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- Quaternion Rotation formula - Mathematics Stack Exchange
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- Quaternions and spatial translations - Mathematics Stack Exchange
$\begingroup$ The alternative is the dual quaternion, which Gerard mentioned (albeit not by name) - they are composed the same way, the "sandwich" (a)(bcb*)(a*) = (ab)c(ba) = (ab)c(ab)*, which means a long sequence on the LHS only needs to be conjugated after the fact to find the RHS So, you don't need to break a long sequence of quaternions
- Understanding quaternions - Mathematics Stack Exchange
Adding two unit quaternions generally does not yield a unit quaternion, so the answer is technically no as written, but the answer is yes if you say "rotating two separate planes by the same angle and rescales " Of course adding two quaternions gives a quaternion, so algebraically this is clear
- Finding the Unit Quaternion - Mathematics Stack Exchange
To normalize the quaternion you do indeed divide by the norm which is $\sqrt{2^2+(-1)^2+2^2+(-3^2)}$ However, you need to divide each component by the norm rather than just the coefficients So your quaternion becomes
- How can one intuitively think about quaternions?
Here is the intuitive interpretation of this Given a particular rotation axis $\omega$, if you restrict the 4D quaternion space to the 2D plane containing $(1,0,0,0)$ and $(0,\omega_x,\omega_y,\omega_z)$, the unit quaternions representing all possible rotations about the axis $\vec \omega$ form the unit circle in that plane
- Real world uses of Quaternions? - Mathematics Stack Exchange
The quaternion algebra shows there as a way of disentangling two Alamouti coded signals transmitted by a pair of antennas The advantages come from the fact that even if the signal from one antenna is lost for a particular receiver (due to sitting in a node for that particular radio wave), then the signal from the other antenna saves the day
- Concise description of why rotation quaternions use half the angle
Every quaternion multiplication does a rotation on two different complex planes When you multiply by a quaternion, the vector part is the axis of 3D rotation The part you want for 3D rotation But you ALSO do a rotation in the complex plane consisting of the axis and the scalar term
- 如何形象地理解四元数？ - 知乎
汉密尔顿定义了一种纯四元数（pure quaternion），其表达式为 qw=(0,wx,wy,wz) 。纯四元数第一项为零，它存在于四维空间的三维超平面上，与三维空间中的三维向量一一对应。
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