Rotation Visualizer

Output

Quaternion [w, x, y, z]

w 1.0000

x 0.0000

y 0.0000

z 0.0000

[x,y,z,w] = 0, 0, 0, 1

Rotation Matrix

100 010 001

✓ SO(3) det = 1.000

Euler ZYX (yaw-pitch-roll)

Z 0.0°

Y 0.0°

X 0.0°

⚠ gimbal lock

Axis-Angle

x 0.0000

y 0.0000

z 1.0000

θ 0.00°

Reference ▼

A rotation is a linear map R: ℝ³ → ℝ³ preserving length and orientation. The set of all valid rotations forms the group SO(3) — Special Orthogonal group in 3D.

SO(3) Constraints R\cdotRᵀ = I (orthonormal columns) det(R) = +1 (orientation-preserving) 9 parameters, 3 DOF (6 constraints)

Quaternion \to Rotation Matrix q = [w, x, y, z], ‖q‖ = 1 R₀₀ = 1 - 2(y²+z²) R₀₁ = 2(xy - wz) R₀₂ = 2(xz + wy) ... 4 params, 3 DOF (unit constraint) Antipodal: q and -q represent the same rotation. Canonical form: w \geq 0.

Rodrigues' Formula (axis-angle) R = I + sin θ\cdot[k]\times + (1-cos θ)\cdot[k]\times² [k]\times = skew-symmetric matrix of axis k Minimal 3-param rep: θ\cdotk̂ Singularity at θ = 0 (axis undefined) and θ = π (two antipodal axes).

Euler ZYX (aerospace) R = Rz(ψ) \cdot Ry(θ) \cdot Rx(φ) ψ = yaw, θ = pitch, φ = roll 3 params, but NOT injective Gimbal lock at θ = \pm90°: rank of Jacobian drops, one DOF lost. This is why spacecraft use quaternions.

Quaternion Composition q₁₂ = q₁ \otimes q₂ (Hamilton product) = [w₁w₂ - v₁\cdotv₂, w₁v₂ + w₂v₁ + v₁\timesv₂] Equivalent to R₁ \cdot R₂