| Element | Name | Meaning |
|---|---|---|
| $w$ | Scalar part | Related to the cosine of half the rotation angle |
| $x,y,z$ | Vector part | Defines the rotation axis and the sine of half the rotation angle |
q=cos(θ/2)+sin(θ/2)(uxi+uyj+uzk)
where:
- θ is the rotation angle
- u=[ux,uy,uz] is a unit vector along the axis of rotation
| Value | Interpretation |
|---|---|
| $w \approx 1$ | Rotation angle $\theta \approx 0$ |
| $[x,y,z] \approx 0$ | Rotation axis contribution is negligible |
| Quaternion $\approx [1,0,0,0]$ | Identity quaternion (no rotation) |
“z is close to 1” means “A 180° rotation around the z-axis”。