Download Geometric Transformations Sequence

GeometricTransformationsSequence
Geometrictransformationsareimplicitlypresentinmathematicsteachingfromthe
veryearlystagesofgradeschoolallthewayuptoadvancedtopicsinsecondaryschool
andtheuniversity.Theyprovideanimportantunifyingconceptfromearlyintuitive
observationstomoreabstractalgebraicnotions.Theycanbeusedtounify
understandingofsimplegeometricideas,theconceptsofadvancedtransformationsin
geometryandlatertheconceptualizationoffunctionsandtheirgraphs.Geometric
transformationsalsoprovideconnectionstocomplexnumbers,matrices,vector
algebra,ingeneral,andtoadvancedmathematicaltopics.Inthisbrief,geometric
transformations(includingrigidtransformations,referredtoasisometries,andthose
thatpreserveshapebutnotnecessarilysize,referredtoasdilations,orsimilarities
wheredilationsarecomposedwithisometries)arepresentedasasequenceofideas
thatevolveanddevelop,independentlyofcurriculaindifferentcountries,todifferent
stagesatdifferentageofstudents.
1. Intheearlygradesofschooling,theveryfirstapproachshouldbeintuitive.
a) Studentsshouldbeginbyobservingsimplesymmetricshapes.
i.
Astheyobserveexamplesofshapesthataresymmetricoveralineorabout
apoint,studentsshouldbechallengedtodescribetheshapesandwhat
makestheshapessymmetric.Infurtherexamplesstudentscanbe
challengedtofilloutincompleteshapestomakethemsymmetric.
ii. Studentsobservecongruentandsimilarfigures,startingwithsimple
shapesthatarenon‐mathematical.Thentheshapesareswitchedto
triangles,rectanglesandothermathematicalgeometricfigures.
b) Constructionsnaturallyevolvefromobservationsofpropertiestohavingthose
propertiesexplainedbytransformations,albeitinformally.
i.
Throughobservationsanddiscussions,studentsdeterminethatcongruent
figurescanbemovedtoexactly“cover”eachother,andsimilarfigurescan
bemovedandenlargedorshrunktocoincidewitheachother.Through
observations,studentsalsodeterminethatinsymmetricandcongruent
figures,correspondingdistancesarepreservedandthatinsimilarfigures,
allthecorrespondingdistancesenlargeorshrinkproportionallywhilelines
remainparallel.Theyalsonotethatmeasuresofcorrespondingangles
remainunchangedinallofthetransformations.
ii. Movementsofcongruentfiguresaredescribedandspecifiedintermsofthe
isometries:translations,rotations,reflectionsandglidereflections.With
thesemovements,correspondingdistancesandanglemeasuresremain
unchanged.Changingthesizebutnottheshapeoffiguresisdescribedin
termsofdilationsorsimilarities.Withthelattertransformations,all
correspondingdistanceschangebythesamefactorandallcorresponding
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iii.
iv.
anglemeasuresremainunchanged.Studentsshouldbechallengedby
combiningtwoormoretransformationsconsecutively.
Studentsbecomefamiliarwithcongruentfiguresobtainedbyisometries
andsimilarities.Studentspracticegeometrictransformationsmore
formallyusingacompassandarulertodrawcongruentandsimilarfigures,
orusingdynamicgeometrysoftware,asrequiredbyspecifictasks.
Studentsbecomeacquaintedwithproblemswhereproofsarerequiredand
whereproportionalreasoningisusedinpracticalgeometricproblems.(See
examples1and2laterinthebrief.)
Exploringisometriesinrelationtogeometryofsolidscanaidindeveloping
students'spatialvisualization.
2. Ananalyticapproachtogeometrictransformationsinacoordinatesystemis
introducedafterstudentsarefamiliarwithcoordinates.
a) Geometrictransformationsstudiedwithoutcoordinatesarereviewedand
interpretedinacoordinatesystem.Allpreviouslyintroducedpropertiesare
interpretedwithinthecoordinatesystem.Nowdistancescanbeunderstoodand
calculatedwithknowledgeofcoordinates(using,forexample,thePythagorean
theorem).Studentsconsiderreflectionsovercoordinatelinesandrotations
abouttheorigin.
b) Algebraicdescriptionsoftransformationsareintroduced.Firstexamplesinclude
arotationof180ºabouttheorigin(sometimescalledareflectionoverthe
origin)addressedfirstwithconcretepoints,forexample(1,1)‐‐>(–1,–1)before
proceedingtothegeneral(x,y)‐‐>(–x,–y).Similarly,reflectionsoverthex–and
y–axesshouldfirstbeconsideredusingconcretepointsandtheningeneral(x,
y)‐‐>(x,–y)forthex–axisand(x,y)‐‐>(–x,y)forthey–axis.Generalnotationfor
thethreementionedtransformationsfollow:RO,180º:(x,y)‐‐>(–x,–y),rx:(x,y)‐‐
>(x,–y)andry:(x,y)‐‐>(–x,y)shouldalsobeintroduced.Translationsshould
startwithsimplenotation,forexample,thetranslation“moveright2units”
shiftsthepoint(3,4)tothepoint(5,4).Movingthepoint(1,1)left2anddown3
wouldgivethepoint(–1,–2).Movingthegeneralpoint(x,y)by(1,2)(right1
andup2)givesthepoint(x+1,y+2).Thenafterevenmoregeneralization
moving(x,y)to(x+a,y+b),thenotationofthetranslationintheformofTa,b:(x,
y)‐‐>(x+a,y+b)isfinallyintroduced.Transformationswithregardtotheorigin
andthecoordinateaxesshouldbeconsideredwithspecialcare;reflectionswith
regardtogenerallinescouldbeconsideredusingthecoordinates.Other
transformationscouldalsobeintroducedstepbystep.Rotationsfor90o,270o
and0o,or360o,canbedescribedforallstudents.Forexamplefortherotation
for90owiththeoriginascenterintheformRO,90o:(x,y)‐‐>(–y,x).Ifpossible,
wherestudentshaveabackgroundintrigonometry,rotationsshouldbe
describedinthegeneralformasRO,φ:(x,y)
‐‐>(xcos(φ)–ysin(φ),xsin(φ)+ycos(φ)).
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c) Specialattentionshouldbegiventothefactthattheformalnotationof
geometrictransformationsmightbequiteabstractforstudents.Thisproblem
shouldbeaddressedbyenoughpracticalexercisesthatdevelopstudents'
understandingofnewnotationsstepbystep.Furtherunderstandingofthe
notationcanbeachievedbyintroducingawidevarietyofdifferent
transformationswithinconcreteproblemsandproofs,bydiscussingissueslike
inversetransformationandfinallybycomposingtwoormoretransformations
byformalcomposition(forexample,Ta,b◦RO,90o).
d) Translatinganddilatingarereviewedwithsimplefiguresinthecoordinate
system.Analysisisshiftedtothequestion,whatwouldhappentoashapelikea
parabolawhentranslationsareapplied?Studentsnoticethemselvesorare
guidedtotherealizationthatthesearejusttheshiftsanddilationsofthegraphs
theyknowanalytically.Studentsvisualizethat,forexample,agraphofa
quadraticfunctiony=(x–p)2+qcorrespondstothetranslationofthegraphofa
functiony=x2punitsrightandqunitsup.Studentsshouldbeexposedtoseveral
similarsituations.Finallyformalnotationfortransformationsshouldbe
introducedinthefollowingform.Thegraphofaquadraticfunctiony=x2
consistsofallthepoints(x,x2)thataretransformedasT(p,q):(x,x2)‐‐>(x+p,x2+
q).Probablythehardeststepforstudentstocomprehendisthatpoints(x+p,x2
+q)fittheequationx2+q=(x+p–p)2+q.Ortaking(X,Y)=(x+p,x2+q),one
hasY=(X–p)2+q.Thisideacanbeusedtoexplorethefactthatdifferent
transformationstransformparabolastoparabolas.Alsoforexample,adilation
DO,k:(x,y)‐‐>(kx,ky)transformsthegraphofaquadraticfunctiony=ax2+bx+c
tothegraphofanewquadraticfunctionwithrelatedcoefficientsasfollows:
.
3. Makingconnectionstoothermathtopics
a) Ifandwhenstudentsarefamiliarwithvectors,transformations(especially
translations)canberepresentedbytheuseofvectors.Theconnectiondeepens
thecomprehensionofbothtopics.Forexample,atranslationright2unitsandup
3unitscanberepresentedassimplyaddingthevector(2,3).Thereforeprevious
notationTa,b:(x,y)‐‐>(x+a,y+b)canbedescribedsimplyintermsof
: → ,where , and , .Seeexample4asanexercise.
b) Ifandwhenstudentsarefamiliarwithcomplexnumbers,aswithvectors,
translationscansimplybepresentedasadditionofcomplexnumbers.Theabove
:
→
.
translationwouldbedescribedas
Adilationcanberepresentedassimplemultiplicationofcomplexnumbersby
realnumbers.Thatisadilationwithcenterattheorigin DO,k (x, y)  (kx, ky) can
berepresentedintheformDO,k:x+iykx+iky.Areflectionoverthex–axisis
givenbyconjugation.Complexnumbersprovidealsoapowerfultoolto
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representrotations.Forexample,arotationby90oabouttheoriginissimply
multiplicationbytheimaginaryuniti.Ifcomplexnumbersareknownto
studentsinmoreadvancedlevelsusingpolarform,therotationofanobject
throughangleφaroundtheorigincanberepresentedbymultiplicationbythe
complexnumbercos(φ)+isin(φ)or,ifthenotationisknowntostudents,bythe
numbereiφ.
c) Ifandwhenstudentsarefamiliarwithmatrices,itiseasytoshowthatadilation
withcenterattheorigincanbepresentedinmatrixformas
0
0
wherethedilationpreviouslynotedasDO,kisnowpresentedasanoperator
0
.Further,arotationabouttheoriginthroughtheangleφisgivenbythe
0
matrix
cos φ
sin φ
sin φ
.
cos φ
Compositionofallmentionedtransformations,excepttranslations,cannowbe
efficientlyrepresentedbyaproductofmatrices.
Translationscannotberepresentedbymultiplicationof2by2matrices,as
matricesrepresentlineartransformationsthatalwayspreservetheorigin,while
translationsdonot.However,translationscanberepresentedasadditionof2x
1matricesintheform
→
.
(Additionally,3x3matricescanbeusedtodepictallisometriesasseeninstep4
below.)
d) Examples5and6helpstudentsacquiredeeperunderstandingofgeometric
transformations.
e) Example7showshowanalgebraicproblemcanbesolvedbythemeansof
geometrictransformations.
f) Example8isgeometricandisusuallysolvedbystudentswiththeuseof
trigonometricequationsthatfollowfromgeometricproperties.However,
transformationsofferanelegantsolution.
4. Extendedperspectiveoftransformationsusing3x3matrices
Geometrictransformationsin2Dcanalsobeconsideredusingspecialcasesof3by
3matrices.Forexample,thematrixbelowrepresentsarotationofφthroughthe
origin.
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 cos( ) sin( ) 0 


 sin( ) cos( ) 0 

0
0
1 

Usinghomogeneouscoordinates,firstdiscussedbyAugustFerdinandMöbiusin
1827,apointwithcoordinates(x,y)maybeidentifiedasthetriple(x,y,1)sothat
thematrixdimensionsworkandthematrixmultiplicationcanbeaccomplishedas
follows:
 cos( ) sin( ) 0 
  x cos( )  y sin( )

 x  
 sin( ) cos( ) 0  y    x sin( )  y cos( )

0
0
1  1  
1






Aprimaryreasonforusing3x3matriceswithgeometrictransformationsisthat
translationsdonothavetobetreatedseparatelywithadifferentformofmatrixand
canbecomposedwithanyothertransformationssimplybymultiplyingmatrices.
Forexampleusing3x3matrices,thetranslationTA,B:(x,y)–>(x+a,y+b)isseen
belowwhereageneralpointintheplanehascoordinates(x,y,1):
1
0
0
0
1
0
a
∙
1
1
=
1
5. Transformationsmayalsobeconsideredwithintheiralgebraicgroupstructures
withtheoperationofcomposition.Forexample,isometriesallhaveinverses,have
anidentityandthepropertyofassociativity,providingthestructureofagroup.
Further,consideringalltherotationsabouttheoriginandallthereflectionsovera
linethroughtheorigin,thesubgroupofallorthogonaltransformationsisobtained.
Inthenotionofmatrices,thisgroupisdefinedbyallorthogonalmatrices,thatisto
saythateachrowandeachcolumninthematrixconsideredasavectorhaslength1
andisorthogonal(perpendicular)totheotherone.Allthesematriceshave
determinant1or–1.Compositionoftransformationsisnotcommutativeingeneral
asstudentscaneasilyseebyconstructionwithinthegeometricinterpretationor
canbeseenbyconsideringthematrixproduct.
Examples:
1. Findtheareaofasquarewithdiagonal6cminFigure1.
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Figure1:Squarewithdiagonal6cm
WhiletheproblemcaneasilybesolvedbythemeansofthePythagoreantheorem,
theproblemcanalsobesolvedbysplittingthesquareintofourisoscelesright
.Anelegantsolution
triangles,eachwithlegs3cm,andthereforetheareais18
canalsobeobtainedbytheuseofgeometrictransformations.AsFigure2below
indicates,therotationofthetriangle(halfofthesquare)producesabiggersquare
withtheside6cm,andhalfofthissquare'sareaisexactlytheareaoftheoriginal
square.Thereforetheareais
∙
18
.
Figure2:Transformationsolutiontotheareaofasquarewhosediagonalis6units
2. Inscribeasquareinanygiventriangle,sothatonesideofthesquareliesononeof
thesidesofthetriangle.
Theproblemcaneasilybesolvedbydilation.Draw'asmallsquare'asseeninFigure
3below.Byadilation(linethroughupperrightcorner)enlargethesquareasseen
inthefigure.Theproofoftheconstructionfollowsfromthefactthatcorresponding
sidesareproportional.
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Figure3:Trianglecontainingasquarewithasideonthetriangle
3. Anequilateraltrianglesitsonasquare(allthesidesareequalinlength,sayequalto
a)asseeninFigure4.FindtheradiusofthecirclecircumscribingthetriangleABC.
Figure4:Squarewithequilateraltriangleatop
Theproblemisnottrivialand(advanced)studentswouldusuallyfindasolution
usingtrigonometricequationsobtainedfromthegeometricpropertiesofthefigure.
Geometrictransformationsofferanelegantandsimplesolution.
BytheuseofatranslationthetriangleismoveddownasseeninFigure5.
Figure5:Squarewithtriangleatopwithatranslation
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Itisobviousthatthe'move'CDofthelengthawherealsoAD=BD=DC=a.
Thereforetheradiusofthecircumscribedcircleisa.
4. Considerthecompositionoftwo180ºrotationsindifferentcenters,AandB.Letthe
twopointsAandBinplanebegivenasinFigure6.IfRA,180ºandRB,180ºarethe
rotationsinpointsAandBrespectively,thenwhatis RB, 180  RA, 180 ?

Bytheuseofvectors, RA, 180 (X)  2A  X usingtheequivalence X  X  O .So,
RB, 180  RA, 180 (X)  RB, 180 (2A  X)  2B  (2A  X)  X  2(B  A) Therefore,the
compositionoftwo180ºrotationsinpointsAandBissimplyatranslationof
2
.
Figure6:Compositionof180ºrotationsintwopointsAandB
5. Showthatanyrigidtransformationcanbeobtainedbycompositionofatmostthree
reflectionsoverlines.
Thesketchoftheproofofthestatementispresentedinthefollowingsequenceof
arguments.
a) Note,thatanyrigidtransformationisgivenbyimageofthreenoncollinear
points.Therefore,assumearigidmotiontransformstriangleABCto
congruenttriangleA'B'C'asseeninFigure7(a)below.
′),triangleABCis
b Byreflection overthebisectorof CC ' (assume
transformedontotriangle
.NotethattrianglesA'B'C'and
are
congruent,asinFigure7(b).
′,triangle
is
c Similarlybyreflection overthebisectorof
transformedontotriangle
.NotethattrianglesA'B'C'and
are
congruent.
′,triangle
istransformed
d Byreflection overthebisectorof
ontotriangleA'B'C'.
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e
Therefore,bythecompositionofthethreereflections ° ° thetriangle
ABCistransformedontotriangleA'B'C',whichprovesthatanyrigid
transformationcanbeobtainedbythecompositionofatmostthree
reflections.
(a)
(b)
Figure7:Reflectionsinlines
6. UsethenotionsinExample5toshowthatanyisometryiseitherareflection,
translation,rotationorglidereflection.
TheproofhereusesExample5.WithExample5,weknowthatanyisometrycanbe
writtenasthecompositionofnomorethan3reflectionsinlines.Ifthereisonlyone
lineneeded,thentheisometryisclearlyareflection.Iftwolinesofreflectionare
used,theneitherthetwolinesareparallelinwhichcaseitisatranslation,orthe
linesintersectinwhichcaseitisarotationaboutthepointofintersection.Inthe
casethatthreelinesarerequired,itmaytakesometimetoshowthiswithcases,but
itcanbeshownthatitisaglidereflection.
7. Findthenumberofsolutionsofthesystemofequationsforspecificvaluesofa.
1
ThegeometricrepresentationsofthetwoequationsproduceFigure8below.The
circlewithcenter(a,0)andradius1correspondingtothesecondequationcanbe
visualizedastranslatedfromrighttoleft,andthenumberofitsintersectionswith
thetwolinescorrespondingtothefirstequationwillgivethenumberofsolutionsof
2,
thesystemofequations.Itcanbededucedthattherearetwosolutionswhen
thattherearethreesolutionswhen
1,fourwhen
2and
1andnone
when
2.
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Figure8:Circlewithtwolines
Contributing authors
Bjorkqvist, Ole; Abo Akademi University,Vasa, Finland
Castellón, Libni Berenice; Universidad Pedagógica Nacional Francisco,Tegucigalpa, Honduras
Govender, Vasuthavan (Nico); University of Fort Hare, Alice,Port Elizabeth, South Africa
Hernendez, Maria; North Carolina School of Science and Mathematics, Durham, North Carolina,
USA
Kobal, Damjan; University of Ljubljana, Ljubljana, Slovenia
Kejzar, Bogdan; Gimnazija Kranj, Kranj, Slovenia
Lott, Johnny; Retired, University of Montana, Oxford, Mississippi, USA
Li, Shiqi; East China Normal University, Shanghai, China
Osterberg, Leif; Korsholms Gymnasium, Kvevlax, Finland
Sambo, Jobe Nhlanhla; Dlumana High School, Hluvukani, South Africa
Soto, Luis; Colonia Miraflores Sur, Tegucigalpa, Honduras
Tao, Yexin (Madeline); Weiyu High School, Shanghai, China
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