# Wolfram Language & System 11.0 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# EulerMatrix

EulerMatrix[{α,β,γ}]
gives the Euler 3D rotation matrix formed by rotating by α around the current axis, then by β around the current axis, and then by γ around the current axis.

EulerMatrix[{α,β,γ},{a,b,c}]
gives the Euler 3D rotation matrix corresponding, first rotating by α around the current a axis, then by β around the current b axis, and finally by γ around the current c axis.

## DetailsDetails

• EulerMatrix is also known as Euler rotation matrix or Euler rotation, and the angles α, β, and γ are often referred to as Euler angles.
• EulerMatrix is typically used to specify a rotation as a sequence of basic rotations around coordinate axes where each subsequent rotation is referring to the current or intrinsic coordinate frame.
• EulerMatrix[{α,β,γ}] is equivalent to EulerMatrix[{α,β,γ},{3,2,3}].
• The default z-y-z rotation EulerMatrix[{α,β,γ},{3,2,3}]:
• EulerMatrix[{α,β,γ},{a,b,c}] is equivalent to , where Rα,a=RotationMatrix[α,UnitVector[3,a]] etc.  »
• The x-y-z rotation EulerMatrix[{α,β,γ},{1,2,3}]:
• The rotation axes a, b, and c can be any integer 1, 2, or 3, but there are only 12 combinations that are general enough to be able to specify any 3D rotation.
• Rotations with the first and last axis repeated:
•  {3,2,3} z-y-z rotation (default) {3,1,3} z-x-z rotation {2,3,2} y-z-y rotation {2,1,2} y-x-y rotation {1,3,1} x-z-x rotation {1,2,1} x-y-x rotation
• Rotations with all three axes different:
•  {1,2,3} x-y-z rotation {1,3,2} x-z-y rotation {2,1,3} y-x-z rotation {2,3,1} y-z-x rotation {3,1,2} z-x-y rotation {3,2,1} z-y-x rotation
• Rotations with subsequent axes repeated still produce a rotation matrix but cannot be inverted uniquely using EulerAngles.

## ExamplesExamplesopen allclose all

### Basic Examples  (2)Basic Examples  (2)

The standard Euler matrix:

 In[4]:=
 Out[4]//MatrixForm=

Rotate an axes-aligned unit cube:

 In[1]:=
 In[2]:=
 Out[2]=