A Simple Algebraic Way to understand Quaternions and Rotations in 3D
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Before doing this study, using quaternions to represent rotation (
It’s easy to verify its correctness by expanding the identity to the vector rotation formula simply using the multiplication rule of quaternions in the language of vector cross and dot products : Proof of the quaternion rotation identity.
I can easily understand the vector rotation formula. Because it can be derived from the geometry by very elementary vector operations.
In fact, the proof shows these two formulas are mathematically equivalent. Its reversed process is called Factorization. We can get the “quaternion rotation formula” from the “vector rotation formula” by the factorization. If the factorization is understandable:
For the first time, I can figure out the quaternion rotation formula from the first place.
P. S.
There is another method. It uses 2D rotation analogy instead of the pure factorization. See:
- 3D Rotations and Quaternion Exponentials: Special Case
- 3D Rotations in General: Rodrigues Rotation Formula and Quaternion Exponentials
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