thought it does because of the fact that the mirror surface lies exactly in the middle of the “apparent position” of my mirror self and my actual self but… that doesn’t work. The mirror image (the outline of the image I see on the mirror) should be half my actual size only if my mirror image is of size zero… or something… or I don’t know!)
Your statement, or the article’s, are somewhat confused.
1.) In a perfectly flat mirror, the image of yourself appears to be the size of a person twice as far away as the mirror.
2.) But, regardless of how far away the mirror is, the mirror only needs to be half your height in order to reflect that image. You can prove this by drawing the way the rays bounce off the mirror.
3.) This doesn’t mean that the image is half your height.
4.) As someone else moves away from the mirror, the angle they subtend will get smaller. It will be the apparent size of a person standing a distance equal to a.) youir distance from the mirror; plus b.( their distance from the mirror.
5.) So the angle subtended by someone else’s image will get smaller as you move away from the mirror. I have no idea how point four, as stated by you, can be true.
It is true. Imagine you are between me and the mirror. As I move away from the mirror (and you) the mirror appears smaller and the proportion of the mirror taken up by your image grows. Eventually I will reach a point where your image occupies the whole mirror.
The size of my mirror image (as viewed by me) is half of my size.
…because the cross-section of a cone parallel to the base of the cone and exactly halfway between its base and its tip is going to be exactly half the size of its base.
The size of my mirror image (as viewed by me) stays the same as I move away from the mirror.
…because I don’t change size as I move away from the mirror.
The size of someone else’s mirror image (as viewed by me, I assume?) shrinks as that person moves away from the mirror.
…for the same reason objects appear to shrink as I move farther away from them–they take up a smaller and smaller percentage of the space available in the plane defined by points equidistant from them to my eye.
4. The size of someone else’s mirror image (as viewed by me?) grows as I move away from the mirror.
…because as I move away from the mirror, the ratio between the percentage of space their mirror-self takes up of the points equidistant from my eye, and on the other hand the percentage of space the mirror itself takes up of the points equidistant from my eye, grows larger.
I do not understand your point. As I move away from the mirror, or as the other person moves away from the mirror, it is as if we are simply moving farther apart, and the anguilar subtense must decrease. That’s quite independent of how it appears relative to the mirror.
But the mirror image (as opposed to the ocular image) grows. The outline you would make on the mirror of the object as you move away from the mirror will grow on that mirror. The object’s image will take up more and more of the mirror as you move away from the mirror.
Go stand close to a mirror, and put one finger on the mirror where you see the top of your head. Put a finger from your other hand on the mirror where you see the tip of your chin. That distance will be half the true distance between the top of your head and you chin. Move back from the mirror, keeping your fingers in place. They will stay where the image of your chin and top of your head are (you may have to move your head up or down, or to the side a little to stay aligned).
I wouldn’t really call it “the size of the image”, but that’s what they are referring to.
Peculiar use of “mirror image” you’ve got there. I’ve never heard it before.
Imagimne that, instead of a mirror, you had a sheet of glass, with an identical you an equal distance away from it. If you were to place bars on the top and the bottom of the mirror where you saw your own top and bottom, those bars would remain the same distance apart, regardless of how far from the mirror you are. You can show this by drawing the rays and using similar triangles.
If you imagine that you are on one side and someone else is on the other (analogous to the case where you’re lookinbg at them in the mirror), then as either you or he get farther away, he subtends a smaller angle, and I’d contend that the csiz of the mirror image is smaller, by the way I understand the term.
But, even considering it as you seem to be using the term, as he stays the same distance away and I step farther back, he is not only subtending a smaller angle at my eye, the limits of the points at which the mirror defines the top and bottom of that subtense are also decreasing. . Try drawing it out.
So I don’t see how the “mirror image” increases by either definition as I move away, but he stands still. The only way he can require more of the mirror in order not to be cut off is if either he or I approach the mirror.
Mkay. Well, I did define it in the OP so its peculiarity seems irrelevant doedn’t?
Or do you think the article means something other than my own concept when it refers to a “mirror image”?
His image on your retina subtends a smaller angle. But his the bars you would place on the plane of the “mirror” in your example would spread further apart. In that sense, he subtends a larger angle on that plane.
Wait. Are you denying that the image of him that I see in the mirror will take up more and more of the mirror as I move away from the mirror?
If I draw an isosceles triangle ABC where AC is the same length as BC, and then draw another isosceles triangle DBC where DB is the same length as DC, and where a ray from D perpindicular to BC intersects A, and I then draw a line parallel to BC which intersects DB, DC, AC and BC, at points E, F, G and H respectively, then EH > FG.
Line segment BC is the apparent object in the mirror. The line EH is the mirror itself. A is the eye at first. D is the eye after it’s moved away from the mirror. Segment FG is the mirror image at first. Segment EH is the mirror image after moving away.
Please respond within 8 minutes as I have to leave after that.
A mirror image of X for Y is a plane section of a mirror which is reflecting light which Y naturally interprets as constituting an image of X.
In other words, a mirror image is a type of image, just as a photograph is a type of image, and in both cases the image is to be defined by a plane section of a surface which the image occupies.
I don’t think so. If a mirror is half my height, and I stand close to it, say an inch or so away, it will not reflect all of me. If my face is close to it, I will not be able to see a reflection of my feet at all.
(Otherwise, what you say seems correct, and makes a lot more sense put your way than than what is in the blog post.)
It sure behaves like a thing, and one of the first physics labs I teach most semesters is measuring it. The trick is to match the image to something else that’s the same size, and then measure that thing (this method can also be used to find the location of the image).
I suggest if you have a full length mirror: stand close put a soap mark at the top of your head and the tip of your chin (as mentioned above) then back away.
