Multiplication by any other quaternion is completely similar.

与任何其他元数乘法完相似。

What I'll show you here is a way to visualize quaternions in their full four-dimensional glory.

我将在这里向您展示一种在完维荣耀中可视化元数方法。

Like all other non-unit quaternions, It's invisible to us.

像所有其他非单位元数一样，它对我们来说是可见。

So the way that this works is slightly more complicated than a single quaternion product.

因此，其工作方式比单个元数乘积稍微复杂一些。

On the other hand, all the unit quaternions with negative real part end up somewhere outside that unit sphere.

另一方面，所有具有负实部单位元数最终都位于该单位球体之外某个地方。

And so far, it's probably not clear how exactly quaternions do describe 3d rotation.

到目前为止，我们可能还清楚元数如何准确地描述 3d 旋转。

That's right, your phone almost certainly has software running inside it that rely on quaternions.

没错，您手机几乎肯定运行着依赖元数软件。

The 20th century also brought quaternions some more love from a completely different direction, quantum mechanics.

20 世纪也从一个完方向——量子力学——给元数带来了更多爱。

Using quaternions to describe 3d rotations is similar, though the look and feel is slightly different.

使用元数描述 3d 旋转是相似，但外观和感觉略有。

What you do is not just a single quaternion product, but a quaternion sandwich, where you multiply by q from the left and the inverse of q from the right.

您所做仅仅是单个元数乘积，而是一个元数三明治， 其中您从左侧乘以 q， 从右侧乘以 q 倒数。

I'll review the method for thinking about quaternion multiplication described last video, explain why half the angle is used, and why you multiply from the right by the inverse.

我将回顾上一个视频中描述思考元数乘法方法，解释为什么使用一半角度，以及为什么从右侧乘以倒数。

The thing is, there are other ways to think about and compute rotations, many of which are way simpler to think about than quaternions.

问题是， 还有其他方法来思考和计算旋转，其中许多方法比元数更容易思考。

It's something analogous to the equator of a 3d sphere, and it represents all of the unit quaternions whose real part is 0. What Hamilton would have described as unit vectors.

它类似于 3d 球体赤道，它代表所有实部为 0 单位元数。 汉密尔顿将其描述为单位向量。

So after his discovery, he pushed hard for quaternions to be the primary language with which we teach students to describe three-dimensional space, even forming an official quaternion society to proselytize his discovery.

因此，在他发现之后， 他极力推动元数成为我们教学生描述三维空间主要语言，甚至组建了一个官方元数协会来宣传他发现。

On the one hand, you could just define quaternion multiplication by giving the rules for how i, j, and k multiply together and saying that everything must distribute nicely.

一方面，您可以通过给出 i、j 和 k 如何相乘规则来定义元数乘法，并规定所有内容必须良好分布。

And multiplying quaternion, q1, by another, q2, has the effect of scaling q2 by the magnitude of q1 followed by a very special type of rotation in four dimensions.

将元数 q1 与另一个 q2 相乘，具有将 q2 按 q1 大小缩放效果，然后进行维旋转非常特殊类型。

And Hamilton used a special word for quaternions that had no real part and just ijk components, a word which was previously somewhat foreign in the lingo of math and physics: " vector" .

汉密尔顿使用了一个特殊词来表示元数，它没有实部，只有 ijk 分量，这个词以前在数学和物理学行话中有点陌生：“向量”。

And just as the magnitude of a complex number—its distance from zero— is the square root of the sum of the squares of its components, that same operation gives you the magnitude of a quaternion.

正如复数大小（距零距离）是其分量平方和平方根一样，样运算也可以得到元数大小。

And indeed, this is how you would tell a computer to perform quaternion multiplication, and the relative compactness of this operation compared to, say, matrix multiplication, is what's made quaternions so useful for graphics programming and many other things.

事实上，这就是告诉计算机执行元数乘法方式，并且与矩阵乘法相比，该操作相对紧凑性使得元数对于图形编程和许多其他事情如此有用。

So j goes to k, k goes to -j, -j goes to -k, and -k goes to j. This gives us a little table for what the number i does to the other quaternions.

因此，j 转到 k， k 转到 -j， -j 转到 -k， -k 转到 j。这为我们提供了一个小表格， 显示 i 对其他元数作用。