Appearance of a Mysterious Reflection: What’s Going On Here? (Part II)
Wherein my book becomes a strange mirror
In Part I of this article I described the strange behavior of the reflective properties of semi-glossy book covers. If you have not read that first article you will want to do so now before continuing with this one in which I explain why some book covers invert the reflected image of a distant object (the expected behavior) and some do not.
You can find that first article here, along with all the photographic evidence of the crime against the laws of nature:
Appearance of a Mysterious Reflection: What’s Going On Here? (Part I)
As a reminder, here is one of the presented images showing the reflected image of a television screen in two nearby books situated between the television and the camera in my cell phone. The cover of one book inverts the image, the other does not:
Anyone who has studied optics will likely be able to figure out what is going on.
I confess to being a little rusty and was quick to dismiss the physical explanation initially because I did not think the effect could be large enough to cause an image inversion.
Let’s start with an explanation of why a flat shiny surface, like the puddle of water in the image at the top of this article, will invert the reflected image coming off it.
It all has to do with one simple law of optics which is that when a light ray reflects off a surface the angle of incidence (angle between the incoming ray and the surface itself) is equal to the angle of reflection (angle between the outgoing ray and the surface).
Let me draw what that looks like, roughly. The rectangle on the left is the television screen, the book cover is the flat white line in the center, and my eye (or the camera) is off to the right.
On the television are two blotches of light, one blue, the other yellow.
The blue blotch is higher up on the screen and emits a ray of light which hits the surface of the book and is reflected into my eye. The angle of the incoming light (with respect to the flat surface) is the same as the outgoing angle.
Now consider the yellow blotch of light which is positioned lower on the screen. To end up in my eye while also obeying the law of reflection (angle of incidence equals angle of reflection) it must strike the surface of the book at a smaller angle of incidence and at a position closer to the television than the reflected blue ray.
You may have to think about this a little, but that is what is going on.
Now consider what my eye sees as it looks at the reflective surface of the book. The reflected yellow blotch appears to be above the reflected blue blotch. So the positions of the blotches are inverted compared to their positions on the television screen.
The same is true of a mountain located on the far side of a still lake. It’s reflection in the water is inverted.
OK, so much for the flat surface.
What explains the book cover which shows the reflected image the right way up, which is to say non-inverted?
The explanation is that the cover is slightly concave, or bowed, with the axis of symmetry parallel to the spine of the book.
If the surface of the cover is bowed this way and the book is sitting sideways between the television and my eye the effect on those impinging rays of light looks more like what is shown in the following image (with the bowed nature of the book cover greatly exaggerated so that you can see the curvature).
I have drawn the flat surface as well, so that you can see how the cover bends up at the outer edges. In reality the cover probably has a reasonably uniform curvature on it, but I have exaggerated the drawing of the curvature towards the left and right edges of the cover so as to make the geometry more visible.
Now, for the light rays leaving the surface of the television, if they are to end up in my eye once again while still obeying the law of reflection (angle of incidence equal to angle of reflection) the positions on the surface of the book where the reflection takes place must change.
Towards the left side of the cover the curvature reduces the angle of incidence, so light striking at a large angle of incidence (blue) will need to bounce off the cover at a position closer to the television screen if it is to end up in my eye.
Likewise for light striking at a small angle of incidence (yellow) the curvature towards the right side of the cover increases the angle of incidence so the light needs to bounce off the cover closer to my eye in order to end up in it.
This effect, which applies to light emitted from all parts of the television screen, causes the apparent positions of the reflected light beams to invert so that for a slightly curved reflective book cover the image appears the right way up.
Or at least that is the only explanation I can think of to explain what is going on.
Am I correct?
What do you think? Is there another explanation which might better explain what is going on here?
And some final thoughts to leave you with on this topic of strange reflections…
If the simple uniformly-curved surface explanation is correct then a few additional questions one might want to try to answer are the following:
● How small does the radius of curvature on the book cover need to be for the image to invert? (Note: the radius of curvature is infinite on a flat cover)● Did I just get lucky and the only reason I was able to see non-inverted images was because of an accident in the value of the radius of curvature of the book cover and the distance of my eye from it… Or is this phenomenon reasonably robust for different curvatures and distances?● What objects other than semi-glossy paperback books might reflect non-inverted images?
By the way, feel free to try this “strange mirror” experiment at home. No safety googles or other hazard gear were deemed necessary during the research phase for this article.
More curious examples of writing for you to consider, courtesy of Mr. Crane:
~ // ~
