Pixar: The math behind the movies - Tony DeRose

Translator: H_L Au Reviewer: 潘 可儿

At Pixar, we're all about telling stories,

喺彼思，我哋做嘅就係講故事

but one story that hasn't been told very much

但係有個故事冇乜人講

is the huge degree to which math is used

就係製作電影嘅時候

in the production of our films.

我哋會幾常用到數學

The math that you're learning in

你初、高中學到嘅數學

middle school and high school

is used all the time at Pixar.

彼思成日都會用到

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

我哋就由一個簡單嘅例子講起

Anybody recognize this guy? (Cheers)

有冇人認得佢 （歡呼聲）

Yeah, so this is Woody from Toy Story,

係呀，反斗奇兵嘅胡迪

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

我哋試下叫胡迪，行過個舞台

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

由左至右，好似咁

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

信唔信都好，頭先你睇到咗好多數學

Where is it?

喺邊呢？

Well, to explain that,

我哋首先要明白

it's important to understand

that artists and designers think in terms of

藝術家同設計師

shape and images

係用圖形同圖像來諗嘢

but computers think in terms of numbers and equations.

而電腦就係用數字同方程式來諗嘢

So, to bridge those two worlds

為咗連繫呢兩個世界

we use a mathematical concept called

我哋要用一個數學概念

coordinate geometry, right?

叫坐標幾何

That is, we lay down a coordinate system

我哋會用個坐標系統

with x describing how far something is to the right

用 x 表示要向右行幾多

and y describing how high something is.

用 y 表示要去幾高

So, with these coordinates we can describe

有咗呢啲坐標，我哋就可以

where Woody is at any instant in time.

無論幾時，都即刻講明胡迪喺邊

For instance, if we know the coordinates of

就好似我哋知道

the lower left corner of that image,

喺左下角幅圖嘅坐標

then we know where the rest of the image is.

咁就知淨低幅圖喺邊

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

頭先見到嗰小小滑動嘅動畫

that motion we call translation,

我哋叫呢個動作做平移

the x coordinate started with a value of one,

x 座標值會由 1 開始

and it ended with a value of about five.

去到大約 5 就停

So, if we want to write that in mathematics,

如果我哋用數學嚟寫

we see that the x at the end is four bigger

最尾個 x

than x at the start.

就大個開頭個 x

So, in other words, the mathematics of translation

所以，平移嘅公式

is addition.

就係加法

Alright?

係咪？

How about scaling?

咁縮放呢？

That is making something bigger or smaller.

就係令一樣嘢變大或者變細

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

估下縮放用咩數學？

Dilation, multiplication, exactly.

擴大、乘法，冇錯啦

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

如果你要令一樣嘢變大兩倍

you need to mulitply the x and the y coordinates

你要將 x 同 y 嘅坐標

all by two.

全部乘 2

So, this shows us that the mathematics of scaling

所以，縮放嘅數學

is mulitiplication.

就係乘法

Okay?

明唔明？

How about this one?

咁呢個呢？

How about rotation? Alright, spinning around.

旋轉又點？轉嚟轉去

The mathematics of rotation is trigonometry.

旋轉嘅數學就係三角函數

So, here's an equation that expresses that.

呢到有個公式可以用嚟表達

It looks a little scary at first.

初初睇會有啲得人驚

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

你可能喺中二或者中三時學過

If you find yourselves sitting in trigonometry class

如果你學緊三角函數

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

諗到底幾時會需要用到呢啲嘢

just remember that any time you see anything rotate

你就記住，喺我哋任何一套戲入面

in one of our films,

任何嘢旋轉

there's trigonometry at work underneath.

背後就有三角函數啦

I first fell in love with mathematics in seventh grade.

我中一嗰時愛上咗數學

Any seventh graders? A few of you? Yeah.

有冇人喺中一呀？有幾個？

My seventh grade science teacher showed me

我中一班嘅科學老師畀我睇過

how to use trigonometry to compute

點樣用三角函數

how high the rockets that I was building was going.

計算我整緊嘅火箭會飛到幾高

I just thought that was amazing,

我覺得好神奇

and I've been enamored with math ever since.

從此就愛上咗數學

So, this is kind of old mathematics.

呢啲數學有啲古老

Mathematics that's been known and, you know,

我哋都認為

developed by the old dead Greek guys.

數學係由好耐以前嘅 古希臘人發明嘅

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

傳聞就係

mathematics has already been figured out,

有趣嘅數學已經全部畀研究晒啦

in fact all of mathematics has been figured out.

又或者，所有數學都已經畀諗晒出来

But the real story is that new mathematics

但事實係，新嘅數學

is being created all the time.

仲係無時無刻咁被創造

And some of it is being created at Pixar.

有啲係喺彼思度創造出嚟嘅

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

我想畀個例子你哋

So, here are some characters

呢度有啲我哋早期嘅電影角色

from some of our early films:

Finding Nemo, Monsters Inc. and Toy Story 2.

《海底奇兵》、 《怪獸公司》、《反斗奇兵 2》

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

有冇人識左上角個藍色角色？

It's Dory. Okay, that was easy.

係多莉，好容易睇出来

Here's a little harder one.

問條難少少嘅

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

有冇人識右下角嘅角色？

Al McWhiggin from Al's Toy Barn, exactly.

「艾爾玩具城」老闆「艾爾」，啱

The thing to notice about these characters

呢啲角色

is they're really complicated.

佢哋其實真係好複雜

Those shapes are really complicated.

啲形狀好複雜

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

比如話個玩具清潔員

the toy cleaner there in the middle,

中間嗰個

here's his hand.

呢隻係佢隻手

You can imagine how fun it was to bring this

帶佢過機場保安

through airport security.

就真係夠晒好玩

His hand is a really complicated shape.

隻手嘅形狀真係好複雜

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

唔只係一堆球體和圓柱體黐埋

And not only is it complicated,

唔只佢本身複雜

but it has to move in complicated ways.

連佢移動嘅方式都好複雜

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

我同你講下，我哋係點做到嘅

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

首先，要講下重點係咩先

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

呢到有兩點，A 同 B

and the line segment between them.

兩點中間有一條線

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

我哋要由二維開始

The midpoint, M, is the point

中點 M 喺呢度

that splits that line segment in the middle, right?

將條線平分

So, that's the geometry.

呢個就係幾何學

To make equations and numbers,

要寫成方程式同數字

we again introduce a coordinate system,

我哋再次用個座標系統

and if we know the coordinates of A and B,

如果我哋知 AB 嘅座標

we can easily compute the coordinates of M

我哋就計到 M 嘅座標

just by averaging.

用平均法就得啦

You now know enough to work at Pixar.

知道呢哋嘢， 已經夠你喺彼思做嘢

Let me show you.

我展示畀你睇

So, I'm going to do something slightly terrifying

我要做一樣有少少得人驚嘅嘢

and move to a live demo here.

喺到做個現場示範

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

我有個四角形喺到

and it's going to be my job

我係要令佢

to make a smooth curve out of this thing.

有圓滑嘅曲線

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

而我淨係用中點就做到

So, the first thing I'm going to do

首先，我會做一個

is an operation I'll call split,

叫做分割嘅操作

which adds midpoints to all those edges.

即係將所有嘅邊加上中點

So, I went from four points to eight points,

4 個中點就變到 8 個點

but it's no smoother.

但係條線仲未夠圓滑

I'm going to make it a little bit smoother

我要令佢圓滑多少少

by moving all of these points from where they are now

將呢啲點移去

to the midpoint of their clockwise neighbor.

原本位置順時針隔籬線嘅中點

So, let me animate that for you.

做個動畫畀你睇

I'm going to call that the averaging step.

我會叫佢做平均步驟

So, now I've got eight points,

咁宜家我有 8 點

they're a little bit smoother,

佢哋圓滑咗少少

my job is to make a smooth curve,

我要成條圓滑嘅曲線

so what do I do?

咁我要點做

Do it again. Split and average.

再嚟一次，分割同平均

So, now I've got sixteen points.

咁宜家我就有 16 點

I'm going to put those two steps,

我會將分割同平均呢兩個步驟

split and average, together into something

合成一樣

I'll call subdivide,

就叫做細分嘅嘢

which just means split and then average.

即係分割之後再摞平均值

So, now I've got 32 points.

宜家我有 32 點

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

如果仲未夠圓滑，我就再整

I'll get 64 points.

就會有 64 點

Do you see a smooth curve appearing here from

見唔見到呢個更圓滑嘅曲線

those original points?

來自呢哋原來嘅點

And that's how we create the shapes

我哋就係咁創造我哋嘅角色嘅外型

of our charcters.

But remember, I said a moment ago

但記住，我頭先講咗

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

淨係了解靜止嘅形狀

the fixed shape.

固定嘅形狀係唔夠嘅

We need to animate it.

我哋要令佢郁起身

And to animate these curves,

而要郁呢啲曲線

the cool thing about subdivision.

就係細分最勁嘅嘢

Did you see the aliens in Toy Story?

見過《反斗奇兵》入面嘅三眼仔？

You know that sound they make,

你知佢哋會咁叫

"Ooh"? Ready?

「噢」準備好未？

So, the way we animate these curves

咁我哋要郁呢啲曲線呢

is simply by animating the original four points.

好簡單咁郁原本嗰 4 點

"Ooh."

「噢」

Alright, I think that's pretty cool,

好，我覺得咁好勁

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

如果你唔覺嘅，門口就喺嗰邊

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

因為冇得再勁啦

This idea of splitting and averaging

分割同平均嘅概念

also holds for surfaces.

適用於所有表面

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

所以分割，平均

I'll split, and I'll average.

分割，平均

Put those together into subdivide,

放埋一齊去細分

and this how we actually create the shapes

咁我哋就創造到

of all of our surface characters in three dimensions.

所有嘅人物嘅3D 外型

So, this idea of subdivision

細分呢個概念

was first used in a short film in 1997

喺 1997 年一套叫《棋局》嘅

called Geri's Game.

短動畫度首次採用

And Geri actually made a cameo apperance

主角基里後尾有喺 《反斗奇兵2》度客串

in Toy Story 2 as the toy cleaner.

佢就係嗰個玩具清潔員

Each of his hands

佢嘅兩隻手

was the first time we ever used subdivision.

就係我哋第一次用細分整嘅

So, each hand was a subdivision surface,

每隻手都係細分嘅表面

his face was a subdivision surface,

佢塊面係細分嘅表面

so was his jacket.

佢件外套都係

Here's Geri's hand before subdivision,

呢隻係佢細分前隻手

and here's Geri's hand after subdivision,

呢隻係佢細分後隻手

so subdivision just goes in and smooths out

所以細分將

all those facets,

所有表面都整到平滑晒

and creates the beautiful surfaces

創造咗你喺螢幕同戲院

that you see on the screen and in the theaters.

睇到哋咁靚嘅表面

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

嗰次之後，我哋就一直咁樣創造人物

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

呢個係《勇敢傳說》嘅主角梅蘭達

Her dress was a subdivision surface,

佢條裙係細分嘅表面

her hands, her face.

佢對手、佢塊面

The faces and hands of all the clansman

全部族人嘅面同手

were subdivision surfaces.

都係細分嘅表面

Today we've seen how addition, multiplication,

今日我哋講到加法、乘法

trigonometry and geometry play a roll in our films.

三角函數同幾何點樣喺電影裡邊應用

Given a little more time,

如果我有多少少時間

I could show you how linear algebra,

我就可以畀你睇到線性代數

differential calculus, integral calculus

微分同積分

also play a roll.

起到嘅作用

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

今日我最主要係想你哋知道

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

你哋宜家所學到嘅所有數學

in high school and actually up through sophomore college

高中到大學二年級嘅

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

每一刻、每一日，喺彼思都會用到

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