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How are rotations represented by quaternions?
Quaternions are hypercomplex numbers
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where
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We define the conjugate
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the dot product
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and the norm
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By associating quaternions with vectors in
, the product of two quaternions
,
can be written as a matrix-vector product
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From the structure of these matrices, we can show the dot product equivalence
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We can embed points
into
as follows:
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It can be shown that unit quaternions represent rotations. In particular, given a rotation axis
where
and an angle
, we can construct
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Then, this rotation can be computed as
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Note that
and
represent the same rotation since both the rotation axis as the magnitude are inverted.
It turns out that
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where
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We define the conjugate

the dot product
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and the norm
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By associating quaternions with vectors in
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From the structure of these matrices, we can show the dot product equivalence
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We can embed points
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It can be shown that unit quaternions represent rotations. In particular, given a rotation axis
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Then, this rotation can be computed as
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Note that
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It turns out that
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Flashcard info:
Author: janisborn
Main topic: Informatik
Topic: Computergrafik
School / Univ.: RWTH Aachen
City: Aachen
Published: 18.05.2022