Why Do Quaternions Double-Cover?
4 days ago
- #4D-geometry
- #3D-rotation
- #quaternions
- Quaternions are 4D numbers used for 3D rotations, unlike complex numbers which are 2D.
- Quaternions double-cover 3D rotations, meaning two distinct quaternions (negatives of each other) represent the same rotation.
- Complex numbers represent 2D rotations, but extending this to 3D requires an extra dimension, leading to quaternions.
- Multiplying by a quaternion rotates two independent planes in 4D, but 3D rotations require restricting to one plane.
- Applying a quaternion twice (left and right multiplication) cancels unwanted rotations and doubles the desired rotation, explaining the half-angle formulas.
- Negating a quaternion (a 180-degree rotation in 4D) results in the same 3D rotation due to the double-cover property.
- Arbitrary 4D rotations require two quaternions (left and right multiplication) to control all six rotational degrees of freedom.