d be unit quaternion let R r - CUHK Mathematics...Unit Quaternions and Rotations in1133 Thed let r...
Transcript of d be unit quaternion let R r - CUHK Mathematics...Unit Quaternions and Rotations in1133 Thed let r...
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Unit Quaternions and Rotations in 1133
The d let r be a unit quaternion let R
be a transformation of R3 defiedBYJ
Rq r g r't R1133 1133 yaf r r
where af is a pure quaternion
Then R is a rotation of a 3 doin l space
of pure quaternions aboutan ax is passing thro
the originCli's specifically if the polar form of
r e's
r ooo us
where u is a pureunitquaternion
Then Rf is the pure quatermainobtained by
rotating q about u by theanglezIEvery rotation of 3 doin'l space
about anaxis
passuiglhro.thenigiudcaubeexpresseduitke.su
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PfofCii
Coset f U q dU HEIR
Then Ru ru rtfoot a suit U Usf Usuif
ucesotu.su f Cao Usui 0
Aso suit Caso using
uceff siufosf ucesosiuf usiuZQ
UCasots.no singles0 asOsso
U
Ru is pure quaternion
and u is fixed point of RAnd immediately we have Rau U
the axis in the direction of U is
fixed by R ru
Case q is perpendicularto u l
In this case
Rq rqr
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Cosotusino queso Usui f
goof tuqsino Caso Usui0
fees tuqsiuooso gucatsuitugusino
Since U f are pure quaternions gtuthen gu Uf C ex6 of HW
2
and hence u que U qu ul Uf
legof
Therefore
Rq gas't t f fuceso suio ufuseifo
qceso tfuq uqjosfsioo fs.us't
of O Sino seriousO Uf
20 ft in 20 Uf
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Note that u f are pure quaternions
Uf U g t Uxq
Uxq since u tf u f o
Ug is also a pure quaternion
i Rq s2oJqtsfui2O3Uf.E1133
purequaternion
Also lug I lull81 181 Ex and
Cupof f gu q by uq fu
flagby ex 6 of HW 2 Ugt of and
relief ul fu Caq u
u tug ex6 of HW 2
Heke ffg Yqf is an orthonormal basis
for the plane perpendicular to U
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Rq 20 got sin20 Ugis the rotation of g thro an angle of20 about the axis in the direction of U
YET
mail8 20 tofu20344g
20 ft iu2O Uf128
Case Generalparequaternions
Note that R is a linear transformation
R fit82 Rg t R 82 t purequaternions
Rhq dRg 8g L8 HR
Similarly a rotation in IRS is also his ear
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Denote the rotation thro an angle of 20about the axis of U
Then any purequaternion p can be written as
p hut of
where a C R and f t U
Rp R Hut fA Rut Rfa Ou t Ogduty Op
I 12 0
Pfoffs are easy Ex
Remarks
T.IS qtr rqr r unitquaternion
Hence tr t the same rotation wi R
Translation if gtb where p e's pure quaternion
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Ch18 19 3 DimensionalEuclideamand
teomty SolidGeometry
Euclidean Solid Geometry
Def let TV L v xity j t z k x y z EIR ft
betheset ofpurequaternions and
IR F TV TV Tv rvr t b
ysane unit quaternion r and
pure quaternionb
be a set of transformationsEuclideantransformations
ot
The pair CTVHR models Euclidean Solid Geometry
that this is well defined i.e elements
in HR are really invertible transformations on TIand HR satisfies the 3 requirements
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Screwmotions
If r of t Usui f U pureunitquaternion
b parallel to U
then TK r v r t b dis called a IIbscrew motion 1
thou Every Euclidean transformation is a screw
motion but centered at different point
temmat Every Euclidean transformation with
a fixedpoint is a rotation
PI Ci If O is a fixedpoint Then
O To r or tb b
b o Tv rvr is a rotation
I If q is a fixed point let S be a
Euclideantransformation such that Sq o
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fa sisstance SV V g ie F t
Then 5151 has 0 as fixed paint
STS 6 STof Sq O
byeSTS is a rotation
Tis a rotation about an axis
passing thro f
Tv ru qtr q
lemme2 let Tv rvr tb E HR
f asf Usui O f EIRu unit purequaternion
If U and b are perpendicular Tun T is
a rotation about an axis parallel to u
PI Stept i voz
r ub is pure quaternion
Pfofstept Saia U b pure Utb
we have Ub U b t Ux q Ux of
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Ub is pure quaternion
Then ptub cosotus.int Ub
O Usui Ub
Dub ucubsuio
gsosubtbsuioo.es purequaternion
Hana vo Hub is also pure quaternion
steps Li bu Ub ExbotHW2
Ci's ur rU note rnotpure
Cii's brt
rbpfofstep.ci UCaotusino UGO serio
6so tuseiogu uceso uau Ouce.cl SEO
Cii's brt bccsotusiso5 bcosf usid
bosf bus no
bosotubsi.aebye's
cesotuseo brb XX
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steps Vo is a fixed point of 1and hence f is a rotation by louamat
pfofstep3i vo rvortb
rztfuo.rub Mtb
z gtr ubr tb
GrErr l 1 Ubr t bZsuiO
by ofsteps oUrbtb1 Iulosotusinoftzsimo bZseisO
UGO sinotzsingb
zUCesotsino b
uoso ukuiojb
bdaso use.no Ub
tub Vo
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Fuialstep e Rotation axis parallel to U
PI Need to show that
Vottafaxed ofay
are fixed points of T t t C C go
To see these
1 Votta r Votta tb
rvor trur tb
Vor b trunkVo turn 4
stop3g f Votta
rift
Proof of the 1hm
let TV FVr tb tr oso tusui.clb pure quaternion
Decompose b b tbz such that
b t U bz 114
Then Tv rvr tb
rvr tb be
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T Trotationwith axb.bz translationparallel to a parallel toll
by lemma 2
Hence T is a screw motion by definition