Current Rotation
q =
1.00
+
0.00
i
+
0.00
j
+
0.00
k
|q| = 1.000
Rotation Matrix
Rotation Controls
Quaternion Operations
q₁ =
×
q₂ =
=
result
Preset Rotations
Euler vs Quaternion Comparison
Gimbal Lock Warning: Euler angles can lose a degree of freedom when two axes become aligned. Quaternions avoid this problem.
Normal
Always Valid
SLERP Interpolation
Interpolates between two rotations along the shortest path
Start: identity
End: 90° X
Rotation Path
Path points: 0
Quaternion Theory
Definition
A quaternion is an extension of complex numbers to 4D:
q = w + xi + yj + zk
where i² = j² = k² = ijk = -1
Rotation Quaternion
For rotation by angle θ around axis (x, y, z):
q = cos(θ/2) + sin(θ/2)(xi + yj + zk)
Advantages over Euler Angles
- No gimbal lock
- Smooth interpolation (SLERP)
- Efficient composition
- Numerical stability
Operations
- Multiplication: Combines rotations
- Conjugate: q* = w - xi - yj - zk (inverse rotation)
- Inverse: q⁻¹ = q*/|q|² (for unit quaternions, q⁻¹ = q*)
- SLERP: Smooth interpolation between rotations