Computing Euler angles from a rotation matrix
Gregory G. Slabaugh
Abstract
This document discusses a simple technique to find all possible Euler angles from
a rotation matrix. Determination of Euler angles is sometimes a necessary step
in computer graphics, vision, robotics, and kinematics. However, the solution
may or may not b e obvious.
Rotation matrices
We start off with the standard definition of the rotations about the three prin-
ciple axes.
A rotation of ψ radians about the x-axis is defined as
R
x
(ψ) =
1 0 0
0 cos ψ − sin ψ
0 sin ψ cos ψ
Similarly, a rotation of θ radians about the y-axis is defined as
R
y
(θ) =
cos θ 0 sin θ
0 1 0
− sin θ 0 cos θ
Finally, a rotation of φ radians about the z-axis is defined as
R
z
(φ) =
cos φ − sin φ 0
sin φ cos φ 0
0 0 1
The angles ψ, θ, and φ are the Euler angles.
Generalized rotation matrices
A general rotation matrix can will have the form,
R =
R
11
R
12
R
13
R
21
R
22
R
23
R
31
R
32
R
33
1
