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

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

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

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

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

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

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

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

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

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

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

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

T Trotationwith axb.bz translationparallel to a parallel toll

by lemma 2

Hence T is a screw motion by definition