2022-06-15 Quaternion multiplication and division

Quaternion Arithmetic

Multiplication of basis quaternion elements

Normalization

Conjugate

Multiplication

Division

Conversion

Quaternion to Rotation Matrix

Given the unit quaternion:
$$Q=(w,x,y,z)$$
The equivalent 3×3 rotation matrix is:

Rotation Matrix to Quaternion

Given a rotation matrix R:
$$R=\left[\begin{array}{ccc}{r}_{11}& r_{12}& r_{13}\\ r_{21}& r_{22}& r_{23}\\ r_{31}& r_{32}& r_{33}\end{array}\right]$$
The equivalent quaternion will be

Quaternion to Euler Angle

Choose Rotation Matrix or Quaternion

Choosing between Euler angles and quaternions is tricky. Euler angles are intuitive for artists, so if you write some 3D editor, use them. But quaternions are handy for programmers, and faster too, so you should use them in a 3D engine core.
The general consensus is exactly that: use quaternions internally, and expose Euler angles whenever you have some kind of user interface.

Attention

Several consecutive quaternion computation will result in accumulated error, thus, periodic normalization is needed for quaternion.

Reference

1. Rotation Matrix: https://en.wikipedia.org/wiki/Rotation_matrix
2. Quaternion: https://en.wikipedia.org/wiki/Quaternion
3. Use Which One: http://www.opengl-tutorial.org/intermediate-tutorials/tutorial-17-quaternions/
4. Quaternions and spatial rotation: https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation
5. Quaternion&Rotation: https://www.zhoulujun.cn/html/theory/Mathematics/Geometry/8149.html
6. Quaternion Physical Signature: https://codeantenna.com/a/FYa8JjhUVg
7. Quaternion Division: https://ww2.mathworks.cn/help/aeroblks/quaterniondivision.html
8. Quaternion Mult.: https://ww2.mathworks.cn/help/aeroblks/quaternionmultiplication.html
9. Quaternion Introduction Book:
   https://mil.ufl.edu/nechyba/www/__eel6667.f2003/course_materials/t3.quaternions/intro_quaternions.pdf