transformation(变换)

Lesson13---TransformationsTransformationIfashapeismovedfromonepositiontoanotheritundergoesatransformation.Therearefourbasicwaysofchangingthepositionandsizeoftwo-dimensionalshapes:atranslation,areflection,arotationoranenlargement.Allofthesetransformations,exceptenlargement,keepshapescongruent.WelcometotheworldofTranslation!1.TranslationShanghaiConcertHallIt’ssplendid,right?Inordertomeettheneedsofurbanplanning,ShanghaiConcerthallmustberemoved……要拆掉？不会吧！可不拆怎么办？Inthefollowingsymbols,canyouguesswhichoneisdesignedby“translation”?Question1Atranslationisatransformationthatmovesthegraphforacertaindistancealongwithafixeddirectioninaplane.1.Definition2.QualityAsfortheobjectandimage,theshapeandsizeareidentical.Asforanytwocorrespondingpointsoftheobjectandimage,theyhavethesamedistanceanddirection.EFGHACBD3.DescriptionofTranslationWedescribetranslationsbyusingcolumnvectors.Acolumnvectorisrepresentedbythecombinationofahorizontalshiftandaverticalshift.1.TranslationRemember,todescribeatranslationfully,youneedtouseacolumnvectorExerciseP4651ABECDFXYInthefollowinggraph,△ABEistranslatedalongtherayXYforafixeddistanceinto△CDF，pleasefindthecongruenttrianglesandtheparallelandequidistantsegments,andthenexpressthemovementof△ABEasthevectorform.Question3Asisshowninthefollowingfigure,thewindowiscomposedofonesemicircleandarectangle,canyoudrawitsimagethroughtranslatingsixunitstotheleft.Question3QAsisshowninthefigure,fortherectangle,thelengthis2cm,widthis1cm,theradiusofsectorBFE,FCDis1cm,canyoucalculatetheareaoftheshadepart?CADEBFApplicationinlife想到答案了吧？以为要拆了再建吧？整体平移就可以了，哈哈！“建筑物的整体平移技术”是将建筑物托换到一个托架上，与地基切断，形成一个可移动体，然后再用牵引设备将它平移到固定的新地基上。AfterTranslation2.ReflectionAreflectiontransformsashapesothatitbecomesamirrorimageofitself.objectimagemirrorline2.ReflectionRemember,todescribeareflectionfully,youneedtouseamirrorline(lineofsymmetry)ExerciseP4661,63.RotationsArotationtransformsashapetoanewpositionbyturningitaboutafixedpointcalledthecentreofrotation.ABCDP(第5题)3.RotationsNote:Thedirectionofangleofrotationisexpressedasclockwiseoranticlockwise.Thepositionofthecentreofrotationisalwaysspecified.Therotation180〫clockwiseand180〫anticlockwisearethesame.Therotationsthatmostoftenappearinexaminationquestionsare90〫and180〫.Page470Example4.3.RotationsTofindthecentreofrotationyoushouldjoincorrespondingpointsontheobjectandimagewithstraightlinesanddrawtheperpendicularbisectorsoftheselines.Thecentreofrotationliesontheintersectionofthesebisectors.Tofindtheangleofrotationyoushouldjoincorrespondingpointsontheobjectandimagetothecentreofrotation.Theanglebetweentheselinesistheangleofrotation.ExerciseP4714,5,6&P47333.RotationsRemember,todescribearotationfully,youneedacentreofrotation,anangleofrotationandthedirectionofrotation.4.EnlargementAnenlargementisatransformationinwhichthedistancebetweenapointontheimageandthefixedpoint(calledthecentreofenlargement)isafactorofthedistancebetweenthecorrespondingpointontheobjectandthefixedpoint.ACBA'B'C'O4.EnlargementTofindthecentreofenlargementyoujoincorrespondingpointsontheobjectandimage---withstraightlines.Thecentreofenlargementliesattheintersectionofthesestraightlines.Youcanfindscalefactor(SF)ofanenlargementasfollows:distancefromthecentreofapointontheimageSF=-----------------------------------------------------------------------------------------------,distancefromthecentreofthecorrespondingpointontheobjectordistancebetweentwopointsontheimageSF=------------------------------------------------------------------------------------------distancebetweentwocorrespondingpointsontheobjectExample5&Example7&Example104.EnlargementAscalefactorlessthan1willhavetheeffectofreducing(orshrinking)theobject.Example9Anegativescalefactorproducesanimageshapeontheoppositesideofthecentreofenlargementtotheoriginalshape.Example84.EnlargementTodescribefullyanenlargementbygivingthepositionofthecentreofenlargementalongwiththefactor(calledthescalefactor).