EulerAngles

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`EulerAngles`

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EulerAngles[r]

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

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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

open allclose all

Basic Examples (2)Summary of the most common use cases

Get Euler angles from the rotation matrix:

In[1]:=1

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

In[2]:=2

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

Out[2]=2

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

In[1]:=1

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

In[2]:=2

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

Out[2]=2

Applications (6)Sample problems that can be solved with this function

Rotation Representations (4)

Convert Euler angles from one rotation order to another:

In[1]:=1

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

In[2]:=2

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

In[3]:=3

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

Out[3]=3

Convert roll-pitch-yaw angles to Euler angles:

In[1]:=1

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

In[2]:=2

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

Out[2]=2

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

In[1]:=1

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

In[2]:=2

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

Out[2]=2

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

In[1]:=1

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

In[2]:=2

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

Out[2]=2

Both perform the same transformation (red):

In[3]:=3

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

In[4]:=4

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

Out[4]=4

Coordinate Systems (2)

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

In[1]:=1

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https://wolfram.com/xid/0b8db5mz6u-bk3406

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:

In[3]:=3

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https://wolfram.com/xid/0b8db5mz6u-c6n7s7

Verify that is the rotation that relates the coordinate systems:

In[4]:=4

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

Out[4]=4

Find the corresponding Euler angles that define :

In[5]:=5

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

Out[5]=5

Visualize:

In[6]:=6

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

In[7]:=7

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

In[8]:=8

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

In[9]:=9

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

Out[9]=9

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:

In[1]:=1