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Math Professor Invents Non-Reversing Mirror

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Re: Math Professor Invents Non-Reversing Mirror

#51

Post by joshdick »

andrewcooke wrote: post #46
joshdick wrote: post #42

Earlier quoted context omitted.

No, that is not what Prof. Hicks made. His mirror is a single curved piece. Source: I am a former undergrad research assistant of his.

huh. thanks. so that's the second type listed at wikipedia? or am i confused? perhaps you or he could fix / clarify the wikipedia article? i guess i could but it feels odd to do it third-hand...

I wish I could give you a link to one of Prof. Hicks's published papers on the topic, but I'm having trouble finding one that isn't behind a paywall. This seems like a nice introduction: http://www.math.drexel.edu/~ahicks/papers/physics-today.pdf

If you have access to journals through a university, you could take a look at some of his papers yourself (His website is http://www.math.drexel.edu/~ahicks/). They're easy enough for a sophomore to recreate the results. (I know, because I did for an independent study with Prof. Hicks.)

Re: Math Professor Invents Non-Reversing Mirror

#52

Post by Someone »

joshdick wrote: post #45

For those interested in learning more about this, including the math behind the mirror's curve, check out Prof. Hicks's website: http://www.math.drexel.edu/~ahicks/

Relevant part (http://www.math.drexel.edu/~ahicks/papers/physics-today.pdf):

"As a second example, take the object plane and the image plane to once again both be at x = 1 and define T(1, u, v) ≡ (1, −αu, αv). As with the flat mirror of the previous example, the transformation scales the image plane, but be- cause of the relative minus sign in the y and z coordinates, the solution surface will not reverse an object as a conven- tional mirror does.

The relative minus sign also means, alas, that no exact solution surface exists. I obtained an approximate solution surface confined to the rectangular volume x = 34 ± 1 cm, y = z = 0 ± 3 cm, and constructed a prototype mirror. With α = 160, the mirror’s field of view is wide enough that I could see myself in the mirror when it was held at arm’s length, as shown in figure 1. To achieve the appropriate ray paths, the mirror is saddle-shaped."

So, this mirror only does its magic in a small volume; if you move your eye closer to the mirror or sideways, or even when you look at it with two eyes, the effect breaks.

From cursory reading, I even get the impression that this mirror is also tailored to the positions and shapes of the objects being photographed.

But as I said: cursory reading. Corrections welcome.

Re: Math Professor Invents Non-Reversing Mirror

#54

Post by mistercow »

>The precise manipulations change the directions light rays are reflected off of the surface in a manner similar to changing the angles of millions of tiny facets on a flattened disco ball, but decreasing the size of each facet until a smooth surface is achieved.

Oh, so it's like a Fresnel lens, but it's a mirror instead of a lens. That's pretty clever. You get the non-reversing property of a curved mirror, but it's still flat.

Edit: Another way of thinking of it is that it's a normal map applied to a real-world object.

Re: Math Professor Invents Non-Reversing Mirror

#55

Post by sliverstorm »

jpdoctor wrote: post #6

> When you look in a normal mirror, the image you see of yourself is in reverse. Funny thing: Mirrors don't reverse left and right. They reverse front and back. So this "non-reversing" mirror is actually mapping y to -y, x to -x (Where the mirror is in front of you in the z-direction. Curious. Edit: I believe tomerv below is correct: x is mapping to -x but y is kept the same.

I still don't intuitively understand how reversing front to back results in what we perceive as a left/right inversion.

Re: Math Professor Invents Non-Reversing Mirror

#56

Post by woah »

rada wrote: post #48
jere wrote: post #38

Earlier quoted context omitted.

Embarrassingly, I listened to a youtube video where a woman read the book aloud (I listened for about 30 pages). She ended up hitting herself in the head about a dozen times because the writing is that awful.

Fifty Shades is pornography for women. IMHO that's the only reason everyone is clutching their pearls about the bad writing. Your typical male pornography storylines/cinematography are much, much worse - and yet they are typically spared the high-brow literary scrutiny.

Yep, must be the patriarchy oppressin' again.

The reason that 50 Shades of Grey is subjected to highbrow literary scrutiny, is that it pretends to be highbrow literature. Male pornography usually makes no pretense, being more along the lines of logjammin' "ich bin here to fix your kable"

Re: Math Professor Invents Non-Reversing Mirror

#57

Post by drcube »

How can he "invent" this? Hasn't it been known for years? It's the same principle as two mirrors at right angles, only instead of a corner, there is a gentle bend. I'm pretty sure I read about this in George Gamow's "One, Two, Three... Infinity".

Re: Math Professor Invents Non-Reversing Mirror

#58

Post by kbolino »

rada wrote: post #48
jere wrote: post #38

Earlier quoted context omitted.

Embarrassingly, I listened to a youtube video where a woman read the book aloud (I listened for about 30 pages). She ended up hitting herself in the head about a dozen times because the writing is that awful.

Fifty Shades is pornography for women. IMHO that's the only reason everyone is clutching their pearls about the bad writing. Your typical male pornography storylines/cinematography are much, much worse - and yet they are typically spared the high-brow literary scrutiny.

In line with woah (minus the snark), male-focused porn is generally spared the scrutiny because it does not seek it. You won't read a review of Modern Whorefare (sadly a real thing) in the New York Times and you won't be able to pick it up at the grocery store. A difference in marketing and distribution amply explains the greater scrutiny, without resorting to implications of sexism.

Re: Math Professor Invents Non-Reversing Mirror

#59

Post by jules »

sliverstorm wrote: post #55
jpdoctor wrote: post #6

> When you look in a normal mirror, the image you see of yourself is in reverse. Funny thing: Mirrors don't reverse left and right. They reverse front and back. So this "non-reversing" mirror is actually mapping y to -y, x to -x (Where the mirror is in front of you in the z-direction. Curious. Edit: I believe tomerv below is correct: x is mapping to -x but y is kept the same.

I still don't intuitively understand how reversing front to back results in what we perceive as a left/right inversion.

It took me a long time to figure this out as a child, since nobody I asked knew the answer! Is it something about the vertical of the mirror being different than the horizontal? No, if you rotate the mirror 90 degrees it still inverts left/right and not top/bottom. Is it something about the Earth that makes horizontal different from vertical? No, if you look into a mirror while your body is in horizontal position, it still inverts left/right, and not top/bottom (viewed from your body). Is it something about our two eyes being on a horizontal line instead of a vertical line? No, if you close one eye it still works all the same. If you trace out how the light goes, you'll see that there is nothing different about horizontal vs vertical. Yet clearly a mirror inverts things horizontally but not vertically. How can this be?

The real answer is that we perceive this because we are used to turning in the horizontal plane. If you see yourself in the mirror and you raise your right hand, the guy in the mirror is raising his left hand. But how do you know that it is his left hand? Mentally you place yourself in his position by rotating yourself and then you check which hand it is, but by doing this rotation your left hand ends up in the position of your right hand in the mirror. So even though the alignment of you and your mirror image appears to be perfect on first sight, it actually isn't. If we were creatures that are used to only turning in the vertical plane, then we'd perceive the same image as top/bottom inversion. You'd mentally rotate yourself vertically, and then instead of your left hand ending up where you see your right hand and vice versa, your head would end up where you see your feet and vice versa.

The fact that our bodies are roughly symmetrical in one direction but not in the other makes this a bit non obvious unfortunately. It may be easier to understand this with an object that's not symmetrical in any direction. If you have a piece of paper with some word written on it, and you look at it via a mirror, why is the word flipped from left to right and not from top to bottom? This question has the same answer: because we're used to turning things in the horizontal plane. To show the paper to yourself via the mirror, you rotated the paper so that the text is now facing away from you, towards the mirror. But how did you rotate it? You rotated it by turning it in the horizontal plane. If you had rotated it in the vertical plane, the word would be flipped vertically. It's just that if you do this nobody is surprised that the word ends up flipped vertically, because that's not how our psychology works.

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