交互式探索：四元数与3D旋转：英语精读讲义

In a moment, I'll point you to a separate website hosting a short sequence of what we're calling explorable videos. 用詞

稍后，我会向大家介绍一个独立的网站，里面托管了一系列我们称之为“可探索视频”的短片。

It was done in collaboration with Ben Eater, 短語搭配

这是与Ben Eater合作完成的，

who some of you may know is that guy who runs the excellent computer engineering channel.

你们中有些人可能知道，他就是那个运营优秀计算机工程频道的家伙。

And if you don't know who he is,

如果你不知道他是谁，

viewers of this channel would certainly enjoy the content of his.

这个频道的观众肯定会喜欢他的内容的。

So do check it out. 講解

所以一定要去看看。

This collaboration was something a little different though,

不过，这次合作有些与众不同，

for both of us.

对我们俩来说都是如此。

And all of the web development that made these explorable videos possible

所有让这些可探索视频成为可能的网页开发工作，

is completely thanks to Ben.

都要完全归功于Ben。

I don't want to say too much about it here.

我不想在这里说太多。

It's really something you have to experience for yourself.

这真的是你需要亲自去体验的东西。

Certainly one of the coolest projects I've had the pleasure of working on. 短語搭配

这绝对是我有幸参与过的最酷的项目之一。

Before that though,

不过在此之前，

if you can contain your excitement, 語氣意圖

如果你能按捺住激动的心情，

I want to use this video as a chance to tee things up with a little bit of surrounding context. 短語動詞

我想借这期视频的机会，通过一些背景知识来做个铺垫。

So to set the stage, 習語

为了切入正题，

last video I described quaternions,

上期视频我介绍了四元数，

a certain four-dimensional number system

这是一个特定的四维数字系统，

that the 19th century versions of Wolverine and the old man from Home Alone

被19世纪版的金刚狼和《小鬼当家》里的老爷爷

called evil for how convoluted it seemed at the time.

称其为“邪恶”，因为在当时它看起来太复杂了。

And perhaps you too are wondering

也许你也会好奇，

why on earth anyone would bother with such an alien seeming number system. 講解

到底为什么有人要费心去弄这种看似外星产物的数字系统。

Well,

那么，

one of the big reasons,

最大的原因之一，

especially for programmers,

特别是对于程序员来说，

is that they give a really nice way for describing 3D orientation,

是它们提供了一种非常好的描述3D朝向的方法，

which is not susceptible to the bugs and edge cases of other methods. 講解

并且不易受到其他方法中的错误和边界情况的影响。

I mean,

我的意思是，

they're interesting mathematically for a lot of reasons,

虽然它们在数学上因很多原因而很有趣，

but this application for computer graphics and robotics and virtual reality

但这种在计算机图形学、机器人技术、虚拟现实

and anything involving 3D orientation is probably the biggest use case for quaternions. 用詞

以及任何涉及3D朝向的领域中的应用，可能是四元数最大的用武之地。

To take one example, 表達脈絡

举个例子，

a friend of mine who used to work at Apple, 語法

我的一位曾在苹果工作过的朋友，

Andy Matuschak,

Andy Matuschak，

delighted in telling me about shipping code to hundreds of millions of devices that uses quaternions to track the phone's model for how it's oriented in space. 用詞

很乐意地告诉我，他们发布到数亿台设备上的代码，就是使用四元数来追踪手机在空间中的朝向模型的。

That's right.

没错。

Your phone almost certainly has software running somewhere inside of it that relies on quaternions.

你的手机里几乎肯定在某个地方运行着依赖四元数的软件。

The thing is, 表達脈絡

问题是，

there are other ways to think about computing rotations,

计算旋转还有其他的思考方式，

many of which are way simpler to think about than quaternions. 語法

其中很多方法都比四元数容易理解得多。

For example,

例如，

any of you familiar with linear algebra will know that 3 x 3 matrices can really nicely describe 3D transformations. 語法

熟悉线性代数的人都知道，3x3矩阵可以很好地描述3D变换。

And a common way that many programmers think about constructing a rotation matrix for a desired orientation

对于所需朝向，许多程序员构建旋转矩阵的一种常见思考方式

is to imagine rotating an object around three easy to think about axes,

是想象将一个物体围绕三个易于理解的轴进行旋转，

where the relevant angles for these rotations are commonly called Euler angles.

这些旋转的相关角度通常被称为欧拉角。

And this mostly works,

这在大多数情况下都管用，

but one big problem is that it's vulnerable to something called gimbal lock,

但一大问题是它容易出现所谓的“万向节死锁”，

where when two of your axes of rotation get lined up, 短語動詞

当你的两个旋转轴重合时，

you lose a degree of freedom. 用詞

你就会失去一个自由度。

And it can also cause difficulties and ambiguities

此外，它还可能引起困难和歧义，

when trying to interpolate between two separate orientations. 短語搭配

当试图在两个独立朝向之间进行插值时。

If you're curious for more of the details,

如果你对更多细节感到好奇，

there are many great sources online for learning about Euler angles and gimbal lock,

网上有很多学习欧拉角和万向节死锁的优秀资料，

and I've left links in the description to a few of them.

我在简介里留下了一些链接。

Not only do quaternions avoid issues like gimbal lock, 語法

四元数不仅避免了万向节死锁等问题，

they give a really seamless way to interpolate between two three-dimensional orientations,

它们还提供了一种在两个三维朝向之间进行插值的无缝方法，

one which lacks the ambiguities of Euler angles, 語法

这种方法不仅没有欧拉角的歧义，

and which avoids the issues of numerical precision and normalization

也避免了在尝试对两个旋转矩阵进行插值时，

that arise in trying to interpolate between two rotation matrices.

出现的数值精度和归一化问题。

To warm up to the idea of how multiplication in some higher-dimensional number system 短語動詞

为了熟悉一下在某种高维数字系统中，

might be used to compute rotations,

如何利用乘法来计算旋转的思路，

take a moment to remember how it is that complex numbers give a slick method for computing 2D rotations. 語法

花点时间回想一下，复数是如何为计算2D旋转提供巧妙方法的。

Specifically,

具体来说，

let's say you have some point in two-dimensional space, 表達脈絡

假设你在二维空间中有一个点，

like 4 1,

比如 (4, 1)，

and you want to know the new coordinates you'd get if you rotate this point 30 degrees around the origin. 語法

你想知道将这个点绕原点旋转30度后得到的新坐标。

Complex numbers give sort of a snazzy way to do this.

复数提供了一种相当时髦的方式来做到这一点。

You take the complex number that's 30 degrees off the horizontal with magnitude one,

你取一个偏离水平方向30度且幅度为1的复数，

cosine 30 degrees plus sine of 30 degrees times I,

即 cos(30°) 加上 sin(30°) 乘以 i，

and then you just multiply this by your point,

然后你只需将它乘以你的点，

represented as a complex number.

这里将点表示为一个复数。

The only rule you need to know to carry out this computation is that I squared equals negative one. 短語動詞

执行这一计算你需要知道的唯一规则是 i 的平方等于 -1。

And then,

然后，

in what might feel like a bit of black magic to those first learning it, 講解

对于刚开始学习的人来说，这可能感觉有点像黑魔法，

carrying out this product from that one simple rule gives the coordinates of a new point,

根据那个简单的规则算出这个乘积后，就得到了一个新点的坐标，

the point rotated 30 degrees from the original.

即那个从原来的点旋转了30度之后的点。

Using quaternions to describe 3D rotations is similar,

使用四元数来描述3D旋转也是类似的，

though the look and feel is slightly different. 短語搭配

尽管外观和感觉略有不同。

Let's say you want to rotate some angle about some axis.

假设你想围绕某个轴旋转某个角度。

You first define that axis with a unit vector,

首先，你用一个单位向量定义那个轴，

which we'll write as having I, J, and K components,

我们把它写成包含 i、j 和 k 三个分量，

normalized so that the sum of the squares of those components is one. 用詞

并进行归一化，使得这些分量的平方和等于1。

Similar to the case of complex numbers, 表達脈絡

与复数的情况类似，

you use the angle to construct a quaternion by taking cosine of that angle as the real part,

你用这个角度来构建一个四元数：取该角度的余弦值作为实部，

plus sine of that angle times an imaginary part,

加上该角度的正弦值乘以一个虚部，

except this time the imaginary part has three components,

只是这次虚部有三个分量，

the coordinates of our axis of rotation.

也就是我们的旋转轴的坐标。

Well, 表達脈絡

嗯，

actually you take half of the angle,

实际上你要取角度的一半，

which might feel totally arbitrary,

这可能会让人感觉完全是任意的，

but hopefully that's going to make some sense by the end of this whole experience.

但希望在整个体验结束时，这能让你觉得是有道理的。

Now let's say you have some 3D point,

现在假设你有一个3D点，

which we'll write with I, J, K components,

我们将其用 i、j、k 分量写出来，

and you want to know the coordinates that you'd get when you rotate this point by your specified angle around your specified axis.

你想知道将这个点围绕你指定的轴，旋转你指定的角度后会得到的坐标。

What you do is not just a single quaternion product, 語法

你所做的不仅仅是单一的四元数乘积，

but a sort of quaternion sandwich, 習語

而是一种“四元数三明治”，

where you multiply by Q from the left 短語搭配

你要从左边乘以 Q，

and the inverse of Q from the right.

并从右边乘以 Q 的逆。

If you know the rules for how I, J, and K multiply amongst themselves,

如果你知道 i、j 和 k 之间如何相乘的规则，

you can carry out these two products by expanding everything out,

你可以通过把所有项展开来完成这两个乘积的计算，

or more realistically by having a computer do it for you.

或者更现实一点，让计算机替你算。

And in what might feel like a bit of black magic,

而在一种可能感觉有点像黑魔法的过程中，

this big computation will return for you the rotated version of the point.

这个庞大的计算将为你返回那个点被旋转后的版本。

Our goal is to break this down 短語動詞

我们的目标是把它分解，

and visualize what's happening with each of these two products.

并可视化这两个乘积中每一个步骤发生的事情。

I'll review the method for thinking about quaternion multiplication described last video

我将回顾上期视频中描述的关于四元数乘法的思考方法，

and explain why half the angle is used

并解释为什么使用的是角度的一半，

and why you would multiply from the right by the inverse.

以及为什么要从右边乘以它的逆。

On the screen now and at the top of the description,

现在屏幕上和简介栏的顶部，

you'll find a link to eater.net/quaternions,

你会找到一个链接 eater.net/quaternions，

which is where Ben set up the explorable video tutorial where I explain what's going on with this rotation computation.

Ben 在那里建立了这个可探索的视频教程，我在里面解释了这个旋转计算到底是怎么回事。

It's just really cool.

这真的非常酷。

Eater did something awesome here.

Eater 在这里做了一件了不起的事情。

So at the very least, 短語搭配

所以，最起码的，

just take a couple minutes to go look at it.

花上几分钟去看看它吧。

But I'd love it if you went through the full experience. 語氣意圖

不过如果你能体验一下完整的过程，那就太好了。