blocks|key|1874566|text|它们比旋转矩阵具有更小的内存占用，并且比矩阵和角度/轴表示更有效。|type|unstyled|depth|inlineStyleRanges|entityRanges|data|1874567|另外：|1874568|与矩阵representations.相比，浮点四元数的|1874569|unordered-list-item|1874570|归一化具有更少的舍入缺陷|1874571|1874572|entityMap^0|0|0|0|0|0|0^^$0|@$1|2|3|4|5|6|7|M|8|@]|9|@]|A|$]]|$1|B|3|C|5|6|7|N|8|@]|9|@]|A|$]]|$1|D|3|E|5|6|7|O|8|@]|9|@]|A|$]]|$1|F|3|-4|5|G|7|P|8|@]|9|@]|A|$]]|$1|H|3|I|5|G|7|Q|8|@]|9|@]|A|$]]|$1|J|3|-4|5|6|7|R|8|@]|9|@]|A|$]]|$1|K|3|-4|5|6|7|S|8|@]|9|@]|A|$]]]|L|$]]

They have a smaller memory footprint than rotation matrices and they are more efficient than both matrix and angle/axis representations.

Also:

<ul>
<li>It's extremely easy to interpolate between two quaternions, which is useful for smooth camera movements etc.</li>
<li>Unit normalisation of floating point quaternions suffers from fewer rounding defects than matrix representations.</li>
</ul>

blocks|key|1874589|text|对于四元数，您还可以处理万向节锁的问题。当您想要执行任意旋转时，它们更容易使用。|type|unstyled|depth|inlineStyleRanges|entityRanges|data|1874590|entityMap^0|0^^$0|@$1|2|3|4|5|6|7|D|8|@]|9|@]|A|$]]|$1|B|3|-4|5|6|7|E|8|@]|9|@]|A|$]]]|C|$]]

With quaternions you also handle the problem of the gimbal lock. And they are easier to work with when you want to perform arbitrary rotations.

blocks|key|615513|text|与Euler角度相比，四元数具有许多优势，并且通常更适合3D旋转：|type|unstyled|depth|inlineStyleRanges|entityRanges|data|615514|615515|四元数(或:方向)之间更容易(且定义明确)的插值:产生的运动围绕单个轴具有恒定的角速度，这通常更美观。这个过程被称为"slerp“，对于动画/旋转混合至关重要。此外，四元数插值不会受到万向节锁的影响。|unordered-list-item|615516|它们很容易重新规范化。|615517|615518|缺点：|615519|615520|的主要缺点是，它们需要更多的数学知识，并且比欧拉/卡尔达尼克角度更不直观。|615521|与仿射变换矩阵相比，四元数只包含旋转，不包含平移和缩放。|615522|615523|entityMap^0|0|0|0|0|0|0|0|0|0|0^^$0|@$1|2|3|4|5|6|7|S|8|@]|9|@]|A|$]]|$1|B|3|-4|5|6|7|T|8|@]|9|@]|A|$]]|$1|C|3|D|5|E|7|U|8|@]|9|@]|A|$]]|$1|F|3|G|5|E|7|V|8|@]|9|@]|A|$]]|$1|H|3|-4|5|6|7|W|8|@]|9|@]|A|$]]|$1|I|3|J|5|6|7|X|8|@]|9|@]|A|$]]|$1|K|3|-4|5|6|7|Y|8|@]|9|@]|A|$]]|$1|L|3|M|5|E|7|Z|8|@]|9|@]|A|$]]|$1|N|3|O|5|E|7|10|8|@]|9|@]|A|$]]|$1|P|3|-4|5|6|7|11|8|@]|9|@]|A|$]]|$1|Q|3|-4|5|6|7|12|8|@]|9|@]|A|$]]]|R|$]]

Quaternions have many advantages over Euler angles and are often preferable for 3D rotations:

<ul>
<li>Easier (and well-defined) interpolation between quaternions (or: orientations): the resulting movement has constant angular velocity around a single axis, which is often aesthetically more pleasing. This process is called "slerp" and critical for animation/rotation blending. Furthermore, quaternion interpolation does not suffer from Gimbal locks.</li>
<li>They are easy to renormalize.</li>
</ul>

Disadvantages:

<ul>
<li>The main disadvantage is that they require a bit more math and are less intuitive than Euler/Cardanic angles. </li>
<li>Compared to affine transformation matrices, Quaternions only contain a rotation, and no translation and scaling.</li>
</ul>

blocks|key|1877085|text|四元数的优点|type|unstyled|depth|inlineStyleRanges|entityRanges|data|1877086|旋转旋转快速multiplication|1877087|Fast+to/from矩阵conversion|unordered-list-item|1877088|
|1877089|Avoid附加(从计算)噪声(缩放，剪切)，并代表纯rotation|1877090|1877091|Simple旋转插值，在自定义情况下可用于实时动画可使用线性interpolation.|1877092|1877093|Some棘手的操作可用，快速旋转集成，旋转旋转decompositions|1877094|
，|1877095|ordered-list-item|1877096|1877097|缺点。|1877098|向量的|1877099|Transformation不如3x3矩阵快。|1877100|包含4个标量，但紧凑旋转表示只能使用3。|1877101|1877102|entityMap^0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0^^$0|@$1|2|3|4|5|6|7|16|8|@]|9|@]|A|$]]|$1|B|3|C|5|6|7|17|8|@]|9|@]|A|$]]|$1|D|3|E|5|F|7|18|8|@]|9|@]|A|$]]|$1|G|3|H|5|6|7|19|8|@]|9|@]|A|$]]|$1|I|3|J|5|F|7|1A|8|@]|9|@]|A|$]]|$1|K|3|H|5|6|7|1B|8|@]|9|@]|A|$]]|$1|L|3|M|5|F|7|1C|8|@]|9|@]|A|$]]|$1|N|3|H|5|6|7|1D|8|@]|9|@]|A|$]]|$1|O|3|P|5|F|7|1E|8|@]|9|@]|A|$]]|$1|Q|3|R|5|6|7|1F|8|@]|9|@]|A|$]]|$1|S|3|-4|5|T|7|1G|8|@]|9|@]|A|$]]|$1|U|3|-4|5|6|7|1H|8|@]|9|@]|A|$]]|$1|V|3|W|5|6|7|1I|8|@]|9|@]|A|$]]|$1|X|3|Y|5|6|7|1J|8|@]|9|@]|A|$]]|$1|Z|3|10|5|T|7|1K|8|@]|9|@]|A|$]]|$1|11|3|12|5|T|7|1L|8|@]|9|@]|A|$]]|$1|13|3|-4|5|6|7|1M|8|@]|9|@]|A|$]]|$1|14|3|-4|5|6|7|1N|8|@]|9|@]|A|$]]]|15|$]]

Pros of quaternions

<ol>
<li>Fast multiplication</li>
<li>Fast to/from matrix conversion</li>
<li>Avoid additional (from computation) noise (scale , shear) , and represent pure rotation</li>
<li>Simple rotation interpolation, in custom case for real time animation can be used linear interpolation.</li>
<li>Some tricky operations available, fast rotation integration, twist swing decompositions</li>
</ol>

Cons.

<ol>
<li>Transformation of vector is not so fast as with 3x3 matrix. </li>
<li>Contains 4 scalar, but compact rotation representation can use only 3.</li>
</ol>

blocks|key|2023567|text|与矩阵相比，四元数的优势不仅是计算速度更快，而且主要是因为围绕任意角度连续旋转的矩阵表示最终会出现可怕的浮点舍入误差，不再表示适当的仿射旋转。“恢复”一个旋转矩阵在计算上比标准化一个四元数要昂贵得多。因此，应该选择四元数而不是纯旋转矩阵。|type|unstyled|depth|inlineStyleRanges|entityRanges|data|2023568|entityMap^0|0^^$0|@$1|2|3|4|5|6|7|D|8|@]|9|@]|A|$]]|$1|B|3|-4|5|6|7|E|8|@]|9|@]|A|$]]]|C|$]]

The advantage of quaternions over matrices is not only faster computation, but mostly because a matrix representation of successive rotations around arbitrary angles eventually give in to dreadful floating-point round-off errors and no longer represent proper, affine rotations. "Restoring" a rotation matrix is computationally more expensive than normalizing a quaternion. Therefore, quaternions should be chosen over pure rotation matrices.

blocks|key|574246|text|与欧拉角相比，它们的合成更简单，并且避免了万向节锁定的问题。|type|unstyled|depth|inlineStyleRanges|entityRanges|data|574247|与旋转矩阵相比，它们在数值上更稳定，并且表示(4个数字)更紧凑。|574248|entityMap^0|0|0^^$0|@$1|2|3|4|5|6|7|F|8|@]|9|@]|A|$]]|$1|B|3|C|5|6|7|G|8|@]|9|@]|A|$]]|$1|D|3|-4|5|6|7|H|8|@]|9|@]|A|$]]]|E|$]]

Compared to Euler angles they are simpler to compose and avoid the problem of gimbal lock.

Compared to rotation matrices they are more numerically stable and the representation (4 numbers) is more compact.

I have been carrying out 2D and 3D operations, including graphics, for many years and have never used quaternions so I don't have a feel for them. I know that they can be used for certain operations that are difficult in Euler angles and also that they can be used to find the rotation required to best fit one set of coordinates (X1, X2...XN, X=(xyz)) onto another (X1', X2'... XN'). 

Are there places where quaternions are essential? And are there places where they make solutions more elegant or more efficient?

When do I need to use quaternions?

翻译质量差，导致语言生硬或混乱。

没有提供实际的解决方法或示例。

解答不清晰，无法理解或解决问题。

页面排版不美观，阅读体验差。

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多年来，我一直在执行2D和3D操作，包括图形，并且从未使用过四元数，因此我对它们没有感觉。我知道它们可以用于欧拉角中困难的某些操作，还可以用来找到将一组坐标(X1，X2...XN，X=(xyz))最佳拟合到另一组坐标(X1'，X2‘)所需的旋转。XN')。有没有四元数必不可少的地方？有没有地方可以让解决方案变得更优雅或更高效？

问什么时候需要使用四元数？
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