Pixar: The math behind the movies - Tony DeRose

譯者: Jesse Chen 陳鉦翰 審譯者: Jheng-Han Chen

At Pixar, we're all about telling stories,

皮克斯擅長說故事

but one story that hasn't been told very much

但有個故事並不廣為人知

is the huge degree to which math is used

就是運用在動畫中

大量的數學

in the production of our films.

The math that you're learning in

你在初中和高中

middle school and high school

所學到的數學

is used all the time at Pixar.

在皮克斯中無所不在

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

舉個很簡單的例子

Anybody recognize this guy? (Cheers)

有人認識這他嗎？

Yeah, so this is Woody from Toy Story,

這是《玩具總動員》裡的「胡迪」

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

讓胡迪走向另一邊

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

就像這樣，從左到右

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

你相信嗎？ 你剛看到了一堆數學

Where is it?

在哪呢？

Well, to explain that,

我解釋一下

it's important to understand

你要知道

that artists and designers think in terms of

藝術家和設計師腦子裡想的

shape and images

都是圖像和意象

but computers think in terms of numbers and equations.

但電腦裡運作的都是數字和方程式

So, to bridge those two worlds

為了要讓兩者結合

we use a mathematical concept called

我們運用了數學概念

coordinate geometry, right?

稱為幾何座標

That is, we lay down a coordinate system

我們設定一個座標系統

with x describing how far something is to the right

x 軸表示物件向右移動的距離

and y describing how high something is.

y 軸則決定物件的高度

So, with these coordinates we can describe

有了座標

就能讓胡迪出現在任何地方

where Woody is at any instant in time.

For instance, if we know the coordinates of

舉例來說 如果我們知道

the lower left corner of that image,

左下角圖像的座標

then we know where the rest of the image is.

那就能知道其他的圖像在哪了

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

剛才看到的動畫

that motion we call translation,

就叫做平移

the x coordinate started with a value of one,

x 座標值從 1 開始

and it ended with a value of about five.

在大約 5 的地方停下

So, if we want to write that in mathematics,

如果用數學的方式表示

we see that the x at the end is four bigger

x 值比剛開始

大四個值

than x at the start.

So, in other words, the mathematics of translation

換言之，平移的數學

is addition.

就是加法

Alright?

對吧？

How about scaling?

那進位法呢？

That is making something bigger or smaller.

功能是讓物件變大或變小

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

猜猜進位法的數學應該是什麼？

Dilation, multiplication, exactly.

擴大、乘法，沒錯

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

如果要把東西放兩倍大

you need to mulitply the x and the y coordinates

x 值和 y 值都要加乘

all by two.

都乘上 2

So, this shows us that the mathematics of scaling

顯示了進位法的數學

is mulitiplication.

就是乘法

Okay?

好的

How about this one?

那這個呢？

How about rotation? Alright, spinning around.

讓圖像旋轉

The mathematics of rotation is trigonometry.

旋轉的數學就是三角函數

So, here's an equation that expresses that.

這是解釋三角函數的方程式

It looks a little scary at first.

乍看有點嚇人

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

你在八或九年級時可能學過

If you find yourselves sitting in trigonometry class

假如你坐在教室裡學三角函數

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

想著以後又不可能用到這東西

just remember that any time you see anything rotate

只要記住 任何你在我們電影裡

in one of our films,

看到會旋轉的東西

there's trigonometry at work underneath.

就關係著三角函數

I first fell in love with mathematics in seventh grade.

我七年級時第一次愛上數學

Any seventh graders? A few of you? Yeah.

現場有七年級生嗎？

有一些是

My seventh grade science teacher showed me

我那時候的自然老師告訴我

how to use trigonometry to compute

怎麼用三角函數

how high the rockets that I was building was going.

計算我正在建造的火箭高度

I just thought that was amazing,

我當時覺得很神奇

and I've been enamored with math ever since.

從此我就對數學死心踏地

So, this is kind of old mathematics.

這有點老數學了

Mathematics that's been known and, you know,

數學是在古希臘時期

developed by the old dead Greek guys.

發展並廣為人知

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

有個傳說

有趣的數學都已被理解

mathematics has already been figured out,

in fact all of mathematics has been figured out.

事實上 是所有數學都已被理解

But the real story is that new mathematics

但新的數學

is being created all the time.

卻是一直持續的創造

And some of it is being created at Pixar.

且有些是皮克斯創造的

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

我要舉個例子

So, here are some characters

這些角色

from some of our early films:

來自一些我們早期的電影

Finding Nemo, Monsters Inc. and Toy Story 2.

《海底總動員》、《怪獸電力公司》、《玩具總動員 2》

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

誰知道左上角那個藍色的角色是誰？

It's Dory. Okay, that was easy.

是「多莉」，這很容易

Here's a little harder one.

這個就比較難了

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

有人知道右下角那個角色是誰嗎？

Al McWhiggin from Al's Toy Barn, exactly.

「艾爾玩具城」的老闆「艾爾」，沒錯

The thing to notice about these characters

注意這些角色

is they're really complicated.

他們都很複雜

Those shapes are really complicated.

他們的形體非常複雜

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

清理玩具的爺爺

我有個樣本

the toy cleaner there in the middle,

畫面中間的是清理玩具的爺爺

這是他的手

here's his hand.

You can imagine how fun it was to bring this

你能想像這隻手過海關

through airport security.

會多麼有趣

His hand is a really complicated shape.

這隻手的形狀很複雜

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

這不只是一堆球體和圓柱體黏在一起

And not only is it complicated,

它不只很複雜

but it has to move in complicated ways.

甚至還要進行複雜的活動

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

我現在告訴你這怎麼做的

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

我要先告訴你什麼是中點

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

這裡有兩個點，A 點和 B 點

and the line segment between them.

兩點間有一條線段

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

我們從兩個面向討論

The midpoint, M, is the point

M 是中點

that splits that line segment in the middle, right?

平分該線段

So, that's the geometry.

這是幾何學

To make equations and numbers,

為了寫出方程式和數字

we again introduce a coordinate system,

我再次介紹座標系統

and if we know the coordinates of A and B,

如果已得知 A 和 B 的座標

we can easily compute the coordinates of M

就可以藉由算出平均

just by averaging.

輕易得到 M 的座標

You now know enough to work at Pixar.

你現在對皮克斯的工作已有足夠了解

Let me show you.

我秀給你看

So, I'm going to do something slightly terrifying

我要做的這個有點嚇人

and move to a live demo here.

移動一個會動的樣本

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

這裡有一個四角形

and it's going to be my job

我現在要把這個

to make a smooth curve out of this thing.

弄成圓滑的曲線

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

只是運用中點的概念

So, the first thing I'm going to do

我首先要做的

is an operation I'll call split,

就是平均分割

which adds midpoints to all those edges.

把各邊都加上中點

So, I went from four points to eight points,

如此 4 個點就變 8 個點

but it's no smoother.

這並沒有使其變圓滑

I'm going to make it a little bit smoother

我現在要讓它圓滑一點

by moving all of these points from where they are now

順時針移動這些點

to the midpoint of their clockwise neighbor.

到鄰近的中點

So, let me animate that for you.

移動一下

I'm going to call that the averaging step.

這就叫做平均線移

So, now I've got eight points,

現在上面有 8 個點

they're a little bit smoother,

變圓滑了一些

my job is to make a smooth curve,

我要把它弄得更圓滑

so what do I do?

我該怎麼做？

Do it again. Split and average.

重複剛才的動作

So, now I've got sixteen points.

現在變16個點

I'm going to put those two steps,

我現在要把平均分割和平均線移

split and average, together into something

兩個步驟合在一起

I'll call subdivide,

稱為細分

which just means split and then average.

意思就是先分割再線移

So, now I've got 32 points.

變 32 個點

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

如果不夠圓滑，就再重複一次

I'll get 64 points.

變 64 個點

Do you see a smooth curve appearing here from

有看到變形後的曲線

those original points?

跟原來四個點的差別嗎？

And that's how we create the shapes

我們就是如此創造出

of our charcters.

各角色的外型

But remember, I said a moment ago

但記得我剛說過

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

只了解這種靜止且固定的形狀

the fixed shape.

是不夠的

We need to animate it.

我們要讓它動起來

And to animate these curves,

讓曲線動起來

the cool thing about subdivision.

這就是細分的厲害

Did you see the aliens in Toy Story?

記得《玩具總動員》裡的三眼外星人嗎？

You know that sound they make,

你認得他們的聲音

"Ooh"? Ready?

「噢～」

So, the way we animate these curves

看好

所以我們要移動這個曲線

is simply by animating the original four points.

就只要移動原本的四個點

"Ooh."

「噢～」

Alright, I think that's pretty cool,

我覺得這很酷

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

如果你不這麼覺得 大門就在那

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

這對我們彼此都好

This idea of splitting and averaging

平均分割與平均線移的概念

also holds for surfaces.

也用在平面外觀

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

先分割再線移

I'll split, and I'll average.

再次分割和線移

Put those together into subdivide,

兩個放在一起就是細分

and this how we actually create the shapes

我們就是如此創造出

of all of our surface characters in three dimensions.

角色的 3D 外型

So, this idea of subdivision

細分的概念

was first used in a short film in 1997

在1997年 我們首次運用在動畫短片

called Geri's Game.

叫做《棋局》

And Geri actually made a cameo apperance

主角「傑瑞」有立體的外表

in Toy Story 2 as the toy cleaner.

他就是《玩具總動員 2》清理玩具的爺爺

Each of his hands

他的兩隻手

was the first time we ever used subdivision.

是我們第一次運用到細分

So, each hand was a subdivision surface,

兩隻手都是細分後的表面

his face was a subdivision surface,

臉部也是

so was his jacket.

他的夾克也是

Here's Geri's hand before subdivision,

這是傑瑞細分前的手

and here's Geri's hand after subdivision,

這是細分之後

so subdivision just goes in and smooths out

所以細分只是讓各個表面

all those facets,

往內縮或往外凸

and creates the beautiful surfaces

而做出漂亮的外觀

that you see on the screen and in the theaters.

就成了你在電影院裡所見的那樣

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

從那之後 我們都是用此法創出角色

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

這是《勇敢傳說》裡的「梅莉達」

Her dress was a subdivision surface,

她穿的衣服就是細分表面

her hands, her face.

雙手、臉都是

The faces and hands of all the clansman

那些族人的臉、手

were subdivision surfaces.

也都是細分表面

Today we've seen how addition, multiplication,

今天看到了我們運用動畫角色上的

trigonometry and geometry play a roll in our films.

加法、乘法、三角函數和幾何學

Given a little more time,

如果再給我一點時間

I could show you how linear algebra,

我就能介紹線性代數

differential calculus, integral calculus

微分和積分

also play a roll.

也如何囊括在內

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

今天我主要是要讓你們知道

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

記得你在高中到大二間

in high school and actually up through sophomore college

所學的數學

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

皮克斯無時無刻都在使用

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