Rotation Matrix to Euler Angles
Calculator for Converting a Rotation Matrix to Euler Angles
Calculate Euler Angles
Instructions
This function converts the Euler angles from a rotation matrix. Enter the values of the matrix whose angles are to be calculated. Then click Calculate.
Euler Angles - Overview
What are Euler Angles?
Euler angles (Yaw, Pitch, Roll) describe the orientation of a 3D object. This calculator extracts these angles from a rotation matrix.
General Solution
Yaw angle (Z-axis):
\(w = \tan^{-1}\left(\frac{m_{21}}{m_{11}}\right) = \text{atan2}(m_{21}, m_{11})\)
Pitch angle (Y-axis):
\(v = -\sin^{-1}(m_{31}) = -\text{asin}(m_{31})\)
Roll angle (X-axis):
\(u = \tan^{-1}\left(\frac{m_{32}}{m_{33}}\right) = \text{atan2}(m_{32}, m_{33})\)
Gimbal Lock Warning
In the special case when the pitch angle (v) = ±90°, a condition occurs that is referred to as "gimbal lock". The pitch angle is still valid, but the other angles are undefined and require special formulas.
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Converting a Rotation Matrix to Euler Angles
General Formulas
The general solution to recovering Euler angles from a rotation matrix uses inverse trigonometric functions:
Yaw angle:
\(\displaystyle w = \tan^{-1}\left(\frac{m_{21}}{m_{11}}\right) = \text{atan2}(m_{21}, m_{11})\)
Pitch angle:
\(\displaystyle v = -\sin^{-1}(m_{31}) = -\text{asin}(m_{31})\)
Roll angle:
\(\displaystyle u = \tan^{-1}\left(\frac{m_{32}}{m_{33}}\right) = \text{atan2}(m_{32}, m_{33})\)
Key Points
- atan2 function is preferred over tan⁻¹ (handles all quadrants)
- Matrix elements are denoted as mij (row i, column j)
- Angles can be expressed in degrees or radians
- Order matters: These formulas assume a specific rotation order
Gimbal Lock Cases
In the special case when the pitch angle (v) = ±90°, a condition occurs that is referred to as "gimbal lock". The pitch angle is still valid, but the other angles are undefined.
Case 1: Pitch v = -90° (m₃₁ = 1)
Yaw angle:
\(\displaystyle w = 0\)
Roll angle:
\(\displaystyle u = \tan^{-1}\left(\frac{-m_{12}}{-m_{13}}\right) = \text{atan2}(-m_{12}, -m_{13})\)
Case 2: Pitch v = 90° (m₃₁ = -1)
Yaw angle:
\(\displaystyle w = 0\)
Roll angle:
\(\displaystyle u = \tan^{-1}\left(\frac{m_{12}}{m_{13}}\right) = \text{atan2}(m_{12}, m_{13})\)
Important Note
When gimbal lock occurs, the rotation can still be represented, but the decomposition into individual Yaw, Pitch, and Roll angles is not unique. The yaw is conventionally set to zero, and all rotation is attributed to roll.
Practical Applications
Aerospace & Robotics:
- Extracting aircraft orientation from sensors
- Converting IMU data to angles
- Robot arm position analysis
- Satellite attitude determination
Computer Graphics & Gaming:
- Camera orientation analysis
- Object rotation decomposition
- Animation keyframe extraction
- VR/AR tracking data conversion
Why Extract Euler Angles?
While rotation matrices provide a complete representation of orientation, Euler angles are more intuitive for humans to understand and easier to visualize. They're commonly used in user interfaces, sensor data, and control systems. However, be aware of gimbal lock when pitch approaches ±90°. In such cases, consider using quaternions or keeping the rotation matrix representation.