Rotation Matrix to Euler Angles

Calculate Euler angles

This function converts the Euler angles from a rotation matrix

Enter the values of the matrix whose angles are to be calculated. Then click on the "Calculate" button

Converting a Rotation Matrix to Euler Angles

The general solution to recovering Euler angles from a rotation matrix is:

Yaw angle: \(\displaystyle w=tan^{-1}\left(\frac{m21}{m11}\right)=atan2(m21,m11)\)

Pitch angle: \(\displaystyle v=-sin^{-1}(m31)= -asin(m31)\)

Roll angle: \(\displaystyle u=tan^{-1}\left(\frac{m32}{m33}\right)=atan2(m32,m33)\)

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. In this case, the following formulas apply:

If the pitch angle v = -90°, (m31 = 1):

Yaw angle: \(\displaystyle w=0\)

Roll angle: \(\displaystyle u=tan^{-1}\left(\frac{-m12}{-m13}\right)=atan2(-m12,-m13)\)

If the pitch angle is v = 90°, (m31 = -1):

Yaw angle: \(\displaystyle w=0\)

Roll angle: \(\displaystyle u=tan^{-1}\left(\frac{m12}{m13}\right)=atan2(m12,m13)\)