Conversion between quaternions and Euler angles: Difference between revisions

Revision as of 23:58, 27 May 2005

Revision as of 21:14, 28 May 2005

Line 1:

Spatial rotations in three dimensions can be parametrized using both [[Euler angles]] and [[Quaternions and spatial rotation|unit quaternions]]. This article explains how to convert betweeenbetween the two representatiosrepresentations.

A unit quaternion can be described as:

Line 7 ⟶ 6:

:<math>|\mathbf{q}|^2 = q_0^2 + q_1^2 + q_2^2 + q_3^2 = 1</math>

SimmilarlySimilarly for Euler angles, we use:

Simmilarly for Euler angles, we use:

* Roll - <math>\phi</math>: rotation about the X-axis

* Pitch - <math>\theta</math>: rotation about the Y-axis

Line 24 ⟶ 22:

The [[orthogonal matrix]] corresponding to a rotation with Euler angles <math>\phi, \theta \, and\, \psi</math>, is given by

:<math>\begin{bmatrix}

\cos\theta \cos\psi & -\cos\phi \sin\psi + \sin\phi \sin\theta \cos\psi & \sin\phi \sin\psi + \cos\phi \sin\theta \cos\psi \\

\cos\theta \sin\psi & \cos\phi \cos\psi + \cos\phi \sin\theta \sin\psi & -\sin\phi \cos\psi + \cos\phi \sin\theta \sin\psi \\

-\sin\theta & \sin\phi \cos\theta & \cos\phi \cos\theta \\

\end{bmatrix}</math>

== Conversion ==

By comparing the temsterms in the two matrices, we get....

:<math> \mathbf{q} = \begin{bmatrix}

\cos_2 \phi \cos_2 \theta \cos_2 \psi + \sin_2 \phi \sin_2 \theta \sin_2 \psi \\

\sin_2 \phi \cos_2 \theta \cos_2 \psi + \cos_2 \phi \sin_2 \theta \sin_2 \psi \\

\cos_2 \phi \sin_2 \theta \cos_2 \psi + \sin_2 \phi \cos_2 \theta \sin_2 \psi \\

\cos_2 \phi \cos_2 \theta \sin_2 \psi + \sin_2 \phi \sin_2 \theta \cos_2 \psi \\

\end{bmatrix}</math>

where <math>\cos_2 \alpha</math> is a notational shortcut for <math>\cos \frac{\alpha}{2}</math>, and <math>\sin_2 \alpha</math> for <math>\sin \frac{\alpha}{2}</math>. And for Euler angles we get:

where <math>cos_2 \alpha</math> is a notational shortcut for <math>cos \frac{\alpha}{2}</math>, and <math>sin_2 \alpha</math> for <math>sin \frac{\alpha}{2}</math>. And for Euler angles we get:

:<math>\begin{bmatrix}

Line 46 ⟶ 43:

\end{bmatrix} =

\begin{bmatrix}

\mbox{atan} \frac {2(e_0 e_1 + e_2 e_3)} {1 - 2(e_1^2 + e_2^2)} \\

\mbox{asin} (2(e_0 e_2 - e_3 e_1)) \\

\mbox{atan} \frac {2(e_0 e_3 + e_1 e_2)} {1 - 2( e_2^2 + e_3^2)}

\end{bmatrix} </math>

=== Singularities ===

One must be aware of singularities in the Euler angle parametrizartion when the pitch approaches <math>\pm 90^o</math> (north/south pole). These cases must be handled specially.

{{compumath-stub}}

[[Category:Abstract algebra]]

[[Category:Computer graphics]]