The 3D point A=(Xa,Ya,Za) is rotated by θ around the axis N=(Nx,Ny,Nz). Extend point A to the quaternion space, then A=(0,Xa,Ya,Za), at this time point A is pure quaternion (that is, the first W component is equal to 0), which is a super three-dimensional space in the four-dimensional space on flat surface. Just like the two-dimensional plane that exists in the three-dimensional space we are in, the point coordinates in the three-dimensional space are (X, Y, Z), and the point coordinates in the two-dimensional plane can be expressed as (0, X, Y) ; Therefore, when W=0 in a point (W, X, Y, Z) in a four-dimensional space, the point is considered to be on the super-three-dimensional plane of the four-dimensional space.
The quaternions used for rotation are generally unit quaternions (that is, normalized, modulo=1). The first is that quaternions are used for rotation and do not care about the modulo length, and the modulo equal to 1 may require calculations; the second Non-unit quaternions may cause errors due to precision in floating-point calculations. Therefore, when using a quaternion, try to normalize it first to make it a unit quaternion.
Next, when point A is rotated about the axis N by an angle of θ, the unit quaternion P(cosθ/2, sinθ/2N) used for the rotation and the conjugate of P $P^*$(cosθ, -sinθN) (because it is P is a unit quaternion, so the conjugate $P^*$ and the inverse $P^{-1}$ are equal) $A^/$ is the rotated point, and the rotation formula is $A^/$=PA $P^{-1}$. This formula is mentioned in the book, please see the specific origin first. Below I will explain this formula in my understanding
First: Quaternion property: The quaternion P multiplied by $P^{-1}$ is equal to 1, which ensures that the rotated point A will not be changed.
Second: when a pure quaternion is multiplied by a unit quaternion, the result is no longer a pure quaternion, and the point A is multiplied by P. At this time, the point A has been transformed into the four-dimensional space, not in the three-dimensional plane. Inside. When multiplying $P^{-1}$ again, because the quaternion P multiplied by $P^{-1}$ is equal to 1, it is guaranteed that the point A is still in the three-dimensional plane. At this time, it explains why it is necessary to multiply by P and $P^{-1}$.
Because it is a unit quaternion, the conjugation and inverse are equal, and the point A times P is rotated by an angle of θ/2 in the positive direction of the axis N. At this time, the point A is rotated to the four-dimensional space, not in the three-dimensional plane. Multiplying by $P^{-1}$ (the inverse and conjugate are equal) is equivalent to multiplying by the conjugate, and the conjugate represents a rotation of θ/2 in the opposite direction to P, at which point A is rotated back to the three-dimensional plane again. Point A is equivalent to having undergone 2 rotations, each time is θ/2, so a total of θ is rotated, which explains why it is θ/2.
So far this is my understanding of the rotation quaternion.