Mirrors, Viewfinders, Sensors, and Orientation

[MGrayson]

Good Morning,

I'm moving a discussion of mirrors and image orientation from Behind the Scenes to its own thread because I feel like I've pulled that miraculous place too much off topic. I'll have the usual verbiage and pictures (computer drawn, alas, not photography) later.

Not sure about the starfish, but isn’t the reason why the image in the camera obscura is not reversed left to right when viewing it a simple matter of perspective of the viewer? In other words, in this case you are viewing the image projected on the wall whereas in the case of the viewfinder you are in essence viewing from the opposite side? I suspect I must be missing something pretty basic here so look forward to being thoroughly disabused of my (il)logic upon reading your essay, abridged for photographers and lowly experimentalists such as myself.

John

John,

You are correct. The reason for the asymmetry is that you must choose how to turn around inside the room. If you were in zero gravity, you could turn around the usual way, or by rotating around your hips and ending up upside-down. In that case, the image would appear reversed left-right, but not up-down.

This choice of rotational axis, and it's mandatory in all odd-dimensional spaces (and three happens to be odd), is the root cause of all mirror and viewfinder related issues.

I don't want to go into four-dimensional photography, but the image on the three-dimensional sensor (that would be a LOT of voxels) , e.g., a tintype, is NOT reversed - merely rotated. The image on the 3D ground glass, OTOH, would be appear rotated and reflected, and not just rotated, as on our 2D versions. To be precise, it's what the world would look like through a corner mirror - three mirrors arranged like the corners of a room - everything upside down and backwards.

That's it - no more higher dimensional or non-Euclidean photography. (My Ph.D. thesis was on counting the bathroom tiles of four-dimensional non-Euclidean creatures. I am not making that up. We called it "Growth Functions of Fundamental Groups of 3-Manifolds", but that's what it amounted to.)

Matt

 

[MGrayson]

I'm writing this at a variety of levels of mathematical sophistication. If it's all nonsense, or if it's eye-rollingly simplistic, or both, I apologize.

Mirrors:

The really interesting thing about mirrors is how they work. I mean at the atomic scale. HOW does light decide to bounce off? Why are the angles of incidence and reflection equal on a mechanistic level - no principles of least action and calculus of variations. If I can't contain myself, I'll get into it, but I'll just assume that mirrors do what we think they do.

Mirrors reverse the directional component of a light ray that is orthogonal to the mirror. If we choose coordinates so that x and y are in the plane of the mirror and z points towards it, then a light ray going in the direction {a,b,c} hits the mirror and thereafter moves in the direction {a,b,-c}. That's it. Full stop. We'll look at the implications below.

NOTE: Mirrors don't do ANYTHING in the plane of the mirror itself, or any plane parallel to it. This is one of those "reread until you mutter it in your sleep" comments. How best to see this? Hold up a transparency in front of you and look at it in the mirror. It and its reflection will be identical. (You're saying "But but but ...". Hold on for now.)

\begin {annoying math}
If things aren't lined up, we have to get fancy and say that the light ray v goes to v - 2 (v•N) N, where N is the outward normal vector to the mirror, and the dot is the dot-product. (v•N) N is just a fancy way of writing "that part of the vector pointing straight at the mirror". The two terms are "how much?" and "which way?" Why the factor of 2? Subtracting only one multiple would eliminate any component pointing towards the mirror and we'd end up moving parallel to it. You need to double the effect to actually reverse the normal component.

And since it comes up a lot, a mirror at 45˚ to the {x, y} plane, say the mirror in a TLR, or SLR (the "R" stands for "Reflex", after all), reverses and interchanges the y and z components. Direction {a, b, c} -> {a, -c, -b}. Using our Annoying Math Formalism, the normal is N = {0, 1, 1}/Sqrt(2), so {a, b, c} -> {a, b, c} - (b+c){0,1,1} = {a, -c, -b}. The factor of 2 gets cancelled by the two factors of Sqrt(2) in the denominator, one from each occurrence of N. It does nothing to x! ("Wait," you say, "that sounds backwards! When I look at my Hasselblad 500 viewfinder, only x is reversed!" )
\end

Lenses and Pinholes.

The other way that our photographic images get messed with is forcing all the light rays to pass through a single point (lenses try to simulate that effect, and stopped down lenses do a pretty good job). This does not change the direction of any light rays, but does limit which ones go into our camera. The effect, as everyone who has used a view camera knows, is an image rotated 180˚. A point in the real world with x-y coordinates {x,y} end up on the ground glass at coordinates a constant times {-x,-y}. That constant is called the magnification.

(Posting this now ... lots to be added ... stay tuned)

 

[MGrayson]

So let's get rid of the most common confusion about mirrors right away. Even if we know that mirrors don't reverse left-right, that notion is so universally assumed that words like "mirror image" or "mirror writing" immediately bring left-right reversal to mind. Why?

The answer is because that's the way we turn around. The thing about the transparency is that you can see through it. But if you had paper blocking your side of that lovely 8x10 Cibachrome, you wouldn't know what it really looked like without either turning it around or turning yourself around to look at it from in front. THAT makes it look different from the image in the mirror.

Convincing experiment: Stand in front of a mirror and look down at your watch. Now rotate your forearm and wrist to point your watch at the mirror. You'll see an image of your watch face that's upside-down, but not reversed left-right. Rotate it back and forth a few times. Three is on the right, nine is on the left, six and twelve switch places depending on whether you're looking at your wrist in the mirror or not. It's because you turned the watch around a horizontal axis. We usually turn books, doors, and especially, ourselves around a vertical axis. So much so that we aren't even aware of doing it. Turn around by doing a handstand (think foosball) and everyone will stare at you. You will be facing the other way, but mirrors will appear to reverse up-down, because you just did the same thing that your wrist did before.

So how about that Hasselblad or Rollei ground glass viewfinder? We're pretty sure that the image is reversed left right, but not up down. Two things are happening. First, the light coming through the lens is doing that view camera reversal thing - the image is rotated 180˚. Then it hits the mirror. We interpret this mirror as reversing up-down because when we look down at the camera and then look up, we're rotating our heads around a horizontal axis (the line between Frankenstein's monster's neck-bolts). So first {x, y} in the outside world goes to {-x, -y} by passing through the lens. But then we decide that the reflex mirror flips it upside down to {-x, y} and THAT is left-right reflection. (Yeah, I suppressed the constant from magnification, in case you were wondering...)

Remember that all the reflex mirror did was interchange y and z (and flip their signs) so light coming towards you (negative z direction) turns into light going up (positive y direction). It changes the direction of the light, but doesn't do any image reversal. That interpretation is on us. Of course, it's hard to imagine a different way of raising your head from looking down at the camera, but if you think of rotating it around a diagonal axis, so that your head is lying on one of your shoulders, then the real world will look rotated 90 degrees. Never mind.

You may well be objecting "Look, light from a point up in the tree comes into my camera and is headed to the bottom of the mirror, but the mirror is angled, so the bottom of the mirror looks like the top of my ground glass, that's where the tree appears, so it doesn't reverse up and down and it has nothing to do with how I move my head!" But what have you just done? You've taken coordinates on your ground glass (up down right left) and matched them up with coordinates in the real world. There's no God-given (what we math types would call "canonical") way to do that. You choose to rotate your ground glass coordinates around a horizontal axis to match them up with the scene in front of you. Rotate a different way, get a different answer.

Here's a really unhelpful picture showing a rotation of the sphere (your head) taking the Hasselblad ground glass (the square on the bottom), rotating it around that strange axis indicated by the arrow, and ending up with a tilted square out in the direction of the real world (the tilted square in the back). The circle shows the path of the rotating square , and it lies in a plane perpendicular to the arrow. As your head comes up, it tilts to the right. Unnatural, maybe, but it's just a different choice of how to look up.



The arrow is pointing slightly towards you, as you can see from this view from above looking straight down on the camera. Yes, you're supposed to be twisting your neck with that arrow as the axis of rotation.



I actually worked out these rotation matrices. I'm so ashamed. But then, I did teach linear algebra at least three times over the decades.

Perhaps a better way to see this is to just hold the camera tilted to the left. Look down and to the right at the screen, look up. You've now rotated your neck through a strange axis and, behold! the image on the glass screen is tilted! "But that's just reflecting across a different line", you say. Exactly!

 

[MGrayson]

And finally, the dust specks on our sensors. This one is easy, as no mirrors are needed. The image on the sensor is, as we've seen, rotated 180˚ from what we want to see in the final image. And, indeed, the camera does that rotation before showing us the image. Suppose we take one of those f/22 images of an OOF white wall and see a nasty dust speck at position {a, b} on the image. Where is that speck on the sensor? Well, it's at {-a, -b}. But to actually remove that speck, we have to (all together now) turn the camera around. Since we usually turn it around a vertical axis, we reverse left-right , and so that changes the speck location to {a, -b}. It's reversed up-down, but not right-left. Of course, if we flipped the camera upside down to see the sensor, it would be reversed left-right and not up-down.

Enough!

Matt

 

[jng]

Matt,

Thanks for the lesson. I think. TBH I wasn't confused about what I *thought* mirrors do until I read your last few posts. But I don't blame you - it's a miracle and perhaps a damning statement about the way that science is taught that I not only passed but received straight A's in college physics (physics for physical science majors, no less, not the course er, watered down for the pre-meds; quantum chemistry, however, just about did me in). Generally speaking, my intuition about the physical world serves me pretty well, until it doesn't. To quote/paraphrase Annie Hall "I just kinda feel it, y'know?"

Can't wait for your next installment...

John

 

[MGrayson]

Matt,

Thanks for the lesson. I think. TBH I wasn't confused about what I *thought* mirrors do until I read your last few posts. But I don't blame you - it's a miracle and perhaps a damning statement about the way that science is taught that I not only passed but received straight A's in college physics (physics for physical science majors, no less, not the course er, watered down for the pre-meds; quantum chemistry, however, just about did me in). Generally speaking, my intuition about the physical world serves me pretty well, until it doesn't. To quote/paraphrase Annie Hall "I just kinda feel it, y'know?"

Can't wait for your next installment...

John

I think you have it right. Mirrors do just what you think they do. What's interesting is how we interpret them, and what unconscious choices we make in coming to those interpretations.

 

[anwarp]

What's interesting is how we interpret them, and what unconscious choices we make in coming to those interpretations.

Which is perhaps why a flat earth with heavenly bodies going around it was the accepted canonical model!

On topic ~ Thank you Matt, for taking the time to write this up. I find your lectures thoroughly enjoyable. Do you have a blog?

Anwar

 

[f6cvalkyrie]

Matt (or is it written Math ???)

When I read your posts, I must say I'm glad that the photographic business seems to move away from SLR and TLR and now favors mirrorless cameras ...

Stay safe,
Rafael

 

[tenmangu81]

Thanks Matt, you are awesome and very good at explaining complex problems, and even better at explaining simple problems !!
Just one more thing ( ) : there is a symmetry operation in geometry and 3D group theory which is "mirror" and results in transforming an "as seen" (not only by eyes) right hand into an "as seen" left hand. And a left-handed helix (like DNA, levogyre) into a right-handed helix.

 

[MGrayson]

Thanks Matt, you are awesome and very good at explaining complex problems, and even better at explaining simple problems !!
Just one more thing ( ) : there is a symmetry operation in geometry and 3D group theory which is "mirror" and results in transforming an "as seen" (not only by eyes) right hand into an "as seen" left hand. And a left-handed helix (like DNA, levogyre) into a right-handed helix.

Yeah. I made a choice not to go there. It’s interesting and relevant, but believe it or not, I try to keep things as simple as possible. Usually unsuccessfully.

 

[MGrayson]

Which is perhaps why a flat earth with heavenly bodies going around it was the accepted canonical model!

On topic ~ Thank you Matt, for taking the time to write this up. I find your lectures thoroughly enjoyable. Do you have a blog?

Anwar

Anwar,
I don’t have a blog, but do have a small (and not particularly well chosen) collection of articles on my website. I’ll post a link if I ever get it cleaned up.
Matt

 

[MGrayson]

Sigh. I was afraid of this. Ok. Here's how mirrors work at the atomic (or molecular, I suppose) level. This explanation is due to Feynman and appears in his magnificent book "QED: The Strange Theory of Light and Matter". In case it's not obvious, Feynman is (professionally, anyway) my hero. Speaking of Bang for the Buck, QED (Quantum Electrodynamics, not Quod Erat Demonstrandum) is the best science book ever written.

Anyway.

We start with an incoming beam of light. It's a laser, actually, so all the peaks and troughs line up. I'm just drawing lines where the waves cross zero. Now atoms are much closer together than the wavelength of visible light, and I don't want to use gamma rays, so we're spacing out or atoms of reflectivity. Weird diffraction things can happen when your reflective bits are spread out (why CD's reflect rainbows), but we're ignoring that.



Soon (VERY soon) the light hits our first mirror atom (pretty sure mirrors are on the periodic table....) and it starts moving with the light.



As it moves, it radiates light - or more properly, radiates the probability of finding light, but we're staying out of the quantum realm. The best analogy, and the one I'm actually drawing here, is water waves coming in and hitting a string of buoys. Each buoy starts bouncing up and down, creating new waves. Sort of.



The second atom gets hit in another picosecond or two





Are you starting to see it?



And here, I'll throw in contributions from the atoms off the left side of the screen, who have been radiating this whole time, but I didn't want to clutter up the image. The predicted (angle of incidence = angle of reflection) line is included.



Pretty cool!

Matt

 

[tenmangu81]

Yes, Feynman books are really fantastic !! You have the impression (just the impression) to understand all physics when reading them.
An other of my favorite lectures is (was...) the Landau and Lifshitz's series. But then you don't have the impression to understand physics at all, and I remember having spent a lot of time thinking about just a footnote....
Thanks again, Matt !

 

[MGrayson]

Yes, Feynman books are really fantastic !! You have the impression (just the impression) to understand all physics when reading them.
An other of my favorite lectures is (was...) the Landau and Lifshitz's series. But then you don't have the impression to understand physics at all, and I remember having spent a lot of time thinking about just a footnote....
Thanks again, Matt !

Landau and Lifshitz write with this "it is inevitable - how could it be any different?" style, but I also don't come away with understanding.

 

[MGrayson]

I particularly liked "Either the universe has a maximum speed or it doesn't." They then derive special relativity from scratch.

 

[tenmangu81]

You probably know the Einstein's sentence about the Universe infinity: "Two things are infinite: the universe and human stupidity; and I'm not yet completely sure about the universe."