This was a super smooth interview. I love how well rehearsed it was, probably because he has explained that stuff dozens and dozens of times. Really nice to watch. (Not to discredit the editing which probably helped too)
Numberphile Thanks for this. I work in 3D Animation and have been wondering about the math behind Subdivision and limit surfaces for quite some time. On a side note, you asked Tony if everything comes out blobized, but in this vid, it doesnt seem like he answered the question the way you were expecting. The answer is no. In the 2d example, if we wanted one corner to be sharper than the others we add 1 or more vertices near the corner vert. The result is the averaging and splitting is tighter where we've added more vertex density resulting in a sharper smoothed corner
Wait a moment... Ed Catmull is the President of Pixar? The Catmull-Clark Catmull? I studied the Catmull-Clark subdivision the other year, but I never imagined the person behind it would also be responsible for some of my favourite movies. You sort of imagine an old dude in a small room of some university, when you think about the name behind an algorithm. You don't connect the people coming up with theories with the ones who put them into practice, although it makes so much sense This is awesome! :-)
Yep, same guy. Pixar was founded as the computer graphics research wing of Lucasfilm, and in those first couple years they laid a ton of the groundwork for computer graphics.
For those who know a little bit of linear algebra and are wondering about where the (A+4B+C)/6 thing comes from . . . Going from step 0 to step 1 can be thought of in terms of matrix multiplication like this: (A1) (4/8 4/8 0/8) (A0) (B1) = (1/8 6/8 1/8) (B0) (C1) (0/8 4/8 4/8) (C0) Or, if we call the matrix M and the vectors x0 and x1, we write x1=M * x0. The eigenvectors of M are (column) vectors v such that M * v is proportional to v. There are standard techniques for finding these eigenvectors and eigenvalues, which I won't go through here unless someone asks. The eigenvectors for this matrix are: v0 = (1,1,1), v1=(2,-1,2), and v2=(1,0,-1), (up to normalization, which we are free to choose however we like) and the eigenvalues (proportionality constants) are a0=1, a1=1/2, and a2=1/4. In other words, the following three equations hold M*v0 = (1)*v0 M*v1=(1/2)*v1 M*v2=(1/4)*v2 Notice that we can write (A0,B0,C0) as A0*(v0 + v1 + 3*v2)/6 + B0*(2*v0 - v1 + 0*v2)/3 + C0*(v0 + v1 - 3*v2)/6 So, what happens when we take M*(A0,B0,C0)? We can distribute M through the expression above and use our egeinvector equations (e.g., replace M*v2 with [1/4]*v2) to get M*(A0,B0,C0) = A0*([1]*v0 + [1/2]*v1 + [1/4]*3*v2)/6 + B0*([1]*2*v0 - [1/2]*v1 + [1/4]*0*v2)/3 + C0*([1]*v0 + [1/2]*v1 - [1/4]*3*v2)/6 We can go again . . . (M^2)*(A0,B0,C0) = A0*([1^2]*v0 + [(1/2)^2]*v1 + [(1/4)^2]*3*v2)/6 + B0*([1^2]*2*v0 - [(1/2)^2]*v1 + [(1/4)^2]*0*v2)/3 + C0*([1^2]*v0 + [(1/2)^2]*v1 - [(1/4)^2]*3*v2)/6 And in general . . . (M^n)*(A0,B0,C0) = A0*([1^n]*v0 + [(1/2)^n]*v1 + [(1/4)^n]*3*v2)/6 + B0*([1^n]*2*v0 - [(1/2)^n]*v1 + [(1/4)^n]*0*v2)/3 + C0*([1^n]*v0 + [(1/2)^n]*v1 - [(1/4)^n]*3*v2)/6 As n gets large, the terms (1/2)^n and (1/4)^n go to zero and 1^n remains equal to 1, so we get limit of (M^n)*(A0,B0,C0) as n -> infinity= A0*(v0)/6 + B0*(2*v0)/3 + C0*(v0)/6 = v0*(A0+4B0+C0)/6 Or, back in vector form (recall v0=(1,1,1)): (A_infinity) ((A0+4B0+C0)/6) (B_infinity) = ((A0+4B0+C0)/6) (C_infinity) ((A0+4B0+C0)/6) So all three points converge to (A0+4B0+C0)/6.
When I calculated some eigenvalues and eigenvectors in the first term of my math studies, I was bored. I _never_ imagined that the beauty of animated movies depended on it! :) Great overview, I'm really hoping for more! :)
Stefan Ehrenhaus Really? It is a fantastic and powerful program. In my case, it is a perfect middle ground because I do some game asset stuff and some CGI VFX stuff. If I was doing all of one or the other, I'd be on a different program. There is also the fact that I've been using 3Ds for 15 years. What is your preference, and why is 3DS 'urgh' for you?
Kyle Biggs Well, my preference is Maya, im a character TD for film. Complex deformation is just more accessible in maya than in 3ds, thanks to python and c++ api. But you sure have your point with 15 years experience, i respect that. Sorry if for any offense.
Alexander Tamayo I use MODO. Ditched 3DSmax for it because it gave me headaches while I was learning how to model. Never looked back since. I'm still crappy at what I do, but now I don't feel the pressure of fighting with the software instead of fighting the problem I need to address. :)
Woah. Today with all the readily available 3D Software it's kinda easy to forget that Pixar actually engineered all this Computer rendering stuff themselves. I can't say I understood what he explained, but still I can't wait for more.
this reminds me of my Graphics elective class back in college where our final project was to create a model (animation optional) in C++ (openGL) Anyway, i remember when our prof was introducing us to Bézier curve, which is what the "midpoint split" you see at the start. Our prof told us the exact same concept is used in carpentry to curve sharp corners
Don't forget Car Design... in fact, the reason they're called Bezier curves wasn't because he made them, but he publicized/popularized use when designing for auto maker Renault. Another designer (from Citroen) started using them a few years earlier... and the basis for these curves was made by Bernstein in the late 1800s.
To answer your question about them all turning into blobs - they can set a "weight" to individual edges, to make them harder rather than softer - where you want more definition in the model instead of a smooth blob
Interesting observations: Each row of pascals triangle is generated by the convolution of the row above it by 1 1. Each progressive row becomes closer and closer to approximating a perfect gaussian curve [or e^(-x^2) ]
Thank the lords I found this video. I had to pick a job that uses science and mathematics for an essay and I chose animating. I only found small things in other videos, but I got so many notes in this video. THANK THE LORDS.
A great illustration of the principle that mathematical infinity doesn't mathematically exist. "B at infinity" = "(1 A at infinity + 4 B at infinity + 1 C at infinity) : 6" = (1 A at zero + 4 B at zero + 1 C at zero) : 6 (accessible without an infinity of steps)
Watched this like 3 times for some reason. Learned something useful about all rendering. Weird this is unknown by many who use this. tried to explain this to friends in AutoCAD/Image Design and ended up just showing this video.
going to hopefully be doing Computer Visualisation and Animation at Bournemouth Uni, the artistic side of computer animation is awesome but the Maths side is so damn interesting, really glad that were going to be looking at the Maths, not sure whether this'll be covered but hope so, seems really interesting:3
Yay it took me about 45 minutes to code a simple program that can do exactly the same (as the 2-D program he showed) ^^. I found the way it could create quite interesting fractal-like patterns very interesting :). I'll experiment a bit more with it, to see if I can create some interesting mathematical art with it.
This was awesome, I learned SuperA, SuperB, SuperC. Turns out, I have an interest in Math but back in High School I encountered teachers that turned me off from learning math. I also knew that computer animation is Math and I have to start learning Python soon, okie dokie :)
This video is so mind blowing 🤯 I work in 3D now and I had no idea who those guys from "Catmull-Clark Algorithm" for subdivision came from!!! And for me, subdividing polygons is only a matter of clicking a button, I had no idea about that "moving points" part and the "weights" part, I just thought it was an average and that's all 😆
OMG Geri's Game was such a sad movie for me. Even though it had the upbeat music playing, and the man seemed like he was happy, it was just so sad for me to watch it.
Fun fact about Toy Story- the character of Andy was allegedly named after Andy Van Dam, a Brown University professor who mentored many people on the animation team. In one scene, you can find Andy Van Dam's book about computer animation on a bookshelf.
That link in the end to subscribe to computerphile was a bit sneaky. I mean, you press a button and then there's things in you're subscriptions feed. Some people might call it "Voodo Witchcraft".
Huh. I had no prior education to 3D modelling when I got started - I taught myself everything up to what I know today. It's interesting to know that what I do after making my low-poly representation to turn it into a higher detail model is a trick used by this guy as well.
I remember in 1995. I was 14 and watched Toy Story at the cinema with my then 12 year old cousin and we were both constantly going "Woah! That must have taken ages to render! Did you see that reflection?" Yes we were, and still are, nerds =)
Quality material as usual. But please, check your microphone levels. This is not the first time that the sound is clipping a whole lot. You did show us that you are checking your sound by asking what your interviewees had for breakfast. So maybe it’s a post processing issue? Anyway great job, but please do something about the sound issues.
I would have liked to have seen the 1-4-1 ran through the computer to see how smooth of a circle it actually produced. I believe you, it's just the only one you didn't actually show.
Any other hard core pixar fans out there? Now I know their magic and can totally animate a movie as good as Monsters Inc. !!!! Looking forward to the new movies that are coming out!!!!
![I.N "HALLUCINATION" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/n5B5q1Hwt_U/mqdefault.jpg)
"My back pocket". The line and his face while saying it were priceless. Best answer ever lol
This was a super smooth interview. I love how well rehearsed it was, probably because he has explained that stuff dozens and dozens of times. Really nice to watch. (Not to discredit the editing which probably helped too)
lesconrads no discredit taken, Tony is just really good!
Numberphile Well he is American.
James Oldfield What does being American have to relate to any of this
ygyuu7 NVuu
ygyuu7 NVuuio
Numberphile Thanks for this. I work in 3D Animation and have been wondering about the math behind Subdivision and limit surfaces for quite some time.
On a side note, you asked Tony if everything comes out blobized, but in this vid, it doesnt seem like he answered the question the way you were expecting. The answer is no. In the 2d example, if we wanted one corner to be sharper than the others we add 1 or more vertices near the corner vert. The result is the averaging and splitting is tighter where we've added more vertex density resulting in a sharper smoothed corner
To infinity and beyond appears less impressive when you realize infinity is just one step away from where you started.
Wait a moment... Ed Catmull is the President of Pixar? The Catmull-Clark Catmull? I studied the Catmull-Clark subdivision the other year, but I never imagined the person behind it would also be responsible for some of my favourite movies. You sort of imagine an old dude in a small room of some university, when you think about the name behind an algorithm. You don't connect the people coming up with theories with the ones who put them into practice, although it makes so much sense This is awesome! :-)
Yep, same guy. Pixar was founded as the computer graphics research wing of Lucasfilm, and in those first couple years they laid a ton of the groundwork for computer graphics.
And, he went to the University of Utah. Go Utes!
"old dude in a university." lol thats also how i thought of him while learning about shaders, so this also blew my mind.
As a computer science student, I'm thrilled
For those who know a little bit of linear algebra and are wondering about where the (A+4B+C)/6 thing comes from . . .
Going from step 0 to step 1 can be thought of in terms of matrix multiplication like this:
(A1) (4/8 4/8 0/8) (A0)
(B1) = (1/8 6/8 1/8) (B0)
(C1) (0/8 4/8 4/8) (C0)
Or, if we call the matrix M and the vectors x0 and x1, we write x1=M * x0.
The eigenvectors of M are (column) vectors v such that M * v is proportional to v. There are standard techniques for finding these eigenvectors and eigenvalues, which I won't go through here unless someone asks. The eigenvectors for this matrix are:
v0 = (1,1,1), v1=(2,-1,2), and v2=(1,0,-1), (up to normalization, which we are free to choose however we like)
and the eigenvalues (proportionality constants) are a0=1, a1=1/2, and a2=1/4. In other words, the following three equations hold
M*v0 = (1)*v0
M*v1=(1/2)*v1
M*v2=(1/4)*v2
Notice that we can write (A0,B0,C0) as
A0*(v0 + v1 + 3*v2)/6 +
B0*(2*v0 - v1 + 0*v2)/3 +
C0*(v0 + v1 - 3*v2)/6
So, what happens when we take M*(A0,B0,C0)? We can distribute M through the expression above and use our egeinvector equations (e.g., replace M*v2 with [1/4]*v2) to get
M*(A0,B0,C0) =
A0*([1]*v0 + [1/2]*v1 + [1/4]*3*v2)/6 +
B0*([1]*2*v0 - [1/2]*v1 + [1/4]*0*v2)/3 +
C0*([1]*v0 + [1/2]*v1 - [1/4]*3*v2)/6
We can go again . . .
(M^2)*(A0,B0,C0) =
A0*([1^2]*v0 + [(1/2)^2]*v1 + [(1/4)^2]*3*v2)/6 +
B0*([1^2]*2*v0 - [(1/2)^2]*v1 + [(1/4)^2]*0*v2)/3 +
C0*([1^2]*v0 + [(1/2)^2]*v1 - [(1/4)^2]*3*v2)/6
And in general . . .
(M^n)*(A0,B0,C0) =
A0*([1^n]*v0 + [(1/2)^n]*v1 + [(1/4)^n]*3*v2)/6 +
B0*([1^n]*2*v0 - [(1/2)^n]*v1 + [(1/4)^n]*0*v2)/3 +
C0*([1^n]*v0 + [(1/2)^n]*v1 - [(1/4)^n]*3*v2)/6
As n gets large, the terms (1/2)^n and (1/4)^n go to zero and 1^n remains equal to 1, so we get
limit of (M^n)*(A0,B0,C0) as n -> infinity=
A0*(v0)/6 +
B0*(2*v0)/3 +
C0*(v0)/6
= v0*(A0+4B0+C0)/6
Or, back in vector form (recall v0=(1,1,1)):
(A_infinity) ((A0+4B0+C0)/6)
(B_infinity) = ((A0+4B0+C0)/6)
(C_infinity) ((A0+4B0+C0)/6)
So all three points converge to (A0+4B0+C0)/6.
This is amazing! Two of my favourite subjects combined - maths and arts :)
I'd love to have a job like his one day.
Impressive video! You can tell that the man was very happy to share with the world how much math means to the animators.
When I calculated some eigenvalues and eigenvectors in the first term of my math studies, I was bored. I _never_ imagined that the beauty of animated movies depended on it! :)
Great overview, I'm really hoping for more! :)
I'm so jealous you visited Pixar!
***** pou tree wq
+Ami Yamato who wouldn't be :P
Ami Yamato didnt expect to see you here.
This is only comment I've seen so far, come from a channel I know about, pop up in the middle of nowhere of youtube.
Never should I hear the word "simple" in a Numberphile video.
THERE IS A COMPUTERPHILE CHANNEL?!
I WASN'T YET SUBBED TO IT?!
There are a few other channels too!
Owen Prescott As a software engineering student, I'm confident these two pander to me the most.
Ozitiho Well programmers can also enjoy the space exploration, philosophy & physics. I'm only a designer and I'm subscribed to all of them. XD
You're one of today's lucky 10k xkcd.com/1053/
Ozitiho Bah, as a software engineering student, I enjoy learning about other things in my spare time!
13:45 He missed an opportunity to say "To Infinity and Beyond!"
as a 3d modeler, this was really awesome to watch. Much thanks!
Thank goodness 3D Studio Max does most of this math for me O___O
urgh... 3ds max... nope.. not for me
Stefan Ehrenhaus Really? It is a fantastic and powerful program. In my case, it is a perfect middle ground because I do some game asset stuff and some CGI VFX stuff. If I was doing all of one or the other, I'd be on a different program. There is also the fact that I've been using 3Ds for 15 years. What is your preference, and why is 3DS 'urgh' for you?
Kyle Biggs Well, my preference is Maya, im a character TD for film.
Complex deformation is just more accessible in maya than in 3ds, thanks to python and c++ api.
But you sure have your point with 15 years experience, i respect that. Sorry if for any offense.
Kyle Biggs Yeah do it, its fun and worth a try for sure.
Alexander Tamayo I use MODO. Ditched 3DSmax for it because it gave me headaches while I was learning how to model. Never looked back since. I'm still crappy at what I do, but now I don't feel the pressure of fighting with the software instead of fighting the problem I need to address. :)
so, negative numbers result in swastikas? :D
As Always: A video that answered questiones I never asked but always wanted to be answered!
Woah. Today with all the readily available 3D Software it's kinda easy to forget that Pixar actually engineered all this Computer rendering stuff themselves.
I can't say I understood what he explained, but still I can't wait for more.
Wow!! Never realised there was so much maths in animation and art. Fantastic stuff!
this reminds me of my Graphics elective class back in college where our final project was to create a model (animation optional) in C++ (openGL)
Anyway, i remember when our prof was introducing us to Bézier curve, which is what the "midpoint split" you see at the start. Our prof told us the exact same concept is used in carpentry to curve sharp corners
Don't forget Car Design... in fact, the reason they're called Bezier curves wasn't because he made them, but he publicized/popularized use when designing for auto maker Renault. Another designer (from Citroen) started using them a few years earlier... and the basis for these curves was made by Bernstein in the late 1800s.
No one seems to have called this out yet, but "Music by A113n Stewart" on the end card
Wow I couldn't believe that 1,4,1 could make you laugh, sad, cry, and be entertained!
No one made a 'to infinity and beyond' joke? I am disappoint. I am very disappoint.
wtf
brady was the first
Well, numberphile did. It's right above your comment.
@wolgercon did
This was SO COOL! We need some more of this "awesome applications of mathematics" stuff on this channel! :D
Very nice! It's refreshing to see a numberphile video with direct, tangible relations to real-world problems, and have it still be utterly elegant.
To answer your question about them all turning into blobs - they can set a "weight" to individual edges, to make them harder rather than softer - where you want more definition in the model instead of a smooth blob
One of the coolest videos yet. Really awesome you got an Interview with pixar
Interesting observations:
Each row of pascals triangle is generated by the convolution of the row above it by 1 1.
Each progressive row becomes closer and closer to approximating a perfect gaussian curve [or e^(-x^2) ]
Ha! They digitized the hand, get it? Digitized? I'll leave now.
TheHammer I gettit
Thank the lords I found this video. I had to pick a job that uses science and mathematics for an essay and I chose animating. I only found small things in other videos, but I got so many notes in this video. THANK THE LORDS.
Thanks brady for bringing so much cool science stuff in a lot of different fields, kudos to you
So nice! Eigenvalues are almost magical in what they allow you to do.
Perfectly explained, thanks!
The shading is what I wanted to see. Shadows can be very mathematically complex. show us a video on it pls
A great illustration of the principle that mathematical infinity doesn't mathematically exist.
"B at infinity" = "(1 A at infinity + 4 B at infinity + 1 C at infinity) : 6"
= (1 A at zero + 4 B at zero + 1 C at zero) : 6 (accessible without an infinity of steps)
Watched this like 3 times for some reason. Learned something useful about all rendering. Weird this is unknown by many who use this. tried to explain this to friends in AutoCAD/Image Design and ended up just showing this video.
HACKED there are 33 likes and 0 dislikes 33+0 = 330 but there are only 301 views
haly shit the 9/11 is behind it
If anybody takes this seriously, i'm gonna have to punch a wall.
nah 330 == "33" + "0"
rsplayer56 nah "330" == "33" + "0"
33+0=33
And now I love both math and cartoons even more.
This was really fascinating. Thank you, Pixar!
No wonder Pixar wins so many awards. They spend years developing a quality film.
Thanks for videos numberphile
going to hopefully be doing Computer Visualisation and Animation at Bournemouth Uni, the artistic side of computer animation is awesome but the Maths side is so damn interesting, really glad that were going to be looking at the Maths, not sure whether this'll be covered but hope so, seems really interesting:3
I agree. Is that what you'll be learning with your major that math behind it or the artistic side?
Omar Arias both, plus some computing:3
I`m amateur 3D animator and i didn`t realize before this video how interesting is Autodesk`s offered smoothing tool for 3ds max and maya.
I love this interview
love the A113 double easter egg at the end!!
Wow, he's really really good at explaining this.
He's explaining in a very clear way. Thanks.
Yay it took me about 45 minutes to code a simple program that can do exactly the same (as the 2-D program he showed) ^^. I found the way it could create quite interesting fractal-like patterns very interesting :). I'll experiment a bit more with it, to see if I can create some interesting mathematical art with it.
That was glorious.
This is a fantastic video, I used to do computer graphics as a hobby and I always wondered how these processes worked.
The averaging of points reminds me of affine combination (in my linear algebra textbook, but not taught in my course).
ty daddy pixar ♥
love your work ♡
Awesome job! One of my favorite Numberphile episodes to date!
Always nice to see a vid of yours!
One of your best videos! Maybe even better then when you ran over the calculator.
Tony DeRose is somehow, mysteriously, a near-exact personification of a Pixar employee I had in my mind since I came to know about Pixar :)
This was awesome, I learned SuperA, SuperB, SuperC. Turns out, I have an interest in Math but back in High School I encountered teachers that turned me off from learning math. I also knew that computer animation is Math and I have to start learning Python soon, okie dokie :)
"Where did you get 1, 4, 1 again?
"From by back pocket."
"Okay!, look here smart ass." Interviewer probably
This makes me so happy and curious. Exciting field.
This video is so mind blowing 🤯 I work in 3D now and I had no idea who those guys from "Catmull-Clark Algorithm" for subdivision came from!!! And for me, subdividing polygons is only a matter of clicking a button, I had no idea about that "moving points" part and the "weights" part, I just thought it was an average and that's all 😆
Another special video, Haran!
Really exciting stuff, Brady! You must have loved it there...
Amazing video, please keep making more of these.
More computer graphics / geometric calculation math videos please.
Especially anything done by Edwin Catmull or similar fields!!!
OMG Geri's Game was such a sad movie for me. Even though it had the upbeat music playing, and the man seemed like he was happy, it was just so sad for me to watch it.
I never would have guessed all this was needed to make a movie.
9:22 is the blue dot the last stadium or what happens when you do the SPLIT once again? It remains the same?
Absolutely brilliant
So cool. Geometry is awesome.
Fun fact about Toy Story- the character of Andy was allegedly named after Andy Van Dam, a Brown University professor who mentored many people on the animation team. In one scene, you can find Andy Van Dam's book about computer animation on a bookshelf.
Was this Andries Van Dam ?
SerBallister
Yeah, that's him.
Wow, interesting fact. One of the first CGI books I studied was one of his.
That was superb!
That link in the end to subscribe to computerphile was a bit sneaky. I mean, you press a button and then there's things in you're subscriptions feed. Some people might call it "Voodo Witchcraft".
Beautifully done!
Huh. I had no prior education to 3D modelling when I got started - I taught myself everything up to what I know today. It's interesting to know that what I do after making my low-poly representation to turn it into a higher detail model is a trick used by this guy as well.
Uh, Ed Catmull invented the trick. See Catmull-Clark subdivision.
This is awesome!!
Really cool video....Tony DeRose also gave a TED talk on this subject...not as detailed though!
Fantastic video, keep them up!!
5:00 : this is not an emergent property, he pretty much described a spline
I'd love the ability to play around with this software, to see exactly how different weights turn out.
So nice!!!
This is awesome!
Great video!
I remember in 1995. I was 14 and watched Toy Story at the cinema with my then 12 year old cousin and we were both constantly going "Woah! That must have taken ages to render! Did you see that reflection?" Yes we were, and still are, nerds =)
The music is fantastic
Whoah that was so interesting! :D Thank you!
Very interesting and well presented.
Quality material as usual. But please, check your microphone levels. This is not the first time that the sound is clipping a whole lot. You did show us that you are checking your sound by asking what your interviewees had for breakfast. So maybe it’s a post processing issue?
Anyway great job, but please do something about the sound issues.
Very interesting.
Very well explained, thank you!
When should we expect the 2nd video in this series? Awesome stuff! Thanks!
I would have liked to have seen the 1-4-1 ran through the computer to see how smooth of a circle it actually produced. I believe you, it's just the only one you didn't actually show.
This is fascinating stuff.
Is there a place where you can play with that software interface? That would be fun for my students (and perhaps educational, as well).
Just so cool!
Truly interesting stuff. Love it.
Somehow this reminded me of your "floating body" video but without the floating.
I was really hoping he was going to discuss the mathematics behind radiosity!
Any other hard core pixar fans out there? Now I know their magic and can totally animate a movie as good as Monsters Inc. !!!! Looking forward to the new movies that are coming out!!!!
Super, thanks for this!
Pretty awe inspiring. What do you need magic for when you got Math.