Quaternion / Euler Angle Converter
🧮 What is a Quaternion?
A quaternion is a four-dimensional complex number extension used to represent 3D rotations in computer graphics, robotics, and aerospace engineering. Discovered by William Rowan Hamilton in 1843, quaternions provide a mathematically elegant and computationally efficient way to encode orientation and rotation without the limitations of Euler angles.
A quaternion is written as Q = w + xi + yj + zk, where w is the scalar (real) component and (x, y, z) form the vector (imaginary) components. For rotation representation, quaternions are normalized such that w² + x² + y² + z² = 1, making them unit quaternions that lie on a 4D hypersphere.
Quaternions excel at avoiding gimbal lock (a singularity where two rotation axes align), provide numerically stable interpolation (SLERP - Spherical Linear Interpolation), and use less memory than rotation matrices while being more efficient for composition of multiple rotations.
⚙️ How This Quaternion Converter Works
This professional-grade tool provides bidirectional conversion between quaternions and Euler angles with industry-standard accuracy. Built on the same mathematical foundations as Three.js, it ensures compatibility with major 3D engines and frameworks.
🔥 Advanced Features:
- Bidirectional Conversion:Convert between quaternions (w, x, y, z) and Euler angles (X, Y, Z rotations) in both directions with full precision
- All Rotation Orders Supported:Complete support for all 6 intrinsic rotation orders (XYZ, XZY, YXZ, YZX, ZXY, ZYX) used across different industries
- Flexible Angle Units:Input and output in radians or degrees with independent control for maximum flexibility
- Automatic Normalization:Input quaternions are automatically normalized to ensure valid rotations, preventing numerical errors
- Real-time Validation:Instant error detection for invalid inputs with clear, actionable error messages
- Independent Input/Output Settings:Configure input and output types, units, and rotation orders independently for complex conversion workflows
- Persistent Settings:Auto-save preferences to Local Storage - your settings are remembered across sessions
- Three.js Compatible:Mathematical formulas verified against Three.js library for industry-standard accuracy
- Click-to-Copy:Instant clipboard copying for all output values - streamlined workflow integration
- Privacy-First:All calculations performed locally in your browser - no data sent to servers
🎯 Configuration Options
Input Settings:
- Input Type: Choose Quaternion or Euler angles as input
- Angle Unit: Radian or Degree (for Euler input)
- Rotation Order: XYZ, XZY, YXZ, YZX, ZXY, or ZYX (for Euler input)
Output Settings:
- Output Type: Choose Quaternion or Euler angles as output
- Angle Unit: Radian or Degree (for Euler output)
- Rotation Order: XYZ, XZY, YXZ, YZX, ZXY, or ZYX (for Euler output)
🔄 Understanding Rotation Orders
Rotation order determines the sequence in which rotations around X, Y, and Z axes are applied. This is critical because rotations are non-commutative - changing the order changes the result. The six rotation orders represent different conventions used across industries:
📊 Rotation Order Comparison
| Order | Application Sequence | Common Usage | Industry Standard |
|---|---|---|---|
| XYZ | X → Y → Z | Default in many 3D engines, CAD software | ⭐⭐⭐⭐⭐ |
| ZYX | Z → Y → X | Aviation (Yaw-Pitch-Roll), robotics | ⭐⭐⭐⭐⭐ |
| YXZ | Y → X → Z | Unity3D default order | ⭐⭐⭐⭐ |
| XZY | X → Z → Y | Specialized applications | ⭐⭐ |
| YZX | Y → Z → X | Specialized applications | ⭐⭐ |
| ZXY | Z → X → Y | Some game engines, animation tools | ⭐⭐⭐ |
⚠️ Critical: Rotation Order Selection
Always verify the rotation order convention used in your application:
- Three.js / Babylon.js:Default XYZ order
- Unity3D:Default YXZ order (Pitch-Yaw-Roll)
- Unreal Engine:Default YXZ order
- Aviation / Aerospace:ZYX order (Yaw-Pitch-Roll)
- Robotics (ROS):Often ZYX order
🆚 Quaternions vs Euler Angles: Deep Dive
Understanding the fundamental differences between quaternions and Euler angles is crucial for choosing the right representation for your application. Both have distinct advantages and limitations that make them suitable for different scenarios.
📊 Comprehensive Comparison
| Aspect | Quaternions | Euler Angles |
|---|---|---|
| Components | 4 values (w, x, y, z) | 3 angles (X, Y, Z rotations) |
| Gimbal Lock | ✓ Never occurs | ✗ Occurs at ±90° pitch |
| Interpolation | ✓ SLERP - smooth, shortest path | △ LERP - can wobble or overshoot |
| Composition | ✓ Simple multiplication (O(16) ops) | △ Matrix conversion needed |
| Human Readability | △ Non-intuitive | ✓ Intuitive angles |
| Memory | 16 bytes (4 floats) | 12 bytes (3 floats) |
| Numerical Stability | ✓ Excellent | △ Can accumulate error |
| Singularities | ✓ None | ✗ At ±90° (gimbal lock) |
| Best Use Case | Animation, real-time 3D, physics | User input, configuration, debugging |
🔍 The Gimbal Lock Problem
Gimbal lock is a phenomenon where two of the three rotation axes align, causing a loss of one degree of freedom. This occurs in Euler angle representations when the middle rotation reaches ±90 degrees (±π/2 radians).
Example of Gimbal Lock (ZYX order):
- When Y rotation = 90° (pitch up): Z and X axes become parallel
- Result: Cannot distinguish between yaw and roll rotations
- Consequence: Impossible to represent certain orientations
- Animation problem: Sudden jumps or unwanted rotations
Why Quaternions Solve This:
- Represent rotation as a point on a 4D unit sphere
- No special cases or singularities anywhere
- Can represent any orientation smoothly
- SLERP naturally finds shortest path between orientations
📚 Step-by-Step Examples
Example 1: 90° Y-Axis Rotation
Quaternion input:
- w = 0.7071, x = 0.0, y = 0.7071, z = 0.0
Euler output (ZYX order, radians):
- X: 0.0 Y: 1.5708 (90°) Z: 0.0
This represents a pure 90-degree rotation around the Y-axis, commonly used for turning objects left/right in 3D space.
Example 2: Identity Rotation
Quaternion (no rotation):
- w = 1.0, x = 0.0, y = 0.0, z = 0.0
Euler angles (any order):
- X = 0.0, Y = 0.0, Z = 0.0
The identity quaternion represents no rotation at all - the default orientation.
Example 3: 180° Z-Axis Rotation
Quaternion input:
- w = 0.0, x = 0.0, y = 0.0, z = 1.0
Euler output (ZYX order):
- X = 0.0, Y = 0.0, Z = 3.1416 (180°)
A 180-degree rotation around Z-axis - commonly used for flipping objects horizontally.
💼 Professional Use Cases & Real-World Applications
1. 🎮 Game Development & Real-Time 3D
Character Animation & Camera Control
Modern game engines extensively use quaternions for character bone rotations, camera orientation, and smooth transitions between animations. Quaternions prevent gimbal lock issues that would cause unnatural character poses or camera flips.
Real Example - Third-Person Camera:
- Current camera orientation: Quaternion (w=0.924, x=0.383, y=0.0, z=0.0)
- Target orientation: Quaternion (w=0.707, x=0.707, y=0.0, z=0.0)
- Use SLERP interpolation for smooth camera rotation
- Result: Smooth 45-degree pitch transition over time
Unity3D & Unreal Engine Integration
Both major game engines use quaternions internally. Unity's Transform.rotation and Unreal's FRotator internally use quaternions, converting to Euler angles only for editor display. This tool helps debug rotation issues by converting between representations.
Common Workflow:
- 1. Get quaternion values from engine inspector/debugger
- 2. Convert to Euler angles (degrees) for human-readable understanding
- 3. Adjust rotation values in intuitive Euler format
- 4. Convert back to quaternion for precise engine application
2. 🚁 Aerospace & Drone Control
Flight Control Systems
Aircraft, drones, and spacecraft use quaternions for attitude representation and control. The ZYX (Yaw-Pitch-Roll) convention is standard in aviation, but internal systems often use quaternions to avoid gimbal lock during extreme maneuvers.
Drone Stabilization Example:
- IMU sensor outputs: Quaternion from onboard AHRS system
- Convert to ZYX Euler angles: Roll, Pitch, Yaw for pilot display
- Control algorithm: Works with quaternions internally
- Result: Stable flight with accurate orientation tracking
Satellite Attitude Determination
Satellites use quaternions for attitude determination and control systems (ADCS). Star trackers and gyroscopes output quaternions, which are then used for pointing accuracy and momentum management.
3. 🤖 Robotics & Industrial Automation
Robot Arm Kinematics
Industrial robots use quaternions for end-effector orientation. ROS (Robot Operating System) geometry_msgs/Quaternion is the standard message type for representing orientation in 3D space.
ROS Navigation Stack:
- tf2 transformation library: Uses quaternions internally
- Sensor data (IMU, odometry): Published as quaternions
- Path planning: Quaternion interpolation for smooth trajectories
- Visualization (RViz): Displays as Euler angles for operators
Computer Vision & SLAM
Visual SLAM systems (ORB-SLAM, LSD-SLAM) use quaternions for camera pose representation. Converting to Euler angles helps with visualization and debugging of camera trajectories.
4. 🎬 Animation & VFX Production
Motion Capture Data Processing
Motion capture systems output rotation data as quaternions. Animators often need to convert to Euler angles for fine-tuning in animation software like Maya, Blender, or MotionBuilder.
Production Pipeline:
- Mocap system: Outputs quaternions for each joint
- Convert to Euler: For animator-friendly graph editor curves
- Manual adjustment: Tweak angles for artistic requirements
- Convert back: Apply corrected rotations to rig
5. 🔧 Development & Testing Scenarios
Debugging Rotation Issues
- Verify Conversions:Test your quaternion/Euler conversion implementations against this tool
- Understand Data:Convert quaternions from logs to human-readable Euler angles
- Cross-Platform Testing:Ensure consistency between different engines and frameworks
- Gimbal Lock Detection:Identify problematic Euler angle configurations
📚 Comprehensive Step-by-Step Tutorials
Tutorial 1: Unity3D Rotation Debugging
Scenario:Your Unity character is rotating incorrectly, and you need to understand the quaternion values.
Step 1: Extract Quaternion from Unity
- In Unity Console, log: Debug.Log(transform.rotation);
- Output example: (0.2706, 0.2706, 0.6533, 0.6533)
- Unity format is (x, y, z, w) - note the order!
Step 2: Convert in Tool
Tool Settings:
- Input Type: Quaternion
- Enter: w=0.6533, x=0.2706, y=0.2706, z=0.6533
- Output Type: Euler
- Angle Unit: Degree
- Rotation Order: YXZ (Unity's default)
Result:
- X: 45° (Pitch)
- Y: 45° (Yaw)
- Z: 60° (Roll)
Step 3: Verify and Fix
Now you can see the object is tilted 45° on X and Y axes. Adjust in Unity's Inspector using these Euler angles for intuitive editing.
Tutorial 2: ROS Robot Orientation Setup
Scenario:Configure robot initial pose in launch file with specific Euler angles.
Step 1: Define Desired Orientation
Requirements:
- Robot should face 30° right (Z-axis rotation)
- With 10° downward tilt (Y-axis rotation)
- No roll (X-axis = 0)
Step 2: Convert Euler to Quaternion
- Set Input Type: Euler
- Set Angle Unit: Degree
- Set Rotation Order: ZYX (ROS standard)
- Enter: X=0, Y=10, Z=30
- Set Output Type: Quaternion
Step 3: Use in ROS Launch File
Generated Quaternion:
<param name="initial_pose_x" value="0.0"/> <param name="initial_pose_y" value="0.0"/> <param name="initial_pose_z" value="0.0"/> <param name="initial_pose_qx" value="0.0448"/> <param name="initial_pose_qy" value="0.0872"/> <param name="initial_pose_qz" value="0.2588"/> <param name="initial_pose_qw" value="0.9615"/>
Tutorial 3: Three.js Camera Positioning
Scenario:Set up camera orientation in Three.js scene.
Step 1: Calculate Target Orientation
You want camera to look down at 45° angle with 30° horizontal rotation.
- Input Type: Euler angles
- Rotation Order: XYZ (Three.js default)
- Angle Unit: Degree
- Enter: X=-45 (looking down), Y=30 (rotated right), Z=0
Step 2: Apply in Three.js Code
// Method 1: Using Euler angles (from tool) camera.rotation.set( -0.7854, // X: -45° in radians 0.5236, // Y: 30° in radians 0.0 // Z: 0° ); // Method 2: Using Quaternion (from tool) camera.quaternion.set( 0.1418, // x 0.3536, // y -0.0581, // z 0.9229 // w );
Tutorial 4: Drone Flight Path Conversion
Scenario:Convert flight log quaternions to understandable Euler angles for analysis.
Step 1: Extract Log Data
Flight controller logs contain quaternion orientation data at each timestamp.
Step 2: Batch Processing Strategy
Process each log entry:
- Input: Quaternion from flight log
- Output: ZYX Euler angles (Yaw, Pitch, Roll)
- Unit: Degrees for pilot-friendly visualization
- Plot: Roll/Pitch/Yaw vs time for flight analysis
Step 3: Identify Issues
By visualizing Euler angles, you can easily spot: excessive roll during turns, unstable pitch oscillations, or incorrect heading (yaw) tracking.
🔧 Advanced Technical Deep Dive
🧮 Mathematical Foundation
Understanding the underlying mathematics helps in debugging and optimizing rotation-heavy applications.
Quaternion Multiplication (Composition):
- Given q₁ = (w₁, x₁, y₁, z₁) and q₂ = (w₂, x₂, y₂, z₂)
- Result q₁ × q₂ = (w, x, y, z) where:
- w = w₁w₂ - x₁x₂ - y₁y₂ - z₁z₂
- x = w₁x₂ + x₁w₂ + y₁z₂ - z₁y₂
- y = w₁y₂ - x₁z₂ + y₁w₂ + z₁x₂
- z = w₁z₂ + x₁y₂ - y₁x₂ + z₁w₂
Note:Quaternion multiplication is not commutative: q₁ × q₂ ≠ q₂ × q₁
⚡ Performance Considerations
Operation Complexity Comparison
Computational Cost (approximate):
- Quaternion Multiplication:16 multiplications, 12 additions = 28 operations
- Matrix Multiplication (3×3):27 multiplications, 18 additions = 45 operations
- Euler to Matrix:6 sin/cos calls + matrix operations = ~100+ operations
- Quaternion Normalization:1 sqrt, 3 multiplications, 4 additions = ~8 operations
For real-time applications with thousands of objects, quaternions provide 40-60% performance improvement over matrices.
🔍 Numerical Precision & Stability
Quaternions maintain numerical stability better than Euler angles or matrices over multiple operations.
Numerical Issues Avoided:
- Gimbal Lock Discontinuity:Euler angles have undefined derivatives at singularities
- Matrix Orthogonality Loss:Rotation matrices accumulate floating-point error, requiring periodic re-orthogonalization
- Euler Wraparound:Angle interpolation across ±180° boundary causes unwanted full rotations
- Quaternion Advantage:Simple normalization (divide by magnitude) restores perfect unit quaternion
🎯 Interpolation Methods
SLERP (Spherical Linear Interpolation)
SLERP provides constant-angular-velocity interpolation between two quaternions, essential for smooth animations.
SLERP Formula:
- Given quaternions q₁ and q₂, parameter t ∈ [0,1]
- cos(θ) = q₁ · q₂ (dot product)
- slerp(q₁, q₂, t) = (sin((1-t)θ)/sin(θ))q₁ + (sin(tθ)/sin(θ))q₂
- For θ ≈ 0 (very close quaternions), use LERP for stability
⚠️ Common Pitfalls & Solutions
Quaternion Double Cover:
- Both q and -q represent the same rotation
- SLERP can take long path if signs differ
- Solution: Check dot product, negate if negative before SLERP
Euler Angle Ambiguity:
- Multiple Euler angle sets can represent same orientation
- Example: (0°, 90°, 0°) ≡ (180°, 90°, 180°) for some orders
- Solution: Normalize angles to consistent range [-180°, 180°]
❓ Comprehensive FAQ & Troubleshooting
Q: Why do my converted Euler angles look different from my game engine?
A: Different engines use different rotation order conventions. Unity uses YXZ by default, Unreal uses YXZ, Three.js uses XYZ. Always verify and match the rotation order setting in this tool to your engine's convention. Also check if your engine displays angles in radians or degrees.
Q: What does 'gimbal lock' actually mean in practice?
A: Gimbal lock occurs when using Euler angles and the middle rotation reaches ±90°. At this point, the first and third rotation axes become parallel, losing one degree of freedom. In practice, this causes: sudden flips in rotation, inability to smoothly interpolate through certain orientations, and unpredictable behavior near the singularity. Quaternions completely avoid this issue.
Q: Can I convert between different rotation orders?
A: Yes! Set input rotation order to your source format and output rotation order to your target format. The tool converts through quaternions as an intermediate step, ensuring accurate transformation between any two rotation order conventions.
Q: Why are my quaternion components not normalized (sum of squares ≠ 1)?
A: Input quaternions are automatically normalized by this tool. If you're seeing non-normalized values from your application, it indicates numerical drift from repeated operations. Always normalize quaternions after multiple compositions to maintain unit length: q_normalized = q / ||q||.
Q: How do I interpolate between two orientations smoothly?
A: Convert both orientations to quaternions, then use SLERP (Spherical Linear Interpolation) with a parameter t from 0 to 1. This provides constant angular velocity and shortest-path rotation. Never interpolate Euler angles directly - this causes gimbal lock and unnatural rotation paths.
Q: What's the relationship between quaternions and rotation matrices?
A: A quaternion q=(w,x,y,z) can be converted to a 3×3 rotation matrix. Both represent the same rotation, but quaternions use 4 numbers vs. 9 for matrices, are faster to compose, and don't suffer from orthogonality loss. Matrices are useful for applying rotations to many points, while quaternions excel at composing and interpolating rotations.
Q: Why does converting quaternion → Euler → quaternion give different results?
A: Due to Euler angle ambiguity and gimbal lock, the round-trip may produce a mathematically equivalent but numerically different quaternion. Both represent the same rotation. If you see sign flips (q vs -q), remember quaternions have double coverage - both represent identical rotations.
Q: How do I validate my own conversion implementation?
A: Test with known cases: (1) Identity rotation: q=(1,0,0,0) → Euler=(0,0,0), (2) 90° rotations around each axis, (3) 180° rotations, (4) Gimbal lock positions (e.g., pitch=90°), (5) Random orientations. Compare your results against this tool, which uses Three.js-verified formulas.
🎯 Professional Tips & Best Practices
- Storage Format:Always store rotations as quaternions in data files and network protocols for consistency and precision
- User Interface:Display Euler angles to users (more intuitive) but convert to quaternions internally for calculations
- Interpolation:Always use SLERP for quaternions, never linearly interpolate Euler angles
- Rotation Composition:Compose rotations using quaternion multiplication, not Euler angle addition
- Normalization:Normalize quaternions after every 10-20 operations to prevent drift from floating-point errors
- Sign Consistency:Before SLERP, ensure quaternions point in same hemisphere (dot product > 0)
- Documentation:Always document rotation order convention in API and configuration files
- Testing:Include gimbal lock cases (±90° pitch) in unit tests to catch Euler angle issues early
- Debugging:Convert quaternions to Euler angles for human-readable logs and visualization
- Performance:For thousands of objects, use quaternions directly; only convert to Euler for UI display
- Coordinate Systems:Be aware of coordinate system differences (left-handed vs right-handed) between engines
- Range Limiting:If you must use Euler angles, implement proper range limiting to prevent wraparound issues
🔗 Related Tools
- Number Base Converter - Convert numbers between binary, decimal, hexadecimal, and octal with precision