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32
How are rotations represented by quaternions?
Quaternions are hypercomplex numbers
![](/pool/data/tex/ab6bf01617e76adeacdf5acfc8c5f12d.gif)
where
![](/pool/data/tex/0c096e11fe9df07f5717107bb1605745.gif)
We define the conjugate
![](/pool/data/tex/cec952b933b0c6641b4ceea9a945590d.gif)
the dot product
![](/pool/data/tex/1244d49aaf3949b6cb15afef9271783f.gif)
and the norm
![](/pool/data/tex/179f59981a9ca43196683bd88514377b.gif)
By associating quaternions with vectors in
, the product of two quaternions
,
can be written as a matrix-vector product
![](/pool/data/tex/6183e4ba3c518fdb0d996f0417548424.gif)
![](/pool/data/tex/96521e156e0aeea656c3ce719d452e7c.gif)
From the structure of these matrices, we can show the dot product equivalence
![](/pool/data/tex/3da0921732526692af7fadf392027361.gif)
![](/pool/data/tex/e0babecae87cf4079bb98a7302eec7b4.gif)
![](/pool/data/tex/6af9e6091c6f3001a51db5883588d47f.gif)
![](/pool/data/tex/10f097e14c1c5d2a7a6d2b93191d2dd1.gif)
![](/pool/data/tex/eae31ca32e8e2d3705b12afc972bbc3d.gif)
We can embed points
into
as follows:
![](/pool/data/tex/2504f18ff73dea89e671a3e401eb296c.gif)
It can be shown that unit quaternions represent rotations. In particular, given a rotation axis
where
and an angle
, we can construct
![](/pool/data/tex/652f54fcffe4000dab052aa39c1e2ea1.gif)
Then, this rotation can be computed as
![](/pool/data/tex/1a03774e3b48cd9b52a9e31447e64cd7.gif)
![](/pool/data/tex/c09ff4ef233fe666b446320d0dacece7.gif)
Note that
and
represent the same rotation since both the rotation axis as the magnitude are inverted.
It turns out that
![](/pool/data/tex/292be9390f8d0da59db3e72b0890b979.gif)
![](/pool/data/tex/ab6bf01617e76adeacdf5acfc8c5f12d.gif)
where
![](/pool/data/tex/0c096e11fe9df07f5717107bb1605745.gif)
We define the conjugate
![](/pool/data/tex/cec952b933b0c6641b4ceea9a945590d.gif)
the dot product
![](/pool/data/tex/1244d49aaf3949b6cb15afef9271783f.gif)
and the norm
![](/pool/data/tex/179f59981a9ca43196683bd88514377b.gif)
By associating quaternions with vectors in
![](/pool/data/tex/dfd26afa6091c61f9775456ee723a190.gif)
![](/pool/data/tex/7694f4a66316e53c8cdd9d9954bd611d.gif)
![](/pool/data/tex/4b43b0aee35624cd95b910189b3dc231.gif)
![](/pool/data/tex/6183e4ba3c518fdb0d996f0417548424.gif)
![](/pool/data/tex/96521e156e0aeea656c3ce719d452e7c.gif)
From the structure of these matrices, we can show the dot product equivalence
![](/pool/data/tex/3da0921732526692af7fadf392027361.gif)
![](/pool/data/tex/e0babecae87cf4079bb98a7302eec7b4.gif)
![](/pool/data/tex/6af9e6091c6f3001a51db5883588d47f.gif)
![](/pool/data/tex/10f097e14c1c5d2a7a6d2b93191d2dd1.gif)
![](/pool/data/tex/eae31ca32e8e2d3705b12afc972bbc3d.gif)
We can embed points
![](/pool/data/tex/982dc08e6186cc0e8209e4eba1b7f995.gif)
![](/pool/data/tex/a829d3f9d2968301849d2bc9557091bf.gif)
![](/pool/data/tex/2504f18ff73dea89e671a3e401eb296c.gif)
It can be shown that unit quaternions represent rotations. In particular, given a rotation axis
![](/pool/data/tex/9d3d61a5218880a4f79281905ecb9dc5.gif)
![](/pool/data/tex/aa80d7b6255d42a18c525f2567257ae6.gif)
![](/pool/data/tex/2554a2bb846cffd697389e5dc8912759.gif)
![](/pool/data/tex/652f54fcffe4000dab052aa39c1e2ea1.gif)
Then, this rotation can be computed as
![](/pool/data/tex/1a03774e3b48cd9b52a9e31447e64cd7.gif)
![](/pool/data/tex/c09ff4ef233fe666b446320d0dacece7.gif)
Note that
![](/pool/data/tex/7694f4a66316e53c8cdd9d9954bd611d.gif)
![](/pool/data/tex/56fec20a5d1887d074ab7c2bb25624dc.gif)
It turns out that
![](/pool/data/tex/292be9390f8d0da59db3e72b0890b979.gif)
![](/pool/img/avatar_40_40.gif)
Flashcard info:
Author: janisborn
Main topic: Informatik
Topic: Computergrafik
School / Univ.: RWTH Aachen
City: Aachen
Published: 18.05.2022