Quaternion Rotation formula - Mathematics Stack Exchange Stack Exchange Network Stack Exchange network consists of 183 Q A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers
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
What does multiplication of two quaternions give? Quaternion inversion (or just conjugate for the normalized case) creates the inverse rotation (the same rotation in the opposite direction) This is arguably easier to compute (on current computers) than to calculate inverse of a rotation matrix (just have to negate w in quaternion, instead of transposing a rotation matrix)
Combining rotation quaternions - Mathematics Stack Exchange If I combine 2 rotation quaternions by multiplying them, lets say one represents some rotation around x axis and other represents some rotation around some arbitrary axis The order of rotation ma