Understanding quaternions - Mathematics Stack Exchange Of course adding two quaternions gives a quaternion, so algebraically this is clear I don't really think it's clear geometrically, however, and with good reason: this is a very exceptional accident that occurs in precisely four dimensions, and no other dimensions
rotations - How do you rotate a vector by a unit quaternion . . . Do one quaternion multiplication and you rotate the circular component just that far around, and the quaternion axis gives you the rest of the location, and the fourth dimension says how far ahead or behind you are in time relative to that fraction of a full orbit All in one operation
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
linear algebra - Conversion of rotation matrix to quaternion . . . One of the quaternion elements is guaranteed to have a magnitude of greater than 0 5 and hence a squared value of 0 25 We can use this to determine the "best" set of parameters to use to calculate the quaternion from a rotation matrix
Apply Quaternion Rotation to Vector - Mathematics Stack Exchange A quaternion can be thought of as a scalar plus a 3D vector (also known as real and imaginary parts) The product of a scalar and a 3D vector is the usual scalar multiplication The product of two vectors produces a quaternion with both scalar and vector components, given by (minus) the dot product and cross product respectively
Confusion with getting a unit quaternion from two vectors Quaternions For me, the quaternions are a 4D algebra $\Bbb H=\Bbb R\oplus\Bbb R^3$ and every quaternion is uniquely expressible as a sum of a scalar and a 3D vector The product of vectors satisfies the "geometric product" formula $ {\bf uv}=- {\bf u}\cdot {\bf v}+ {\bf u}\times {\bf v}$, and thus the product of quaternions can be FOILed
四元数 (Quaternions) 数学背景 四元数变换 (Quaternion Transforms) 连接两个四元数 矩阵和四元数相互转换 球面线性插值 从一个向量旋转到另一个向量)Rotation from One Vector to Another 1 数学背景 (Mathematical Background) 四元数定义: 一个四元数 可以被定义以下形式，相互等价。 其中， 为四元数