Difference between revisions of "Realization of transformations"
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{{Navigation|before=[[Composition of rotations]]|overview=[[Quaternions]]|next=[[???]]}} | {{Navigation|before=[[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 <math>\mathbf{R}</math> and a translation vector <math>\vec{\mathbf{p}}</math>. Now a new notation is introduced to represent a transformation using two quaternions <math>e</math> and <math>p</math>: | Up to now transformations have been defined by homogeneous matrices combining a rotation matrix <math>\mathbf{R}</math> and a translation vector <math>\vec{\mathbf{p}}</math>. Now a new notation is introduced to represent a transformation using two quaternions <math>e</math> and <math>p</math>: | ||
:<math> | :<math> |
Revision as of 16:02, 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: