EulerAngles

EulerAngles[r]

gives Euler angles {α,β,γ} corresponding to the rotation matrix r.

EulerAngles[r,{a,b,c}]

gives Euler angles {α,β,γ} with rotation order {a,b,c}.

Details

  • EulerAngles[r,{a,b,c}] gives angles {α,β,γ} such that EulerMatrix[{α,β,γ},{a,b,c}]r.
  • EulerAngles[r] is equivalent to EulerAngles[r,{3,2,3}], the z-y-z rotations.
  • The default z-y-z angles EulerAngles[r,{3,2,3}] decomposes rotation into three steps:
  • The rotation axes a, b, and c can be any integer 1, 2, or 3. But there are only twelve 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 may not be invertible, since these are not capable of representing all possible rotations in 3D.

Examples

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Basic Examples  (2)

Get Euler angles from the rotation matrix:

Get Euler angles from the rotation matrix with the given rotation order:

Applications  (6)

Rotation Representations  (4)

Convert Euler angles from one rotation order to another:

Convert roll-pitch-yaw angles to Euler angles:

Get Euler angles for a 3D rotation in the plane given by t{1,1,1} + s{1,–2,1}:

Find a single set of Euler angles from a composition of Euler rotations:

Both perform the same transformation (red):

Coordinate Systems  (2)

Let and be the coordinate axes for two orthogonal coordinate systems that are rotated from each other:

Given TemplateBox[{{{, {{x, _, 2}, ,, {y, _, 2}, ,, {z, _, 2}}, }}}, Transpose]=R.TemplateBox[{{{, {{x, _, 1}, ,, {y, _, 1}, ,, {z, _, 1}}, }}}, Transpose], where the rotated axis is given by etc., one then finds that R=TemplateBox[{{{, {{x, _, 2}, ,, {y, _, 2}, ,, {z, _, 2}}, }}}, Transpose].{x_1,y_1,z_1}, since TemplateBox[{{{, {{x, _, 1}, ,, {y, _, 1}, ,, {z, _, 1}}, }}}, Transpose] is an orthogonal matrix and its inverse is its transpose:

Verify that is the rotation that relates the coordinate systems:

Find the corresponding Euler angles that define :

Visualize:

The right-handed, z-up coordinate system is standard for Cartesian coordinates in mathematics. However, in computer graphics applications, different systems, such as right-handed, y-up may be used. Using the previous example, find the Euler angles that transform a z-up to a y-up coordinate system:

Obtain the rotation matrix and the corresponding Euler angles that define :

Use those angles to transform the y-up coordinate system and visualize (z-up system, y-up system, and transformed y-up system):

Properties & Relations  (1)

EulerAngles returns angles for which EulerMatrix gives the same rotation matrix:

The angles need not be the same:

However, both sets of angles produce the same rotation matrix:

Possible Issues  (1)

EulerMatrix allows equal consecutive axes and this generates a rotation matrix:

However, EulerAngles requires consecutive axes to be distinct:

This is because with consecutive axes equal, some rotation matrices cannot be represented:

Wolfram Research (2015), EulerAngles, Wolfram Language function, https://reference.wolfram.com/language/ref/EulerAngles.html.

Text

Wolfram Research (2015), EulerAngles, Wolfram Language function, https://reference.wolfram.com/language/ref/EulerAngles.html.

CMS

Wolfram Language. 2015. "EulerAngles." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/EulerAngles.html.

APA

Wolfram Language. (2015). EulerAngles. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EulerAngles.html

BibTeX

@misc{reference.wolfram_2025_eulerangles, author="Wolfram Research", title="{EulerAngles}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/EulerAngles.html}", note=[Accessed: 03-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_eulerangles, organization={Wolfram Research}, title={EulerAngles}, year={2015}, url={https://reference.wolfram.com/language/ref/EulerAngles.html}, note=[Accessed: 03-July-2025 ]}