Dual Quaternions and 4x4 Homogeneous Transforms
It might be helpful, especially in rigid body motion, to represent dual quaternions as homogeneous matrices. As given above a dual quaternion can be written as: where r and d are both quaternions. The r quaternion is known as the real or rotational part and the quaternion is known as the dual or displacement part. A 3 dimensional position vector,
can be transformed by constructing the dual-quaternion representation,
then a transformation by is given by
- .
The rotation part can be given by
where is the angle of rotation about axis . The rotation part can be expressed as a 3×3 orthogonal matrix by
The displacement can be written as
- .
Translation and rotation combined in one transformation matrix is:
Where the left upper 3×3 matrix is the rotation matrix we just calculated.
Read more about this topic: Dual Quaternion
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