Mathematical Proof 3D movies don’t work

So, doing movies in “3D” seems to be a big thing right now, everyone seems to think they need to do it to their new movie. The problem is, we just can’t do it to make it work, at least not for everyone at the theater. In fact I will claim (and shortly show for the most part) that it can only work correctly for one person in the theatre and the pictures get more wrong the further away you are from where that person is.

How does 3D work in the first place? Close one eye, and look at the scene around your right now, like your desk and computer. Then open it, and close the other eye. Notice that things don’t look the same to each of your eyes. Your brain takes these images and puts them together to create a 3D scene for you. When you watch a normal movie though you don’t get that effect on the objects on the screen, both eyes see the exact same parts of that car that’s being chased down the highway by the aliens. In 3D movies, there’s 2 pictures on the screen, but you wear glasses so each lens blocks one of the images making each eye only see one, then your brain puts them together and you get the 3D effect.

So the idea of what is put up on the screen at the movie theatre is this, what would each eye see if the objects on the screen were real and behind (or in front of) the movie screen. This idea I have no problem with at all, however, here’s something to think about, what about the person sitting beside you? The movie is showing what you would see, but that person is somewhere else, they would see something different, but they’re still seeing the same two images you are, and what about the person who came late and got stuck in the front row at the very edge, or the couple sitting in the back row so they can make out?

Everyone in the theatre is seeing the same images, but only one person can be sitting in the spot where the images would combine into the proper scene from where the person is sitting. So what do the people in the non-optimal positions see? That’s what I wanted to find out, do they see the same objects in the same proportions, maybe just in slightly different positions than they should be (best case)? Does the object get warped in all sorts of ways including straight lines no longer being straight? I decided I would sit down and do some math to partially answer these questions. I got lazy and uncreative with my naming conventions so my example only uses a specific general case, but I think it shows enough evidence that 3D movies don’t work I’m not going to do the general case (which is really easy to do from here, it would just get very messy really fast)

A quick note before I start with the math: The math here isn’t very complicated, it’s actually at the grade 9 or 10 level, and thus would be a great test (there’s too many steps for it to really be a single questions, but for an entire test or exam or assignment this would be perfect). In the spirit of a high school math test I’m going to word it all like a problem you would see there too. My example is also missing a dimension, however, that doesn’t really change the way things work, and if you’re upset at me handwaving that dimension away, then let’s just say that they’re only looking at one horizontal line of the screen. The units are also kind of unrealistic, but I’ll discuss the implecations of that later.

Xavier and Yvette went to see a 3D movie, however when they got there they couldn’t find 2 seats beside each other so they had to sit in different parts of the theatre. Xavier managed to get the perfect seat right in the middle of the theatre (0,0), whereas Yvette ended up in front of him and to his left at point (-5,4), imagine their eyes are 1 unit to the left or right of their centres (so Xavier’s left eye at (-1,0) and right at (1,0) and Yvette’s at (-6,4) and (-4,4)). The screen is 10 units in front of Xavier (at y=10). At one scene in the movie Xavier sees the characters Alice at position (4,12), Bob at (4,18) and Carol at (-4,12). At what position does Yvette see the characters?

I have the answer, but I said “proof” in the title of this post, so I think I should explain how I got the answer. Feel free to double check my numbers if you want, but I’m not going to transcribe them all for you right now.

First, find equations for the 6 lines that go between a character and one of Xavier’s eyes. Then find the intersection point of those lines with the screen. Now, take the points on the screen that correspond to Xavier’s left eye, and find the 3 lines that go through them and Yvette’s left eye, and the same for the right eye. You should now have 6 lines which each correspond to a character’s screen position and one of Yvette’s eyes. The last thing you need to do is find the intersection points between the lines that correspond between each character, and one of the eyes, the point of intersection should be where Yvette sees the character. This past paragraph probably sounded very confusing, and so to hopefully make it make more sense here’s a diagram (not to Scale) that sort of shows the process for one character.

So what are the answers to my question? Well, Yvette sees Alice at (5,11.2), Bob at (8, 14.8) and Carol at (-3,11.2). What’s the most obvious thing to note between this set of numbers and the original set ( (4,12), (4,18) and (-4,12) )? Well the original set formed a right angled triangle, whereas the Yvette data does not. Distances (and proportional distances) are also not the same (that said, it’s not possible for one of angles or distances to not work and the other one to work). Where Xavier sees a square, Yvette does not, and if in the movie Carol threw 2 balls, one to Alice, and one to Bob, if Xavier saw them going at the same speed, Yvette would see the ball going to Bob as being faster than the one to Alice.

Alright, I’m almost done now. To sort of conclude, this data shows that in terms of viewing a 3D movie, angles and distances in the image are relative to the observers based on their positions, unless you are sitting in the absolutely ideal position (and there is only going to be one) then the objects will be warped.

Disclaimers: The scale used in the example has the screen only 5 times the distance between eyes from the furthest person’s face, only 3 eye lengths from the closer person, although warping is still bound to happen it’s not going to be as severe as my data showed, that said, I would not be surprised to hear of an uncanny valley like effect where the things that are almost right but not quite are more unsettling than those that are more obviously wrong.  Final disclaimer, the thought process I used here I came up with independently of any other similar ideas if they exist, I’m sure they must because it seemed so obvious to me, but I’ve yet to encounter them, if you know of it let me know, I bet they were able to phrase things more eloquently than I did.