Difference between revisions of "Realization of transformations"
Line 42: | Line 42: | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
+ | Thus the combination of two transformations can be denoted in quaternion notation as | ||
+ | :<math> | ||
+ | \left\{e_c,p_c\right\}=\left\{e_2,p_2\right\}\left\{e_1,p_1\right\}=\left\{e_2e_1\ , \ e_2p_1e_2^*+p_2\right\} | ||
+ | </math> | ||
+ | Using the knowledge about [[Addition of quaternions|addition of quaternions]] and [[Rotations using quaternions|rotations]] and [[Combination of rotations|combinations of rotations]] using quaternions this can already be seen regarding the homogeneous transformation matrix <math>\mathbf{T}_c</math>. |
Revision as of 16:31, 15 October 2015
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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 . The resulting equation can be solved using the distributive law for quaternions to determine
and
:
Thus the combination of two transformations can be denoted in quaternion notation as
Using the knowledge about addition of quaternions and rotations and combinations of rotations using quaternions this can already be seen regarding the homogeneous transformation matrix .