Interactive Apps 3D Kinematics & Rotations

3D Euler Angles, Rotation Sequences & Gimbal Lock Visualizer

Interactive 3D laboratory for exploring all 6 Proper Euler and 6 Tait-Bryan (Roll-Pitch-Yaw) rotation sequences, intrinsic and extrinsic equivalence, symbolic & numerical 3×3 rotation matrices, dual atan2 solutions, and Gimbal Lock singularities.

1. Select Rotation Sequence (6 Proper Euler + 6 Tait-Bryan)

Angle triplet: (ϕ, θ, ψ)  |  Matrix product: R = R_z(ϕ) · R_y(θ) · R_z(ψ)
Proper (Classic) Euler Sequences (6) 1st & 3rd axes identical (a₁ = a₃); singular at θ = 0°, 180°
Tait-Bryan (Roll-Pitch-Yaw / Cardan) Sequences (6) All 3 axes distinct (a₁ ≠ a₂ ≠ a₃); singular at θ = ±90°
Intrinsic (Current / Body-Attached Frame) Z(ϕ) → Y'(θ) → Z''(ψ) Post-multiply: 1st rotate by ϕ about ẑ, then θ about current ŷ', then ψ about current ẑ''.
≡
Equivalent Extrinsic (Fixed / World Frame) Fixed Z(ψ) → Y(θ) → Z(ϕ) Reversing the rotation order and switching to the fixed frame simultaneously produces the exact same matrix R!
Center Object:
Mouse Mode:
Z-Y-Z: R = R_z(ϕ) · R_y(θ) · R_z(ψ)
3 DoF Full Rank
Drag Object to rotate 3D orientation (solves ϕ, θ, ψ) · Drag Gimbal Rings for single angles · Click ViewCube for standard views
1st Angle: ϕ about ẑ
+40.0°
°
0.70 rad
2nd Angle: θ about ŷ'
+55.0°
°
0.96 rad
3rd Angle: ψ about ẑ''
+30.0°
°
0.52 rad
Step Simulation
Step:
Step 3.00 (Complete): Full Composite Orientation F₃

2. Elementary & Composite 3×3 Rotation Matrices

Elementary Rotation Matrices Product:
Resultant Composite Rotation Matrix R = [rᵢⱼ] ∈ SO(3) (★ marks the Pivot Entry governing θ and Gimbal Lock):

3. Gimbal Lock & Singularity Analysis

Degrees of Freedom & Gimbal Alignment: |sin(θ)| = 0.819 (81.9%)
Angle Between 1st Axis (ê₁) & 3rd Axis (ê₃''): ∠(ê₁, ê₃'') = 55.0°
Interactive Null-Space Explorer (Coupled ϕ & ψ):
At Gimbal Lock, Axis 1 (ê₁) and Axis 3 (ê₃'') become collinear (ϕ + ψ = 70.00°). Drag the slider below or click Spin Null-Space: the outer and inner gimbal rings rotate continuously in 3D while the center object and matrix R stay 100% stationary!

4. Solving for Euler Angles from R (Dual Solutions via atan2)

Every non-singular 3D orientation R ∈ SO(3) is represented by two distinct Euler angle triplets. Clicking Swap Sliders to Alternate Solution Triplet flips the gimbal rings to the second solution while keeping the 3D object and matrix R identical.

Solution Set 1 (Primary Branch)
Solution Set 2 (Alternate Branch)