Difference between revisions of "Realization of transformations"
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\mathbf{T}_c = \mathbf{T}_2\mathbf{T}_1 = | \mathbf{T}_c = \mathbf{T}_2\mathbf{T}_1 = | ||
\left[\begin{array}{cccc} & & & \\ & \mathbf{R}_2\mathbf{R}_1 & & \vec{\mathbf{p}}_2+\mathbf{R}_2\vec{\mathbf{p}}_1 \\ & & & \\ 0 & 0 & 0 & 1\end{array}\right] | \left[\begin{array}{cccc} & & & \\ & \mathbf{R}_2\mathbf{R}_1 & & \vec{\mathbf{p}}_2+\mathbf{R}_2\vec{\mathbf{p}}_1 \\ & & & \\ 0 & 0 & 0 & 1\end{array}\right] | ||
+ | = | ||
+ | \left[\begin{array}{cccc} & & & \\ & \mathbf{R}_c & & \vec{\mathbf{p}}_c \\ & & & \\ 0 & 0 & 0 & 1\end{array}\right] | ||
\quad \equiv \quad | \quad \equiv \quad | ||
− | |||
− | |||
\left\{e_c,p_c\right\} | \left\{e_c,p_c\right\} | ||
\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 individual transformations? |
Revision as of 16:07, 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 individual transformations?