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.

Details

  • 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.

Examples

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Basic Examples  (2)

The standard Euler matrix:

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

Rotate an axes-aligned unit cube:

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

Scope  (6)

Applications  (6)

Properties & Relations  (11)

Possible Issues  (1)

Neat Examples  (1)

See Also

EulerAngles  RollPitchYawMatrix  RollPitchYawAngles  RotationMatrix  RotationTransform  AnglePath3D

Introduced in 2015
(10.2)