PAGE

PAGE

1

Student’sName:

Student’sIDNo.:

CollegeName:

ThestudyofQuaternions

Abstract

Findingthedefinitionofquaternions,operationsofquaternions,andpropertiesofquaternions.Todiscusstheproblemifthesetofquaternionstogetherwiththeoperationsofquaternionsisavectorspaceovertherealnumberfield.Todiscusstheproblemifthesetofquaternionstogetherwiththeoperationsofquaternionsisafield.

Introduction

Searchthedefinitionofquaternions,anddiscusssomepropertiesofthem.Thendiscusstheapplicationsusedbyquaternions.

MainResults

AnswersofQ1

1.1Thedefinitionofquaternion:

Quaternionisthemost

simple

hyper-complex

number.

Thecomplex

iscomposedofa

real

plusthe

elementsofI,

including

i^2=-1.

Similarly,quaternion

iscomposedof

realnumber

plusthree

elementsI,

J,

K,

andthey

havethefollowing

relationship:

i^2=j^2=k^2=ijk=-1,

$foureach

number

isalinearcombinationof

1,

I,

J

andK,

thatisquaternion

itcanbeexpressedasa+bi+cj+dk,

wherea,

B,

C,

Disarealnumber.

1.2Operationsofquaternion

1)Quaternionaddition:p+q

With

complexnumbers,

vectorsandmatrices,

thesumoftwo

quaternion

needto

combinedifferentelements

together.

The

addition

followsthe

commutativeandassociativelaws

ofrealandcomplexnumber.

2)Quaternionmultiplication:pq

Betweentwo

toquaternion

inthenumberof

non-commutative

product

usuallyisGlassman

(Hermann

Grassmann)

iscalledthe

product,

the

product

abovehasbeenbriefly

introduced,

complete

type

it

is:

Becauseof

quaternionmultiplicationcannotbechanged

,

pqisnot

equaltoqp.

Glassmanproduct

used

inthedescriptionof

manyother

algebraicfunction.

Thevector

product

is

partofqp:

3)Quaterniondotproduct:

p·q

Thedotproduct

iscalledthe

Euclidean

innerproduct,

quaternion

dotproductisequivalenttoa

four-dimensionalvector

dotproduct.

Thedotproduct

valueis

thecorrespondingelement

numericalvalue

ofeachelementinthep

and

q

.

Thisisbetweenquaternion

canchangetheproduct

number,

andreturnsa

scalar.

Thedotproduct

canuse

Glassmanproduct

form:

This

product

isusefulfor

theelementsof

isolated

fromquaternion

.

Forexample,

ican

comeout

fromp

extraction:

4)Quaternionouterproduct:Outer(p,q)

TheEuclideanouterproduct

isnot

commonlyused;However,

because

theouterproductand

the

product

formofthe

Glassmaninnerproduct

similarity,

theyarealways

to

bementioned:

5)Quaternionevenproduct:Even(p,q)

Quaternionevenproductisnot

commonlyused,

butit

willbementioned,becauseofitssimilarwithodd

product.

Itisapure

symmetricproduct;therefore,

itiscompletely

interchangeable.

6)Quaternioncrossproduct:p×q

Quaternion

crossproduct

alsoknownas

odd

product.

It

is

equivalenttothecrossproductofvectors

,

and

onlyreturn

onevectorvalue:

7)Quaterniontransposition:

Quaterniontransposition’sdefinitionisby.The

sameway

to

constructcomplex

inversestructure:

Aquaternionitselfdotmultiplicationisascalar.quaterniondividedby

ascalar

isequivalentto

the

scalar

multiplicationonthe

countdown,

buttomakeevery

elementofthequaternion

isdividedby

a

divisor.

8)Quaterniondivision:

Quaternion’sunchangeablepropertyleadtothedifferenceofand.Thismeansthatunlessthe

pisa

scalar,

otherwise

youcannotusetheq/p.

9)QuaternionScalar

Department:Scalar(p)

10)Quaternionvectordepartment:Vector(p)

11)QuaternionModulus:|p|

12)Quaternionsignalnumber:Sgn(p)

13)Quaternionargument:Argu(p)

1.3Propertiesofquaternion

Quaternionis

shapedlikea

numberofai+bj+ck+d,

a,

b,c,disarealnumber.

AnswersofQ2

2.Therearetwoways

to

thematrixrepresentationof

quaternion.

Justascomplexnumberscanbe

\o"Complexnumber"

representedasmatrices

,socanquaternions.Thereareatleasttwowaysofrepresentingquaternionsas

\o"Matrix(mathematics)"

matrices

insuchawaythatquaternionadditionandmultiplicationcorrespondtomatrixadditionand

\o"Matrixmultiplication"

matrixmultiplication

.Oneistouse2?×?2

\o"Complexnumber"

complex

matrices,andtheotheristouse4?×?4

\o"Realnumber"

real

matrices.Ineachcase,therepresentationgivenisoneofafamilyoflinearlyrelatedrepresentations.Intheterminologyof

\o"Abstractalgebra"

abstractalgebra

,theseare

\o"Injectivefunction"

injective

\o"Homomorphism"

homomorphisms

from

H

tothe

\o"Matrixring"

matrixrings

M(2,

C)

and

M(4,

R),respectively.

Using2?×?2complexmatrices,thequaternion

a

+

bi

+

cj

+

dk

canberepresentedas

Thisrepresentationhasthefollowingproperties:

Constraininganytwoof

b,

c

and

d

tozeroproducesarepresentationof

\o"Complexnumber"

complexnumbers

.Forexample,setting

c

=

d

=0

producesadiagonalcomplexmatrixrepresentationofcomplexnumbers,andsetting

b

=

d

=0

producesarealmatrixrepresentation.

Thenormofaquaternion(thesquarerootoftheproductwithitsconjugate,aswithcomplexnumbers)isthesquarerootofthe

\o"Determinant"

determinant

ofthecorrespondingmatrix.

[20]

Theconjugateofaquaternioncorrespondstothe

\o"Conjugatetranspose"

conjugatetranspose

ofthematrix.

Byrestrictionthisrepresentationyieldsa

\o"Groupisomorphism"

isomorphism

betweenthesubgroupofunitquaternionsandtheirimage

\o"SU(2)"

SU(2)

.Topologically,theunitquaternionsarethe

\o"3-sphere"

3-sphere

,sotheunderlyingspaceofSU(2)isalsoa3-sphere.ThegroupSU(2)isimportantfordescribing

\o"Spin(physics)"

spin

in

\o"Quantummechanics"

quantummechanics

;see

\o"Paulimatrices"

Paulimatrices

.

Using4?×?4realmatrices,thatsamequaternioncanbewrittenas

Inthisrepresentation,theconjugateofaquaternioncorrespondstothe

\o"Transpose"

transpose

ofthematrix.Thefourthpowerofthenormofaquaternionisthe

\o"Determinant"

determinant

ofthecorrespondingmatrix.Aswiththe2?×?2complexrepresentationabove,complexnumberscanagainbeproducedbyconstrainingthecoefficientssuitably;forexample,asblockdiagonalmatriceswithtwo2?×?2blocksbysetting

c

=

d

=0.

AnswersofQ3

Becausethevectorpartofaquaternionisavectorin

R3,thegeometryof

R3

isreflectedinthealgebraicstructureofthequaternions.Manyoperationsonvectorscanbedefinedintermsofquaternions,andthismakesitpossibletoapplyquaterniontechniqueswhereverspatialvectorsarise.Forinstance,thisistruein

\o"Electrodynamics"

electrodynamics

and

\o"3Dcomputergraphics"

3Dcomputergraphics

.

Fortheremainderofthissection,

i,

j,and

k

willdenotebothimaginary

[18]

basisvectorsof

H

andabasisfor

R3.Noticethatreplacing

i

by?i,

j

by?j,and

k

by?k

sendsavectortoitsadditiveinverse,sotheadditiveinverseofavectoristhesameasitsconjugateasaquaternion.Forthisreason,conjugationissometimescalledthe

spatialinverse.

Choosetwoimaginaryquaternions

p

=

b1i

+

c1j

+

d1k

and

q

=

b2i

+

c2j

+

d2k.Their

\o"Dotproduct"

dotproduct

is

Thisisequaltothescalarpartsof

pq?,

qp?,

p?q,and

q?p.(Notethatthevectorpartsofthesefourproductsaredifferent.)Italsohastheformulas

The

\o"Crossproduct"

crossproduct

of

p

and

q

relativetotheorientationdeterminedbytheorderedbasis

i,

j,and

k

is

(Recallthattheorientationisnecessarytodeterminethesign.)Thisisequaltothevectorpartoftheproduct

pq

(asquaternions),aswellasthevectorpartof?q?p?.Italsohastheformula

Ingeneral,let

p

and

q

bequaternions(possiblynon-imaginary),andwrite

where

ps

and

qs

arethescalarparts,and

and

arethevectorpartsof

p

and

q.Thenwehavetheformula

Thisshowsthatthenoncommutativityofquaternionmultiplicationcomesfromthemultiplicationofpureimaginaryquaternions.Italsoshowsthattwoquaternionscommuteifandonlyiftheirvectorpartsarecollinear.

Forfurtherelaborationonmodelingthree-dimensionalvectorsusingquaternions,see

\o"Quaternionsandspatialrotation"

quaternionsandspatialrotation

.ApossiblevisualisationwasintroducedbyAndrewJ.Hanson.

AnswersofQ4

1）Applicationofquaternionsinthe

attitudeofarigidbody

simulation

With

symmetricgyroscope

asanexample,

discussestheexisting

applicationandthe

quaternions

inthe

attitudeofarigidbody

simulation

problemin.

Thatattitude

withquaternionsdescription

hasasolution

quickly,

won't

appearsingular

advantages,

but

implied

quaternions

equation

constraint

isdifferentialforms,

whichleadtoa

strictlimitonthe

simulation

timestep,

whichlimitsits

applicationin

acertainextent.

Finally

discussestheimplementationofattitudedescription

uniqueness

problem

withquaternions,

and

putforwardtheconceptof"standard"

quaternions.

2）Applicationofunit

quaternionsin

aerialphoto-grammetry

solution

Researchon

unit

quaternionsmethod

in

aerial

applicationof

aerialtriangulation

ineachstep

of

the

algorithm,

andthe

stabilityandapplicability

isevaluated.

Thefirstdescribesthe

methodofunit

quaternions

tectonicrotation

matrixbasedon

relativeorientation,

establishing

modeland

basedon

thenumberofunits

quaternionssettlement

methodforthe

modelisconstructed

basedonthebeammethod;

regional

networkunit

quaternionsrientationandbundleblockadjustment

test,

and

withthetraditional

Eulerangle

toconstructtherotationmatrix

basedschemesarecompared.

Thetestresultsshowthat,

inthe

relativeorientation

test,

iftake

P-Halgorithm,

whichrequiresonlyminimalcontrolpointsto

ensurethatall

testdata

canobtain

thecorrectsolution.

Whilein

thebundleadjustmentmethod,

methodofunitqu