# Rotation Matrix to Euler Angles

Calculator for Converting a 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

Calculate Euler angles

 Input
 Unit of angles Degree Radians Decimal places 0 1 2 3 4 6 Result Yaw Pitch Roll

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