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Rotations in 3D
A Nexus mindmap
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This actually works
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Angular velocity
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There is only 1 plane
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(or any other 3 independent planes)
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xy
xz
yz
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There are 3 planes
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tx
ty
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xy
xz
yz
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There are 6 planes
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Can't create a suitable vector space
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This is a curiosity of 3D!
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This argument can be extended
to non-colinear rotation axes
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Example
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Rotations are not Abelian!
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We saw that there was a really nice correspondence
between rotational motion in 2D and linear motion.
Can this correspondence be extended to 3D?
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This is called 'Abelian' under summation.
It's a defining proverty of vector spaces.
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A natural extension of a location
parameter to higher dimensions
is by using a vector
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Extending rotations to 3D
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Cross products
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Trick to derive quickly
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Rotations in 2D
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N.B. to stick to RH coordinate system
we have to rotate this one 'backwards'
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2D rotations in
x-y/y-z/x-z planes
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27 posibilities for choosing
3 angles to rotate by
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3 of these are all around same axis
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12 combinations involve
2 sequential rotations
about same axis
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In the end have only 27-3-12=12 legitimate
combinations that could describe any 3D rotation
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Combinations
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Rotations in 3D
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[Morin,
Introduction to Classical Mechanics
]
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Rotations in 3D