Difference between revisions of "Realization of transformations"
Line 28: | Line 28: | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
− | But how can the two quaternions <math>e_c</math> and <math>p_c</math> of the quaternion notation be calculated based on the individual transformations? | + | But how can the two quaternions <math>e_c</math> and <math>p_c</math> of the quaternion notation be calculated based on the quaternions of individual transformations? The first transformation leads to |
+ | :<math> | ||
+ | q' = e_1\ q\ e_1^*+ p_1 | ||
+ | </math> | ||
+ | Now the second transformation is applied on <math>q'</math>: | ||
+ | :<math> | ||
+ | \begin{align} | ||
+ | q'' &= e_2\ q'\ e_2^*+ p_2 \\ | ||
+ | &= e_2\Big(e_1\ q\ e_1^*+ p_1\Big)\ e_2^*+ p_2 | ||
+ | \end{align} | ||
+ | </math> |
Revision as of 16:14, 15 October 2015
← Back: Composition of rotations | Overview: Quaternions | Next: ??? → |
Quaternion notation for general transformations
Up to now transformations have been defined by homogeneous matrices combining a rotation matrix and a translation vector
. Now a new notation is introduced to represent a transformation using two quaternions
and
:
The quaternion is equivalent to
and describes the rotation while
is defined as
and so equivalent to the translation.
Applying such a transformation to a quaternion is done by first rotating
with
corresponding to the rotation equation and then adding
:
Combination of transformations
It is known that a combination of transformations is defined as:
But how can the two quaternions and
of the quaternion notation be calculated based on the quaternions of individual transformations? The first transformation leads to
Now the second transformation is applied on :