EulerAngles
✖
EulerAngles
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
open allclose allBasic Examples (2)Summary of the most common use cases
Get Euler angles from the rotation matrix:

https://wolfram.com/xid/0b8db5mz6u-8lvk4l

https://wolfram.com/xid/0b8db5mz6u-g4sesa

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

https://wolfram.com/xid/0b8db5mz6u-hl0z3o

https://wolfram.com/xid/0b8db5mz6u-8n9w92

Applications (6)Sample problems that can be solved with this function
Rotation Representations (4)
Convert Euler angles from one rotation order to another:

https://wolfram.com/xid/0b8db5mz6u-0lox75

https://wolfram.com/xid/0b8db5mz6u-rxy0ql

https://wolfram.com/xid/0b8db5mz6u-7zzt2k

Convert roll-pitch-yaw angles to Euler angles:

https://wolfram.com/xid/0b8db5mz6u-i92sb0

https://wolfram.com/xid/0b8db5mz6u-l9rf5g

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

https://wolfram.com/xid/0b8db5mz6u-4fc3bh

https://wolfram.com/xid/0b8db5mz6u-x2rq0m

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

https://wolfram.com/xid/0b8db5mz6u-xuxuhn

https://wolfram.com/xid/0b8db5mz6u-olum2s

Both perform the same transformation (red):

https://wolfram.com/xid/0b8db5mz6u-jsdrek

https://wolfram.com/xid/0b8db5mz6u-v07w09

Coordinate Systems (2)
Let and
be the coordinate axes for two orthogonal coordinate systems that are rotated from each other:

https://wolfram.com/xid/0b8db5mz6u-bk3406
Given , where the rotated
axis is given by
etc., one then finds that
, since
is an orthogonal matrix and its inverse is its transpose:

https://wolfram.com/xid/0b8db5mz6u-c6n7s7
Verify that is the rotation that relates the coordinate systems:

https://wolfram.com/xid/0b8db5mz6u-h1hmq

Find the corresponding Euler angles that define :

https://wolfram.com/xid/0b8db5mz6u-ywuyg


https://wolfram.com/xid/0b8db5mz6u-d3mfw1

https://wolfram.com/xid/0b8db5mz6u-clgbgx

https://wolfram.com/xid/0b8db5mz6u-ijg0vy

https://wolfram.com/xid/0b8db5mz6u-oi1ey

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:

https://wolfram.com/xid/0b8db5mz6u-dnd8m
Obtain the rotation matrix and the corresponding Euler angles that define
:

https://wolfram.com/xid/0b8db5mz6u-owjcs4

https://wolfram.com/xid/0b8db5mz6u-ijup21

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

https://wolfram.com/xid/0b8db5mz6u-hn6zvt

https://wolfram.com/xid/0b8db5mz6u-nuckdw

https://wolfram.com/xid/0b8db5mz6u-b1nap6

https://wolfram.com/xid/0b8db5mz6u-ing7lr

https://wolfram.com/xid/0b8db5mz6u-eh7zpu

Properties & Relations (1)Properties of the function, and connections to other functions
EulerAngles returns angles for which EulerMatrix gives the same rotation matrix:

https://wolfram.com/xid/0b8db5mz6u-baosc0

https://wolfram.com/xid/0b8db5mz6u-ce5rhy

The angles need not be the same:

https://wolfram.com/xid/0b8db5mz6u-jmeupz

https://wolfram.com/xid/0b8db5mz6u-gh9g1a

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

https://wolfram.com/xid/0b8db5mz6u-4l7ji

Possible Issues (1)Common pitfalls and unexpected behavior
EulerMatrix allows equal consecutive axes and this generates a rotation matrix:

https://wolfram.com/xid/0b8db5mz6u-dhrn8d


https://wolfram.com/xid/0b8db5mz6u-bt34u0

However, EulerAngles requires consecutive axes to be distinct:

https://wolfram.com/xid/0b8db5mz6u-mypgyo


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

https://wolfram.com/xid/0b8db5mz6u-hd2fbb


https://wolfram.com/xid/0b8db5mz6u-xvumf


https://wolfram.com/xid/0b8db5mz6u-l3oy2l

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.
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.
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
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: 09-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: 09-July-2025
]}