计算机图形学computer graphics课件11

PPT&Programs&Labcourse2/share/ComputerGraphics/12TransformationsShandongUniversitySoftwareCollegeInstructor:ZhouYuanfengE-mail:yuanfeng.zhou@3ObjectivesIntroducestandardtransformationsRotationTranslationScalingShearDerivehomogeneouscoordinatetransformationmatricesLearntobuildarbitrarytransformationmatricesfromsimpletransformationsWhytransformation?4BeforeAfterAfterBeforeWhytransformation?5Snowconstruction6GeneralTransformationsAtransformationmapspointstootherpointsand/orvectorstoothervectorsQ=T(P)v=T(u)7AffineTransformationsLineartransformation+translationHowever,anaffinetransformationhasonly12degreesoffreedom

because4oftheelementsinthematrixarefixedandareasubsetofallpossible4x4lineartransformations8AffineTransformationsLinepreservingCharacteristicofmanyphysicallyimportanttransformationsRigidbodytransformations:rotation,translationScaling,shearImportanceingraphicsisthatweneedonlytransformendpointsoflinesegmentsandletimplementationdrawlinesegmentbetweenthetransformedendpoints9PipelineImplementationtransformationrasterizer

uv

u

vTT(u)T(v)T(u)T(u)T(v)T(v)verticesverticespixelsframebuffer(fromapplicationprogram)10NotationWewillbeworkingwithbothcoordinate-freerepresentationsoftransformationsandrepresentationswithinaparticularframe

P,Q,R:pointsinanaffinespaceu,v,w:vectorsinanaffinespace

a,b,g:scalars

p,q,r:representationsofpoints -arrayof4scalarsinhomogeneouscoordinatesu,v,w:representationsofvectors -arrayof4scalarsinhomogeneouscoordinates11TheWorldandCameraFramesWhenweworkwithrepresentations,weworkwithn-tuplesorarraysofscalarsChangesinframearethendefinedby4x4matricesInOpenGL,thebaseframethatwestartwithistheworldframeEventuallywerepresententitiesinthecameraframebychangingtheworldrepresentationusingthemodel-viewmatrixInitiallytheseframesarethesame(M=I)12MovingtheCameraIfobjectsareonbothsidesofz=0,wemustmovecameraframeM=13TranslationMove(translate,displace)apointtoanewlocationDisplacementdeterminedbyavectorvThreedegreesoffreedomP’=P+vPP’v14Howmanyways?Althoughwecanmoveapointtoanewlocationininfiniteways,whenwemovemanypointsthereisusuallyonlyonewayobjecttranslation:everypointdisplacedbysamevector15TranslationUsingRepresentationsUsingthehomogeneouscoordinaterepresentationinsomeframep=[xyz1]Tp’=[x’y’z’1]Tv=[vxvyvz0]THencep’=p+vor

x’=x+vx

y’=y+vy

z’=z+vznotethatthisexpressionisinfourdimensionsandexpressespoint=vector+point16TranslationMatrixWecanalsoexpresstranslationusinga4x4matrixTinhomogeneouscoordinatesp’=TpwhereT=T(vx,vy,vz)=Thisformisbetterforimplementationbecauseallaffinetransformationscanbeexpressedthiswayandmultipletransformationscanbeconcatenatedtogether17Rotation(2D)Considerrotationabouttheoriginby

q

degreesradiusstaysthesame,angleincreasesbyqx’=xcos

q

–ysinqy’=xsinq+ycosq18RotationaboutthezaxisRotationaboutzaxisinthreedimensionsleavesallpointswiththesamezEquivalenttorotationintwodimensionsinplanesofconstantzorinhomogeneouscoordinatesp’=Rz(q)px’=xcosq–ysinqy’=xsinq+ycos

qz’=z19RotationMatrixR=Rz(q)=20RotationaboutxandyaxesSameargumentasforrotationaboutzaxisForrotationaboutxaxis,xisunchangedForrotationaboutyaxis,yisunchangedR=Rx(q)=R=Ry(q)=21ScalingS=S(sx,sy,sz)=x’=sxxy’=syyz’=szzp’=SpExpandorcontractalongeachaxis(fixedpointoforigin)22Reflectioncorrespondstonegativescalefactorsoriginalsx=-1sy=1sx=-1sy=-1sx=1sy=-123InversesAlthoughwecouldcomputeinversematricesbygeneralformulas,wecanusesimplegeometricobservationsTranslation:T-1(dx,dy,dz)=T(-dx,-dy,-dz)Rotation:R

-1(q)=R(-q)HoldsforanyrotationmatrixNotethatsincecos(-q)=cos(q)andsin(-q)=-sin(q)R

-1(q)=RT(q)Scaling:S-1(sx,sy,sz)=S(1/sx,1/sy,1/sz)

24ConcatenationWecanformarbitraryaffinetransformationmatricesbymultiplyingtogetherrotation,translation,andscalingmatricesBecausethesametransformationisappliedtomanyvertices,thecostofformingamatrixM=ABCDisnotsignificantcomparedtothecostofcomputingMpformanyverticespThedifficultpartishowtoformadesiredtransformationfromthespecificationsintheapplication25OrderofTransformationsNotethatmatrixontherightisthefirstappliedMathematically,thefollowingareequivalentp’=ABCp=A(B(Cp))Notemanyreferencesusecolumnmatricestorepresentpoints.Intermsofcolumnmatrices((AB)T=BTAT)p’T=pTCTBTAT26GeneralRotationAbouttheOriginqxzyvArotationbyqaboutanarbitraryaxiscanbedecomposedintotheconcatenationofrotationsaboutthex,y,andzaxesR(q)=Ry(-qy)Rx(-qx)Rz(qz)Ry(qy)Rx(qx)qxqyqzarecalledtheEuleranglesNotethatrotationsdonotcommuteWecanuserotationsinanotherorderbutwithdifferentangles27RotationAboutaFixedPointotherthantheOriginMovefixedpointtooriginRotateMovefixedpointbackM=T(pf)R(q)T(-pf)28InstancetransformationInmodeling,weoftenstartwithasimpleobjectcenteredattheorigin,orientedwiththeaxis,andatastandardsizeWeapplyaninstancetransformationtoitsverticesto Scale Orient Locate29ShearHelpfultoaddonemorebasictransformationEquivalenttopullingfacesinoppositedirections30ShearMatrixConsidersimpleshearalongxaxisx’=x+ycotqy’=yz’=zH(q)=

RotationaroundarbitraryvectorR(Ax,Ay,Az)Ifthenv=w×u31R=Rz(q)=32OpenGLMatricesInOpenGLmatricesarepartofthestateMultipletypesModel-View(GL_MODELVIEW)Projection(GL_PROJECTION)Texture(GL_TEXTURE)(ignorefornow)Color(GL_COLOR)(ignorefornow)SinglesetoffunctionsformanipulationSelectwhichtomanipulatedbyglMatrixMode(GL_MODELVIEW);glMatrixMode(GL_PROJECTION);33CurrentTransformationMatrix(CTM)Conceptuallythereisa4x4homogeneouscoordinatematrix,thecurrenttransformationmatrix(CTM)thatispartofthestateandisappliedtoallverticesthatpassdownthepipelineTheCTMisdefinedintheuserprogramandloadedintoatransformationunitCTMverticesverticespp’=CpC34CTMoperationsTheCTMcanbealteredeitherbyloadinganewCTMorbypostmutiplicationLoadanidentitymatrix:C

ILoadanarbitrarymatrix:C

MLoadatranslationmatrix:C

TLoadarotationmatrix:C

RLoadascalingmatrix:C

SPostmultiplybyanarbitrarymatrix:C

CMPostmultiplybyatranslationmatrix:C

CTPostmultiplybyarotationmatrix:C

C

RPostmultiplybyascalingmatrix:C

C

S35RotationaboutaFixedPointStartwithidentitymatrix:C

IMovefixedpointtoorigin:C

CTRotate:C

CRMovefixedpointback:C

CT-1Result:C=TRT–1

whichisbackwards.Thisresultisaconsequenceofdoingpostmultiplications.Let’stryagain.36ReversingtheOrderWewant

C=T–1RTsowemustdotheoperationsinthefollowingorderC

IC

CT-1C

CRC

CTEachoperationcorrespondstoonefunctioncallintheprogram.Notethatthelastoperationspecifiedisthefirstexecutedintheprogram37CTMinOpenGLOpenGLhasamodel-viewandaprojectionmatrixinthepipelinewhichareconcatenatedtogethertoformtheCTMCanmanipulateeachbyfirstsettingthecorrectmatrixmode38Rotation,Translation,ScalingglRotatef(theta,vx,vy,vz)glTranslatef(dx,dy,dz)glScalef(sx,sy,sz)glLoadIdentity()Loadanidentitymatrix:Multiplyonright:thetaindegrees,(vx,vy,vz)defineaxisofrotationEachhasafloat(f)anddouble(d)format(glScaled)39ExampleRotationaboutzaxisby30degreeswithafixedpointof(1.0,2.0,3.0)RememberthatlastmatrixspecifiedintheprogramisthefirstappliedDemoglMatrixMode(GL_MODELVIEW);glLoadIdentity();glTranslatef(1.0,2.0,3.0);glRotatef(30.0,0.0,0.0,1.0);glTranslatef(-1.0,-2.0,-3.0);40ArbitraryMatricesCanloadandmultiplybymatricesdefinedintheapplicationprogramThematrixmisaonedimensionarrayof16elementswhicharethecomponentsofthedesired4x4matrixstoredbycolumnsInglMultMatrixf,mmultipliestheexistingmatrixontherightglLoadMatrixf(m)glMultMatrixf(m)Matrixmultiply//沿Y轴向上平移10个单位

glTranslatef(0,10,0);

//画第一个球体

DrawSphere(5);

//沿X轴向左平移10个单位

glTranslatef(10,0,0);

//画第二个球体

DrawSphere(5);41procedureRenderScene();

begin

glMatrixMode(GL_MODELVIEW);

//沿Y轴向上平移10个单位

glTranslatef(0,10,0);

//画第一个球体

DrawSphere(5);

//加载单位矩阵

glLoadIdentity();

//沿X轴向上平移10个单位

glTranslatef(10,0,0);

//画第二个球体

DrawSphere(5);

end;42MatrixStacksInmanysituationswewanttosavetransformationmatricesforuselaterTraversinghierarchicaldatastructuresAvoidingstatechangeswhenexecutingdisplaylistsOpenGLmaintainsstacksforeachtypeofmatrixAccesspresenttype(assetbyglMatrixMode)byglPushMatrix()glPopMatrix()GL_PROJECTIONGL_MODEVIEWMatrixStacksprocedureRenderScene();

begin

glMatrixMode(GL_MODELVIEW);

//pushmatrixstack

glPushMatrix;

//translate10alongYaxis

glTranslatef(0,10,0);

//drawthefirstsphere

DrawSphere(5);

//comebacktothelastsavedstate

glPopMatrix;

//translate10alongXaxis

glTranslatef(10,0,0);

//drawthesecondsphere

DrawSphere(5);

end;4344ReadingBackMatricesCanalsoaccessmatrices(andotherpartsofthestate)byqueryfunctionsFormatrices,weuseasglGetIntegervglGetFloatvglGetBooleanvglGetDoublevglIsEnabledfloatm[16];glGetFloatv(GL_MODELVIEW,m);45UsingTransformationsExample:useidlefunctiontorotateacubeandmousefunctiontochangedirectionofrotationStartwithaprogramthatdrawsacube(colorcube.c)inastandardwayCenteredatoriginSidesalignedwithaxesWilldiscussmodelinginnextlecture46main.cvoidmain(intargc,char**argv){glutInit(&argc,argv);glutInitDisplayMode(GLUT_DOUBLE|GLUT_RGB|GLUT_DEPTH);glutInitWindowSize(500,500);glutCreateWindow("colorcube");glutReshapeFunc(myReshape);glutDisplayFunc(display);glutIdleFunc(spinCube);glutMouseFunc(mouse);glEnable(GL_DEPTH_TEST);glutMainLoop();}47IdleandMousecallbacksvoidspinCube(){ theta[axis]+=2.0; if(theta[axis]>360.0)theta[axis]-=360.0; glutPostRedisplay();}voidmouse(intbtn,intstate,intx,inty){char