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Moving Remy in Harmony
April 15, 2010 8:44 AM   Subscribe

Moving Remy in Harmony - Pixar's Use of Harmonic Functions.

The AMS has been on a roll with excellent feature columns recently:
  Crypto Graphics (on the 2D data matrices used on stamps and other places)
  Puzzling Over Exact Cover Problems (on Kanoodle, Sudoku, and the dancing links algorithm)
  Mathematics and Sports (tournament scheduling)
posted by Wolfdog (38 comments total) 14 users marked this as a favorite

 
β1(x, y) = ax + by + c

by declaring


β1(v1) = 1
β1(v2) = 0
β1(v3) = 0


Like, NO DUH!!!
posted by spicynuts at 8:57 AM on April 15, 2010 [1 favorite]


This is clearly very cool and interesting, and it's really a shame that I got confused immediately and skimmed the entire article because I'm a dunce.
posted by shakespeherian at 8:59 AM on April 15, 2010 [3 favorites]


Me, I was just in the football thread learning confusing technical stuff I don't know much about. That's why I like this place.
posted by Wolfdog at 9:01 AM on April 15, 2010 [1 favorite]


Yeah, but can you draw this turtle?
posted by swift at 9:01 AM on April 15, 2010


Math and I have have always had a contentious relationship. I can handle most web development tasks but start throwing in calculus functions and geometric splines and tangent coefficients (I realize those last two are probably meaningless nonsense) and I'm completely lost. But I'm glad there are math nerds at Pixar willing to tackle the tough stuff for the benefit of moviegoers everywhere. I don't have to understand it to appreciate it.
posted by The Winsome Parker Lewis at 9:03 AM on April 15, 2010 [1 favorite]


Ow, ow, ow...

Gawd, as if I needed more evidence that math and spatial rotations are not my strong suits...
posted by Scattercat at 9:03 AM on April 15, 2010


This seems amazing, but alas, I'm too dumb to fully appreciate/understand it.
posted by Lutoslawski at 9:08 AM on April 15, 2010


Quickie translation: "It makes our characters squish nicer".
posted by shino-boy at 9:09 AM on April 15, 2010


β1(x, y) = ax + by + c

by declaring


β1(v1) = 1
β1(v2) = 0
β1(v3) = 0


this is a nice example of slightly obfuscating mathematical writing:

A straight line between v_2 and v_3 is defined by an equation of the form:

β_1 (x,y) = ax +by +c = 0

where β1(v2) = 0 and β1(v3) = 0. This is just another way of talking about the fact that a straight line is defined by choosing two points on it.

However, this does not uniquely define {a,b,c} e.g.

x + 2y + 3 = 0 and 2x + 4y + 6 = 0 define the same line.

So, β_1(v1) = 1 is an added condition that defines {a,b,c} uniquely... which essentially proves this somewhat baffling universal statement:

Since linear functions on the plane are determined by their values at three non-collinear points, this is enough information to determine β1 uniquely.

the other interesting thing is that you are much more likely to encounter this kind of geometry in the computer science department than the math department. in math one generally encounters "barycentric coordinates" as this somewhat baffling and definitely archaic topic in the beginning of an elementary algebraic topology class and never after... however, the AMS wants the kids to think that math is about making cool Pixar movies.
posted by ennui.bz at 9:14 AM on April 15, 2010 [2 favorites]


Very interesting. More importantly: I now have incontrovertible evidence that being an artist is hard.
posted by 1f2frfbf at 9:23 AM on April 15, 2010


When is it going to stop being cool to declare how ignorant one is of math?

If you think it is interesting but don't understand it, learn.
posted by DU at 9:30 AM on April 15, 2010 [6 favorites]


dammit Wolfdog, I'm still working through your last post and my mathalyzer is pretty rusty. Hold off, will ya?
posted by Quietgal at 9:38 AM on April 15, 2010


Oh yeah and this is an awesome post.
posted by DU at 9:39 AM on April 15, 2010


I was gonna say what ennui.bz said but I just decided it was self-evident and obvious to everyone.
posted by Babblesort at 9:41 AM on April 15, 2010


tl;du
posted by gottabefunky at 10:09 AM on April 15, 2010 [3 favorites]


If you think it is interesting but don't understand it, learn.

What I'd like is to be able to go back in time to some point starting around 4th grade (so I was about 10) and be able to have teachers teach me in a way that allowed me to grok concepts I couldn't easily grok. I could do homework problems if they were set up in exactly the same way as they had been in class, but was unable to extrapolate - clearly, I was unable to extrapolate because I didn't really understand the fundamental concepts; I could only parrot back what I'd learned.

You think I feel cool, not being able to grasp this stuff? I don't. I feel embarrassed and stupid.
posted by rtha at 10:12 AM on April 15, 2010 [4 favorites]


I was gonna say what ennui.bz said but I just decided it was self-evident and obvious to everyone.

the thing is that if i had any doubts about what he was talking about, i would be intimidated by this statement:

Since linear functions on the plane are determined by their values at three non-collinear points, this is enough information to determine β1 uniquely.

it seems very general and, to be honest, I'm not even sure what a "linear function on the plane" is... except that it means just a function of the form ax +by + c (since he doesn't want to about projective coordinates.)

basically, he could of just said, "check this yourself: these conditions uniquely define {a,b,c}" but instead he kind of makes it seem like you need to be a smart mathematician to know what he's talking about... when what he is talking about is very elementary.

(I think this is all-in-all a nice little essay, but I don't think it's surprising that people respond to being intimidated by giving up... )
posted by ennui.bz at 10:13 AM on April 15, 2010


When is it going to stop being cool to declare how ignorant one is of math?

If you think it is interesting but don't understand it, learn.


I'm normally with you on that one, but this is relatively high-level mathematics. It's no different than being like, "I don't quite understand the science behind that phenomenon, but it looks damn cool."
posted by explosion at 10:16 AM on April 15, 2010 [2 favorites]


What I'd like is to be able to go back in time to some point starting around 4th grade...

You don't literally have to go back in time, of course. I was like you, although my "back in time" point was freshman year in college. I just went back and did it myself. You can to. The really great part is once you know how to teach yourself math, learning more gets easier and easier. MeMail me for help if you want.
posted by DU at 10:24 AM on April 15, 2010


Math, even advanced math, is pretty simple stuff conceptually. Sadly, its most frequent manifestation is shrouded in jargon and sprinkled with glyphs that only several years of study proffer a subconscious grasp of. I don't know if there's a solution for that problem, because math requires jargon and glyphology in order to express itself on paper clearly and concisely. It's beautiful stuff, though.
posted by seanmpuckett at 10:37 AM on April 15, 2010


Actually, I don't find that jargon and glyphs are what make math confusing. People can learn new vocabulary pretty easily.

What makes math confusing is the pathological elegance. Like, did you really need to express B1 + B2 + B3 as ∑Bi? Sometimes, sure. But in an article meant for a lay audience, it just takes more time to unpack.
posted by DU at 10:40 AM on April 15, 2010 [2 favorites]


When is it going to stop being cool to declare how ignorant one is of math?

I for one certainly wasn't trying to position myself as cool. I said (and meant) that it's a shame how quickly I got confused. I also called myself a dunce.
posted by shakespeherian at 10:42 AM on April 15, 2010


When is it going to stop being cool to declare how ignorant one is of math?

Yeah, I'm with shakespeherian. Defo not trying to be cool. I'm ashamed of how rusty my math skills have become.
posted by Lutoslawski at 10:46 AM on April 15, 2010


This is not an article intended for a lay audience; it's intended for an audience that is comfortable with linear algebra, calculus, and differential equations, and the content of the article could not be conveyed without either assuming that background or providing an entire course in it. However, it is a well-structured article, and if you read for ideas, I think it's not hard to understand the problems involved and get a sense of what mathematical tools are used to help with them. And, possibly, to find even a little more awe and marvel in Pixar's movies - rather than spoiling things, I find this is like pulling back the curtain and finding the wizards behind it are actually really and truly wizards equipped with awesome wizardry.
posted by Wolfdog at 10:51 AM on April 15, 2010 [3 favorites]


I just went back and did it myself. You can to.

I've actually started to go through some of the Khan Academy videos (starting back even before the 4th grade stuff, because I probably missed some things there, too), and they're helpful. But I'm not a very good candidate for teaching myself math. Teach myself history? I can do that. But math, or things that require math - physics, chemistry - I do better in a classroom, and better yet with a good teacher who has multiple ways of explaining something, in case I can't get at the concepts the first 10 ways they're explained to me.
posted by rtha at 10:58 AM on April 15, 2010


The first link seems to be broken.
posted by valrus at 12:36 PM on April 15, 2010


When is it going to stop being cool to declare how ignorant one is of math?...
posted by DU at 9:30 AM on April 15


You think I feel cool, not being able to grasp this stuff? I don't. I feel embarrassed and stupid.
posted by rtha at 10:12 AM on April 15


Yeah, I think a lot of the people-bad-at-math thing is embarrassment mixed with a shared experience of having bad instructors.

The greatest math teacher I've had, a grad student named Dave ("Call me Dave."), made an announcement after he heard a student talking down about her math intelligence—"Listen, no one in this class is stupid! If you are having trouble, that's on me—that is because I am not teaching it to you the right way—remember, I'm hear to teach you." Guess what, people did really well in his class.

Cut to a couple of years later when our chemistry teacher (a PhD, "You will call me doctor!") tells a disabled student (in front of the class) who was having trouble with the formulas "Come on, this isn't kindergarten, this is 'baby' chemistry!"

There's a lot of shame in people not knowing their various maths, but you know, a lot of that shame was actually put there.
posted by blueberry at 2:57 PM on April 15, 2010 [3 favorites]


I wasn't trying to be "cool" either. I could go out and start brushing up on my high-level math today, but my hours are limited and I can't justify investing them in something that helps me appreciate random posts on MetaFilter but has no application in my daily life. I'm not a computer animator, or a physicist, or an engineer. There are a million knowledges I could stop complaining about and just learn already(!!) but I can't learn all of them, and there's no point in my learning most of them of them anyway. I have plenty of interests I already do pursue for fun, and I'm happy with my choices.

Appreciation of something from a distance is not an unforgivable sin. Even math.
posted by The Winsome Parker Lewis at 3:15 PM on April 15, 2010 [1 favorite]


I thought most animators used skeletal (or 'Inverse Kinematic') models, not these 'control cage' things. That's what they were doing when I was in college, have things changed since then?
posted by delmoi at 5:53 PM on April 15, 2010


What I'd like is to be able to go back in time to some point starting around 4th grade (so I was about 10) and be able to have teachers teach me in a way that allowed me to grok concepts I couldn't easily grok.

Hmm... Have you tried learning on your own? I read an article in slate once about a woman who couldn't figure out 13 – 5 and over a few months using the Kumon system that her daughter was using learned to calculate arithmetical formulas "(3 1/6 - 1 19/24) ÷ (4 3/4 - 0.9)" mostly in her head. Now, that's certainly not something I could do. But I don't think raw arithmetic is that important for "knowing math"


But I get kind of annoyed when I hear people say "Oh, my teachers were terrible" when talking about why they can't do math. I had crappy math teachers in elementary school I'm not sure I learned anything worthwhile mathematically at all between 3rd and 7th grade. But since I enjoyed math, it was easy for me to learn more, even on my own.

(And think about it: Elementary school can seem like a formative experience, a foundation of your life, but in reality it was only a few years. Three years between 30 and 33 doesn't seem like a lot of time, compared to three years between 5 and 8. But in theory you can learn at least the same amount of stuff, if not far more due your advanced brain. )

Of course, if you don't "enjoy" math the way I did, if math isn't it's own reward then what? The fact is, people don't really need to know any math to get along in society. People pretend like it's important but it's not. *kanye shrug*. Especially in the era of computers.

The only problem, really, is all the B.S. economic arguments floating around that people vote. It would be nice if people knew statistics, economic modeling (and the limits of economic modeling) before they voted.
posted by delmoi at 6:22 PM on April 15, 2010


Well, it may not be exactly "cool" to display your math ignorance, but...

Look at it this way -- if someone posted an article about cars, would you then have ten comments saying "I've always thought driving is cool but I'm the suckiest driver around, I get dizzy if I have to make two right turns one after another"?" No. It's very uncool to admit to being a bad driver, but it's never uncool to not understand math.

(I used to be a math professor and those kinds of comments drive stakes through my heart.)
posted by phliar at 7:09 PM on April 15, 2010


I didn't get what was "harmonic" about this, until I came across this poking around wikipedia:

"A solution to Laplace's equation has the property that the average value over a spherical surface is equal to the value at the center of the sphere"

The desired goal of the activity here is to control one point with your little pointer, and have that influence a lot of related or connected points, so you don't have to go in there and fuss with each one separately with your little pointer.

The Laplacian math tools have to do with kind of spreading out of the tip of your little pointer, in what might be imagined as radiating spheres of influence. Once you get into all those overlayed spherical factors, then you involve decomposition into what could be imagined as orbiting and circling traces, with harmonic oscillations.

That's all deep in the math. To the user I don't think there is any "harmonic" quality that is recognized or important.
posted by StickyCarpet at 9:08 PM on April 15, 2010


Look at it this way -- if someone posted an article about cars, would you then have ten comments saying "I've always thought driving is cool but I'm the suckiest driver around, I get dizzy if I have to make two right turns one after another"?" No. It's very uncool to admit to being a bad driver, but it's never uncool to not understand math.

I'm really good at math. I was two years ahead all through school and I got very good scores on aptitude tests. That was several years ago, and I haven't done anything beyond basic algebra since, because I haven't had any reason to, which is too bad. When I looked at the linked page, I wanted to follow it because it looked very interesting, but I was unable to. I wanted to express my gratitude to the OP for the link, but not posture as someone who was able to get through the thing. So I said what I said. I was attempting to thank the OP, not say anything about math or myself, really.
posted by shakespeherian at 9:19 PM on April 15, 2010


delmoi: I get kind of annoyed when I hear people say "Oh, my teachers were terrible" when talking about why they can't do math.

I entered a new school system in the 7th grade, and couldn't have cared less about math. But my math teacher would thumb tack an extra-credit math puzzle, hand-written on colored construction paper, to the bulletin board each day. I quickly discovered that the answer was always written in pencil in the upper left of the reverse side, and submitted my solutions accordingly.

Getting every puzzle question right led to my being placed in AP math, and only then did I start to care about it, because the AP math teachers were so much more enthusiastic about the beauty of math.
posted by StickyCarpet at 9:30 PM on April 15, 2010


I'm normally with you on that one, but this is relatively high-level mathematics.

I hate to be the one to say this, explosion, but this is pretty low-level mathematics. As mentioned above, these equations are taught in high school (albeit in different forms).

It's the jargon that's throwing a lot of people off the scent. "Jargon" is sometimes used to mean "obfuscation"; in this case, it actually means "professional shorthand". If all the pro's settle on a specific jargon, they don't need to waste time explaining what they mean by their commonly-accepted terminology, and can reduce longer strings of characters like "B1 + B2 + B3" to the shorter "∑Bi".

This article begins with the heading "Feature Column: Monthly Essays on Mathematical Topics." It isn't meant for a lay audience.
posted by IAmBroom at 6:55 AM on April 16, 2010


The fact is, people don't really need to know any math to get along in society. People pretend like it's important but it's not.

The whole of mathematics consists in the organization of a series of aids to the imagination in the process of reasoning.-Alfred North Whitehead
posted by Enron Hubbard at 7:38 AM on April 16, 2010


The whole of mathematics consists in the organization of a series of aids to the imagination in the process of reasoning.-Alfred North Whitehead

Can you prove this is true? Or do you merely imagine this to be true?
posted by sebastienbailard at 1:22 PM on April 16, 2010


It's the jargon that's throwing a lot of people off the scent.

There's a journalistic trick for productively writing prose: tell the story with a natural flow, and when you come to a missing piece, don't stop to research that name, date, or whatever. Just insert the letters TK, (To Kome,) and keep rolling, then fill in the blanks later.

This is how I read math I don't understand. Mentally flag the unfamiliar terms and operators with TK, and try to keep going with the flow of the argument, even with a very partial understanding -- get a feel for the arc of it, and look for what things you can recognize. Next, Google the unfamiliar terms.

Then go to sleep and have a dream about it, then call up someone who actually knows this stuff and ask him to explain it.
posted by StickyCarpet at 6:53 PM on April 16, 2010 [1 favorite]


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