Pixar: The math behind the movies - Tony DeRose

翻译人员: Fen Guan 校对人员: Peipei Xiang

At Pixar, we're all about telling stories,

在皮克斯动画，我们一直在讲故事，

but one story that hasn't been told very much

但有一个故事一直没怎么讲，

is the huge degree to which math is used

那就是数学在我们的电影生产中的

in the production of our films.

巨大应用。

The math that you're learning in

大家在初中、高中学的

middle school and high school

数学

is used all the time at Pixar.

在皮克斯动画中无处不在。

So, let's start with a very simple example.

让我们从一个简单的例子开始。

Anybody recognize this guy? (Cheers)

有人认识这个人物么？（欢呼声）

Yeah, so this is Woody from Toy Story,

对，这是《玩具总动员》里的胡迪。

and let's ask Woody to, say, walk across the stage

现在让胡迪穿过舞台荧幕，

from, say, left to right, just like that.

从左到右，就像这样。

So, believe it or not, you just saw a ton of mathematics.

你信不信，你刚刚看到了很多很多的数学。

Where is it?

在哪儿呢？

Well, to explain that,

要解释这一点，

it's important to understand

很重要的一点是明白

that artists and designers think in terms of

艺术家和设计者

shape and images

是用形状和图案来构思，

but computers think in terms of numbers and equations.

而电脑则是通过数字和方程来工作。

So, to bridge those two worlds

所以，要连接这两个世界，

we use a mathematical concept called

我们要用一个叫做

coordinate geometry, right?

坐标几何的概念，对吧？

That is, we lay down a coordinate system

就是说，我们制定一个坐标系统，

with x describing how far something is to the right

用 x 形容物体在右边的距离，

and y describing how high something is.

用 y 形容物体的高度。

So, with these coordinates we can describe

这样，通过这些坐标我们就可以形容

where Woody is at any instant in time.

任意时刻胡迪的位置。

For instance, if we know the coordinates of

比如，如果我们知道

the lower left corner of that image,

图形左下角的坐标，

then we know where the rest of the image is.

我们就知道整个图形的位置。

And in that little sliding animation we saw a second ago,

在之前看到的短动画中，

that motion we call translation,

那个动作我们叫做平移，

the x coordinate started with a value of one,

x 坐标从 1 开始

and it ended with a value of about five.

以 5 结束。

So, if we want to write that in mathematics,

所以，如果我们用数学书写，

we see that the x at the end is four bigger

我们看到 x 最后比

than x at the start.

起始位置大了 4 个单位。

So, in other words, the mathematics of translation

换句话说，数学平移

is addition.

是加法，

Alright?

对吧？

How about scaling?

那么什么是比例缩放？

That is making something bigger or smaller.

比例缩放是把物体放大或缩小，

Any guesses as to what the mathematics of scaling might be?

那么缩放用数学怎么表达？

Dilation, multiplication, exactly.

扩大，乘法，完全正确。

If you're going to make something twice as big,

如果你想把物体放大两倍，

you need to mulitply the x and the y coordinates

需要把 x 和 y 坐标都

all by two.

乘以 2。

So, this shows us that the mathematics of scaling

这就是说比例缩放用数学表示

is mulitiplication.

就是乘法。

Okay?

好。

How about this one?

那这个呢？

How about rotation? Alright, spinning around.

旋转呢？这样，转圈。

The mathematics of rotation is trigonometry.

旋转的数学是三角学。

So, here's an equation that expresses that.

这是描述旋转的方程。

It looks a little scary at first.

第一眼看起来有些吓人，

You'll probably get this in eighth or ninth grade.

你可能会在八或九年级的时候学到这个。

If you find yourselves sitting in trigonometry class

如果你怀疑你在三角学的课上

wondering when you're ever going to need this stuff,

学的那些东西到底什么时候能用到，

just remember that any time you see anything rotate

就想想我们的电影，

in one of our films,

你看到的每一次旋转

there's trigonometry at work underneath.

都是利用三角学的知识。

I first fell in love with mathematics in seventh grade.

我在七年级的时候爱上了数学。

Any seventh graders? A few of you? Yeah.

这里有七年级学生么？有一些？有。

My seventh grade science teacher showed me

我的七年级科学老师告诉我

how to use trigonometry to compute

怎么用三角学计算

how high the rockets that I was building was going.

我做的火箭可以飞行的高度。

I just thought that was amazing,

我当时就觉得太不可思议了，

and I've been enamored with math ever since.

从那开始我就被数学迷住了。

So, this is kind of old mathematics.

那么这些是相对较老的数学，

Mathematics that's been known and, you know,

就是我们所知道的

developed by the old dead Greek guys.

由古希腊人发明的数学。

And there's a myth out there that all the interesting

有传言说所有的有趣的

mathematics has already been figured out,

数学计算都已经被发现，

in fact all of mathematics has been figured out.

甚至所有的数学都已经被研究完了。

But the real story is that new mathematics

但事实是新的数学

is being created all the time.

在不断地被创造出来。

And some of it is being created at Pixar.

皮克斯就创造了一些。

So, I'd like to give you an example of that.

我可以给你一个例子。

So, here are some characters

这里有一些我们早期电影

from some of our early films:

的一些人物：

Finding Nemo, Monsters Inc. and Toy Story 2.

海底总动员，怪兽电力公司，玩具总动员 2。

Anybody know who the blue character in the upper left is?

有人知道左上方的蓝色人物么？

It's Dory. Okay, that was easy.

是多利。好，这个很简单。

Here's a little harder one.

来一个难点的，

Anybody know who's the character in the lower right?

有人知道左下方的人物么？

Al McWhiggin from Al's Toy Barn, exactly.

艾尔玩具仓的主人艾尔，非常正确。

The thing to notice about these characters

值得注意的是，这些人物

is they're really complicated.

其实很复杂。

Those shapes are really complicated.

那些形状真的很复杂。

In fact, the toy cleaner, I have an example,

事实上，那个玩具清理工，我有个模型，

the toy cleaner there in the middle,

中间的玩具清理工，

here's his hand.

这是他的手。

You can imagine how fun it was to bring this

你可以想象带这个过机场安检

through airport security.

是怎样的情形。

His hand is a really complicated shape.

他的手是一个非常复杂的结构。

It's not just a bunch of spheres and cylinders stuck together, right?

它不仅仅是一些球体和圆柱体粘在一起那么简单，是吧？

And not only is it complicated,

它不仅复杂，

but it has to move in complicated ways.

它还有很复杂的移动。

So, I'd like to tell you how we do that,

我想告诉你我们是怎么移动它的，

and to do that I need to tell you about midpoints.

在这之前，我得告诉你中点的定义。

So, here's a couple of points, A and B,

这是两个点，A 和 B，

and the line segment between them.

和连接两点的线段。

We're going to start out first in two dimensions.

我们从二维平面开始。

The midpoint, M, is the point

中点 M 是

that splits that line segment in the middle, right?

把线段均分为两段的点，对吧？

So, that's the geometry.

这是几何学。

To make equations and numbers,

要用方程和数字表达，

we again introduce a coordinate system,

我们需要建立坐标系统，

and if we know the coordinates of A and B,

如果我们知道 A 和 B 的坐标，

we can easily compute the coordinates of M

我们可以通过求平均很容易算出

just by averaging.

M 点的坐标。

You now know enough to work at Pixar.

大家现在已经具备足够在皮克斯工作的知识了。

Let me show you.

让我证明给你看。

So, I'm going to do something slightly terrifying

接下来我会做一个让人稍微有些紧张的

and move to a live demo here.

现场示范。

So, what I have is a four-point polygon here,

看屏幕上，我有一个四点的多边形，

and it's going to be my job

接下来我的工作就是将它变成

to make a smooth curve out of this thing.

一段光滑曲线。

And I'm going to do it just using the idea of midpoints.

我只需要通过取中点就可以做到。

So, the first thing I'm going to do

所以第一件事就是

is an operation I'll call split,

进行一个分裂的操作，

which adds midpoints to all those edges.

就是在所有边上加中点。

So, I went from four points to eight points,

这样，我从四个点得到八个点，

but it's no smoother.

但这并没有让图形更光滑。

I'm going to make it a little bit smoother

我将把所有点

by moving all of these points from where they are now

从当前位置移到它们的顺时针相邻中点，

to the midpoint of their clockwise neighbor.

把它变得更光滑。

So, let me animate that for you.

就如这个动画展示。

I'm going to call that the averaging step.

我称这一步为平均步骤。

So, now I've got eight points,

现在我有八个点，

they're a little bit smoother,

有一点光滑了，

my job is to make a smooth curve,

我的目标是做成光滑曲线，

so what do I do?

那接下来该做什么呢？

Do it again. Split and average.

重复以上步骤。分裂、平均。

So, now I've got sixteen points.

这样，我得到了16个点。

I'm going to put those two steps,

把上面两个步骤，

split and average, together into something

分裂和平均，合在一起

I'll call subdivide,

我称之为“细分”，

which just means split and then average.

就是分裂再平均的意思。

So, now I've got 32 points.

现在我有32个点。

If that's not smooth enough, I'll do more.

如果还不够光滑，那就再重复。

I'll get 64 points.

现在变成64个点。

Do you see a smooth curve appearing here from

看到如何从最初四点得到

those original points?

一段光滑曲线了么？

And that's how we create the shapes

这就是我们创造我们的

of our charcters.

人物外形的方法

But remember, I said a moment ago

但记住我之前说的，

it's not enough just to know the static shape,

仅仅知道静态图形、

the fixed shape.

固定图形是不够的，

We need to animate it.

我们需要让它动起来。

And to animate these curves,

要让这些曲线动起来，

the cool thing about subdivision.

这也是细分厉害的地方。

Did you see the aliens in Toy Story?

看过玩具总动员的外星人吧？

You know that sound they make,

记得他们的声音

"Ooh"? Ready?

“哇喔”吗？准备好了么？

So, the way we animate these curves

让这些曲线动起来的方法

is simply by animating the original four points.

很简单，就是移动最初的四个点，

"Ooh."

“哇喔”。

Alright, I think that's pretty cool,

好了，我觉得这非常了不起。

and if you don't, the door is there,

如果你不这么觉得，你可以离开了，

it doesn't get any better than that, so.

因为没有比这更酷的了。

This idea of splitting and averaging

分裂和平均的方法

also holds for surfaces.

对于三维表面也成立。

So, I'll split, and I'll average.

分裂，平均。

I'll split, and I'll average.

分裂，平均。

Put those together into subdivide,

不断细分，

and this how we actually create the shapes

这就是我们创造所有

of all of our surface characters in three dimensions.

三维人物外形的方法。

So, this idea of subdivision

细分的方法

was first used in a short film in 1997

最先用在1997年的一个叫做

called Geri's Game.

《棋逢对手》的短片里。

And Geri actually made a cameo apperance

里面的格里在玩具总动员 2 里

in Toy Story 2 as the toy cleaner.

特别演出了玩具清理工一角。

Each of his hands

他的双手

was the first time we ever used subdivision.

是我们的第一个运用细分的案例。

So, each hand was a subdivision surface,

每只手都是一个细分的表面，

his face was a subdivision surface,

他的脸也是细分表面，

so was his jacket.

还有他的外套。

Here's Geri's hand before subdivision,

这是细分前格里的手，

and here's Geri's hand after subdivision,

这是细分后的手，

so subdivision just goes in and smooths out

所以细分让

all those facets,

所有表面更光滑，

and creates the beautiful surfaces

细分创造了你在屏幕和电影院

that you see on the screen and in the theaters.

看到的漂亮外形。

Since that time, we've built all of our characters this way.

从那开始，我们用这种方法创造了我们的所有人物。

So, here's Merida, the lead character from Brave.

这是《勇敢传说》里的主要人物，梅丽达。

Her dress was a subdivision surface,

她的连衣裙是细分的表面，

her hands, her face.

她的手，她的脸也是。

The faces and hands of all the clansman

所有族人的脸和手

were subdivision surfaces.

都是细分表面。

Today we've seen how addition, multiplication,

今天我们看到了，加法，乘法，

trigonometry and geometry play a roll in our films.

三角学和几何学在我们电影里的作用。

Given a little more time,

如果再给我一点时间，

I could show you how linear algebra,

我还可以告诉大家线性代数，

differential calculus, integral calculus

微分，积分

also play a roll.

发挥的作用。

The main thing I want you to go away with today is

今天想让大家记住的最主要的就是，

to just remember that all the math that you're learning

你在高中到大二所学的

in high school and actually up through sophomore college

所有数学，

we use all the time, everyday, at Pixar. Thanks.

我们在皮克斯一直在用，每天都在用。谢谢