The Euler Angle Calculator computes the three angles that describe the orientation of a rigid body in 3D space. These angles—commonly known as roll (ϕ), pitch (θ), and yaw (ψ)—are derived from either a rotation matrix or a quaternion. The calculator is widely used in aerospace, robotics, mechanical engineering, and computer graphics to convert rotational data into a format that is easy to interpret and apply.
This tool supports various rotation sequences, but a common standard is the Z–Y–X (yaw–pitch–roll) convention. It enables precise analysis and visualization of object orientations, particularly in simulations, virtual environments, and motion tracking systems.
Formula of Euler Angle Calculator
Given a 3×3 Rotation Matrix:
R =
| r₁₁ r₁₂ r₁₃ |
| r₂₁ r₂₂ r₂₃ |
| r₃₁ r₃₂ r₃₃ |
The Euler angles for the Z–Y–X (yaw–pitch–roll) sequence are computed as:
- θ (pitch) = arcsin(−r₃₁)
- ϕ (roll) = arctan2(r₃₂, r₃₃)
- ψ (yaw) = arctan2(r₂₁, r₁₁)
Notes:
- Angles are typically returned in radians. Use standard conversion to degrees if needed: degrees = radians × (180 / π)
- The arctan2 function ensures correct quadrant handling for full 360° coverage
- The input rotation matrix must be orthonormal—meaning it represents a valid rotation without distortion or scaling
This method works best for systems that rely on real-time orientation interpretation, such as navigation, drone control, and motion capture.
Helpful Reference Table
The table below shows commonly used values and their corresponding Euler angles in the Z–Y–X convention:
| Rotation Matrix Element | Resulting Euler Angle | Description |
|---|---|---|
| r₃₁ = 0 | θ = 0° | No pitch |
| r₃₁ = −1 | θ = 90° | Straight upward tilt |
| r₃₁ = 1 | θ = −90° | Straight downward tilt |
| r₃₂ = 0, r₃₃ = 1 | ϕ = 0° | No roll |
| r₂₁ = 0, r₁₁ = 1 | ψ = 0° | No yaw |
This reference helps users verify expected outputs during testing or simulation development.
Example of Euler Angle Calculator
Let’s compute the Euler angles for a rotation matrix:
R =
| 0.866 −0.5 0 |
| 0.5 0.866 0 |
| 0 0 1 |
Step 1: Identify elements
- r₃₁ = 0
- r₃₂ = 0
- r₃₃ = 1
- r₂₁ = 0.5
- r₁₁ = 0.866
Step 2: Apply the formulas
- θ = arcsin(−0) = 0
- ϕ = arctan2(0, 1) = 0
- ψ = arctan2(0.5, 0.866) ≈ 0.5236 radians ≈ 30°
Result:
- Roll (ϕ): 0°
- Pitch (θ): 0°
- Yaw (ψ): 30°
This shows a pure rotation around the Z-axis, typical in many simplified 2D to 3D transformations.
Most Common FAQs
Euler angles are used to represent the orientation of a body or coordinate system in 3D space. They’re applied in animation, flight dynamics, navigation, and robotics to describe rotations in a human-readable format.
Euler angles are easier to interpret but can suffer from “gimbal lock,” a condition where rotational axes become aligned. Quaternions avoid this issue and are more stable in simulations but harder to visualize. Euler angles are often derived from quaternions for interpretation.
Yes. Depending on the quadrant and rotation direction, the angles can be positive or negative. Most calculators can output angles in a full −180° to 180° or 0° to 360° range, depending on the setting.