交互式計算機圖形學(xué)(第五版)1-7章課后題答案

交互式計算機圖形學(xué)(第五版)1-7章課后題答案交互式計算機圖形學(xué)(第五版)1-7章課后題答案交互式計算機圖形學(xué)(第五版)1-7章課后題答案xxx公司交互式計算機圖形學(xué)(第五版)1-7章課后題答案文件編號：文件日期：修訂次數(shù)：第1.0次更改批準審核制定方案設(shè)計，管理制度Angel:InteractiveComputerGraphics,FifthEditionChapter1SolutionsThemainadvantageofthepipelineisthateachprimitivecanbeprocessedindependently.Notonlydoesthisarchitectureleadtofastperformance,itreducesmemoryrequirementsbecauseweneednotkeepallobjectsavailable.Themaindisadvantageisthatwecannothandlemostglobaleffectssuchasshadows,reflections,andblendinginaphysicallycorrectmanner.WederivethisalgorithmlaterinChapter6.First,wecanformthetetrahedronbyfindingfourequallyspacedpointsonaunitspherecenteredattheorigin.Oneapproachistostartwithonepointonthezaxis(0,0,1).Wethencanplacetheotherthreepointsinaplaneofconstantz.Oneofthesethreepointscanbeplacedontheyaxis.Tosatisfytherequirementthatthepointsbeequidistant,thepointmustbeat(0,2p2/3,?1/3).Theothertwocanbefoundbysymmetrytobeat(?p6/3,?p2/3,?1/3)and(p6/3,?p2/3,?1/3).Wecansubdivideeachfaceofthetetrahedronintofourequilateraltrianglesbybisectingthesidesandconnectingthebisectors.However,thebisectorsofthesidesarenotontheunitcirclesowemustpushthesepointsouttotheunitcirclebyscalingthevalues.Wecancontinuethisprocessrecursivelyoneachofthetrianglescreatedbythebisectionprocess.InExercise,wesawthatwecouldintersectthelineofwhichthelinesegmentispartindependentlyagainsteachofthesidesofthewindow.Wecoulddothisprocessiteratively,eachtimeshorteningthelinesegmentifitintersectsonesideofthewindow.Inaone–pointperspective,twofacesofthecubeisparalleltotheprojectionplane,whileinatwo–pointperspectiveonlytheedgesofthecubeinonedirectionareparalleltotheprojection.Inthegeneralcaseofathree–pointperspectivetherearethreevanishingpointsandnoneoftheedgesofthecubeareparalleltotheprojectionplane.Eachframefora480x640pixelvideodisplaycontainsonlyabout300kpixelswhereasthe2000x3000pixelmovieframehas6Mpixels,orabout18timesasmanyasthevideodisplay.Thus,itcantake18timesasmuchtimetorendereachframeifthereisalotofpixel-levelcalculations.TherearesinglebeamCRTs.Oneschemeistoarrangethephosphorsinverticalstripes(red,green,blue,red,green,....).Themajordifficultyisthatthebeammustchangeveryrapidly,approximatelythreetimesasfastaeachbeaminathreebeamsystem.Theelectronicsinsuchasystemtheelectroniccomponentsmustalsobemuchfaster(andmoreexpensive).Chapter2SolutionsWecansolvethisproblemseparatelyinthexandydirections.Thetransformationislinear,thatisxs=ax+b,ys=cy+d.Wemustmaintainproportions,sothatxsinthesamerelativepositionintheviewportasxisinthewindow,hencex?xminxmax?xmin=xs?uw,xs=u+wx?xminxmax?xmin.Likewiseys=v+hx?xminymax?ymin.Mostpracticaltestsworkonalinebylinebasis.Usuallyweusescanlines,eachofwhichcorrespondstoarowofpixelsintheframebuffer.Ifwecomputetheintersectionsoftheedgesofthepolygonwithalinepassingthroughit,theseintersectionscanbeordered.Thefirstintersectionbeginsasetofpointsinsidethepolygon.Thesecondintersectionleavesthepolygon,thethirdreentersandsoon.Therearetwofundamentalapproaches:vertexlistsandedgelists.Withvertexlistswestorethevertexlocationsinanarray.Themeshisrepresentedasalistofinteriorpolygons(thosepolygonswithnootherpolygonsinsidethem).Eachinteriorpolygonisrepresentedasanarrayofpointersintothevertexarray.Todrawthemesh,wetraversethelistofinteriorpolygons,drawingeachpolygon.Onedisadvantageofthevertexlististhatifwewishtodrawtheedgesinthemesh,byrenderingeachpolygonsharededgesaredrawntwice.Wecanavoidthisproblembyforminganedgelistoredgearray,eachelementisapairofpointerstoverticesinthevertexarray.Thus,wecandraweachedgeoncebysimplytraversingtheedgelist.However,thesimpleedgelisthasnoinformationonpolygonsandthusifwewanttorenderthemeshinsomeotherwaysuchasbyfillinginteriorpolygonswemustaddsomethingtothisdatastructurethatgivesinformationastowhichedgesformeachpolygon.Aflexiblemeshrepresentationwouldconsistofanedgelist,avertexlistandapolygonlistwithpointerssowecouldknowwhichedgesbelongtowhichpolygonsandwhichpolygonsshareagivenvertex.TheMaxwelltrianglecorrespondstothetrianglethatconnectsthered,green,andblueverticesinthecolorcube.Considerthelinesdefinedbythesidesofthepolygon.Wecanassignadirectionforeachoftheselinesbytraversingtheverticesinacounter-clockwiseorder.Oneverysimpletestisobtainedbynotingthatanypointinsidetheobjectisontheleftofeachoftheselines.Thus,ifwesubstitutethepointintotheequationforeachofthelines(ax+by+c),weshouldalwaysgetthesamesign.Thereareeightverticesandthus256=28possibleblack/whitecolorings.Ifweremovesymmetries(black/whiteandrotational)thereare14uniquecases.SeeAngel,InteractiveComputerGraphics(ThirdEdition)orthepaperbyLorensenandKlineinthereferences.Chapter3SolutionsThegeneralproblemishowtodescribeasetofcharactersthatmighthavethickness,curvature,andholes(suchasinthelettersaandq).Supposethatweconsiderasimpleexamplewhereeachcharactercanbeapproximatedbyasequenceoflinesegments.Onepossibilityistouseamove/linesystemwhere0isamoveand1aline.Thenacharactercanbedescribedbyasequenceoftheform(x0,y0,b0),(x1,y1,b1),(x2,y2,b2),.....wherebiisa0or1.ThisapproachisusedintheexampleintheOpenGLProgrammingGuide.Amoreelaboratefontcanbedevelopedbyusingpolygonsinsteadoflinesegments.Thereareacoupleofpotentialproblems.Oneisthattheapplicationprogramcanmapdifferentpointsinobjectcoordinatestothesamepointinscreencoordinates.Second,agivenpositiononthescreenwhentransformedbackintoobjectcoordinatesmaylieoutsidetheuser’swindow.Eachscanisallocated1/60second.Foragivenscanwehavetotake10%ofthetimefortheverticalretracewhichmeansthatwestarttodrawscanlinenat.9n/(60*1024)secondsfromthebeginningoftherefresh.Butallocating10%ofthistimeforthehorizontalretraceweareatpixelmonthislineattime.81nm/(60*1024).Whenthedisplayischanging,primitivesthatmoveorareremovedfromthedisplaywillleaveatraceormotionbluronthedisplayasthephosphorspersist.Longpersistencephosphorshavebeenusedintextonlydisplayswheremotionblurislessofaproblemandthelongpersistencegivesaverystableflicker-freeimage.Chapter4SolutionsIfthescalingmatrixisuniformthenRS=RS(α,α,α)=αR=SRConsiderRx(θ),ifwemultiplyandusethestandardtrigonometricidentitiesforthesineandcosineofthesumoftwoangles,wefindRx(θ)Rx(φ)=Rx(θ+φ)BysimplymultiplyingthematriceswefindT(x1,y1,z1)T(x2,y2,z2)=T(x1+x2,y1+y2,z1+z2)Thereare12degreesoffreedominthethree–dimensionalaffinetransformation.Considerapointp=[x,y,z,1]Tthatistransformedtop_=[x_y_,z_,1]TbythematrixM.Hencewehavetherelationshipp_=MpwhereMhas12unknowncoefficientsbutpandp_areknown.Thuswehave3equationsin12unknowns(thefourthequationissimplytheidentity1=1).Ifwehave4suchpairsofpointswewillhave12equationsin12unknownswhichcouldbesolvedfortheelementsofM.Thusifweknowhowaquadrilateralistransformedwecandeterminetheaffinetransformation.Intwodimensions,thereare6degreesoffreedominMbutpandp_haveonlyxandycomponents.Henceifweknow3pointsbothbeforeandaftertransformation,wewillhave6equationsin6unknownsandthusintwodimensionsifweknowhowatriangleistransformedwecandeterminetheaffinetransformation.Itiseasytoshowbysimplymultiplyingthematricesthattheconcatenationoftworotationsyieldsarotationandthattheconcatenationoftwotranslationsyieldsatranslation.Ifwelookattheproductofarotationandatranslation,wefindthattheleftthreecolumnsofRTaretheleftthreecolumnsofRandtherightcolumnofRTistherightcolumnofthetranslationmatrix.IfwenowconsiderRTR_whereR_isarotationmatrix,theleftthreecolumnsareexactlythesameastheleftthreecolumnsofRR_andtheandrightcolumnstillhas1asitsbottomelement.Thus,theformisthesameasRTwithanalteredrotation(whichistheconcatenationofthetworotations)andanalteredtranslation.Inductively,wecanseethatanyfurtherconcatenationswithrotationsandtranslationsdonotalterthisform.Ifwedoatranslationby-hweconverttheproblemtoreflectionaboutalinepassingthroughtheorigin.Frommwecanfindananglebywhichwecanrotatesothelineisalignedwitheitherthexoryaxis.Nowreflectaboutthexoryaxis.FinallyweundotherotationandtranslationsothesequenceisoftheformT?1R?1SRT.ThemostsensibleplacetoputtheshearissecondsothattheinstancetransformationbecomesI=TRHS.Wecanseethatthisordermakessenseifweconsideracubecenteredattheoriginwhosesidesarealignedwiththeaxes.Thescalegivesusthedesiredsizeandproportions.Theshearthenconvertstherightparallelepipedtoageneralparallelepiped.Finallywecanorientthisparallelepipedwitharotationandplaceitwheredesiredwithatranslation.NotethattheorderI=TRSHwillworktoo.R=Rz(θz)Ry(θy)Rx(θx)=?????cosθycosθzcosθzsinθxsinθy?cosθxsinθzcosθxcosθzsinθy+sinθxsinθz0cosθysinθzcosθxcosθz+sinθxsinθysinθz?cosθzsinθx+cosθxsinθysinθz0?sinθycosθysinθxcosθxcosθy00001?????Onetestistousethefirstthreeverticestofindtheequationoftheplaneax+by+cz+d=0.Althoughtherearefourcoefficientsintheequationonlythreeareindependentsowecanselectonearbitrarilyornormalizesothata2+b2+c2=1.Thenwecansuccessivelyevaluateax+bc+cz+dfortheothervertices.Avertexwillbeontheplaneifweevaluatetozero.Anequivalenttestistoformthematrix?????1111x1x2x3x4y1y2y3y4z1z2z3z4?????foreachi=4,...Ifthedeterminantofthismatrixiszerotheithvertexisintheplanedeterminedbythefirstthree.Althoughwewillhavethesamenumberofdegreesoffreedomintheobjectsweproduce,theclassofobjectswillbeverydifferent.Forexampleifwerotateasquarebeforeweapplyanonuniformscale,wewillshearthesquare,somethingwecannotdoifwescalethenrotate.Thevectora=u×visorthogonaltouandv.Thevectorb=u×aisorthogonaltouanda.Hence,u,aandbformanorthogonalcoordinatesystem.Usingr=cosθ2+sinθ2v,withθ=90andv=(1,0,0),wefindforrotationaboutthex-axisr=√22(1,1,0,0).Likewise,forrotationabouttheyaxisr=√22(1,0,1,0).Possiblereasonsinclude(1)object-orientedsystemsareslower,(2)usersareoftencomfortableworkinginworldcoordinateswithhigher-levelobjectsanddonotneedtheflexibilityofferedbyacoordinate-freeapproach,(3)evenasystemthatprovidesscalars,vectors,andpointswouldhavetohaveanunderlyingframetousefortheimplementation.Chapter5SolutionsEclipses(bothsolarandlunar)aregoodexamplesoftheprojectionofanobject(themoonortheearth)ontoanonplanarsurface.Anytimeashadowiscreatedoncurvedsurface,thereisanonplanarprojection.Allthemapsinanatlasareexamplesoftheuseofcurvedprojectors.Iftheprojectorswerenotcurvedwecouldnotprojecttheentiresurfaceofasphericalobject(theEarth)ontoarectangle.SupposethatwewanttheviewoftheEarthrotatingaboutthesun.Beforewedrawtheearth,wemustrotatetheEarthwhichisarotationabouttheyaxis.NextwetranslatetheEarthawayfromtheorigin.FinallywedoanotherrotationabouttheyaxistopositiontheEarthinitsdesiredlocationalongitsorbit.ThereareanumberofinterestingvariantsofthisproblemsuchastheviewfromtheEarthoftherestofthesolarsystem.Yes.Anysequenceofrotationsisequivalenttoasinglerotationaboutasuitablychosenaxis.Onewaytocomputethisrotationmatrixistoformthematrixbysequenceofsimplerotations,suchasR=RxRyRz.Thedesiredaxisisaneigenvectorofthismatrix.Theresultfollowsfromthetransformationbeingaffine.Wecanalsotakeadirectapproach.Considerthelinedeterminedbythepoints(x1,y1,z1)and(x2,y2,z2).Anypointalongcanbewrittenparametricallyas(_x1+(1?_)x2,_y1+(1?_)y2,_z1+(1?_)z2).Considerthesimpleprojectionofthispoint1d(_z1+(1?_)z2)(_x1+(1?_)x2,_y1+(1?_)y2)whichisoftheformf(_)(_x1+(1?_)x2,_y1+(1?_)y2).Thisformdescribesalinebecausetheslopeisconstant.Notethatthefunctionf(_)impliesthatwetraceoutthelineatanonlinearrateas_increasesfrom0to1.Thespecificationusedinmanygraphicstextisoftheanglestheprojectormakeswithx,zandy,zplanes,theanglesdefinedbytheprojectionofaprojectorbyatopviewandasideview.Anotherapproachistospecifytheforeshorteningofoneortwosidesofacubealignedwiththeaxes.TheCOREsystemusedthisapproach.Retainedobjectswerekeptindistortedform.Anytransformationtoanyobjectthatwasdefinedwithotherthananorthographicviewtransformedthedistortedobjectandtheorthographicprojectionofthetransformeddistortedobjectwasincorrect.Ifweuse_=_=45,weobtaintheprojectionmatrixP=2666410?1001?100000000137775Allthepointsontheprojectionofthepoint,z)inthedirectiondx,dy,dz)areoftheform(x+_dx,y+_dy,z+_dz).Thustheshadowofthepoint(x,y,z)isfoundbydeterminingthe_forwhichthelineintersectstheplane,thatisaxs+bys+czs=dSubstitutingandsolving,wefind_=d?ax?by?czadx+bdy+cdz.However,whatwewantisaprojectionmatrix,Usingthisvalueof_wefindxs=z+_dx=x(bdy+cdx)?dx(d?by?cz)adx+bdy+cdzwithsimilarequationsforysandzs.Theseresultscanbecomputedbymultiplyingthehomogeneouscoordinatepoint(x,y,z,1)bytheprojectionmatrixM=26664bdy+cdz?bdx?cdx?ddx?adyadx+cdz?cdy?ddy?adz?bdzadx+bdy?ddz000adx+bdy+cdz37775.Supposethattheaverageofthetwoeyepositionsisat(x,y,z)andtheviewerislookingattheorigin.WecouldformtheimagesusingtheLookAtfunctiontwice,thatisgluLookAt(x-dx/2,y,z,0,0,0,0,1,0);/*drawscenehere*//*swapbuffersandclear*/gluLookAt(x+dx/2,y,z,0,0,0,0,1,0);/*drawsceneagain*//*swapbuffersandclear*/Chapter6SolutionsPointsourcesproduceaveryharshlighting.Suchimagesarecharacterizedbyabrupttransitionsbetweenlightanddark.Theambientlightinarealsceneisdependentonboththelightsonthesceneandthereflectivitypropertiesoftheobjectsinthescene,somethingthatcannotbecomputedcorrectlywithOpenGL.ThePhongreflectiontermisnotphysicallycorrect;thereflectionterminthemodifiedPhongmodelisevenfurtherfrombeingphysicallycorrect.Ifweweretotakeintoaccountalightsourcebeingobscuredbyanobject,wewouldhavetohaveallpolygonsavailablesoastotestforthiscondition.Suchaglobalcalculationisincompatiblewiththepipelinemodelthatassumeswecanshadeeachpolygonindependentlyofallotherpolygonsasitflowsthroughthepipeline.Materialsabsorblightfromsources.Thus,asurfacethatappearsredunderwhitelightappearssobecausethesurfaceabsorbsallwavelengthsoflightexceptintheredrange—asubtractiveprocess.Tobecompatiblewithsuchamodel,weshouldusesurfaceabsorbtionconstantsthatdefinethematerialsforcyan,magentaandyellow,ratherthanred,greenandblue.Letψbetheanglebetweenthenormalandthehalfwayvector,φbetheanglebetweentheviewerandthereflectionangle,andθbetheanglebetweenthenormalandthelightsource.Ifallthevectorslieinthesameplane,theanglebetweenthelightsourceandtheviewercanbecomputereitherasφ+2θoras2(θ+ψ).Settingthetwoequal,wefindφ=2ψ.Ifthevectorsarenotcoplanarthenφ<2ψ.Withoutlossofgenerality,wecanconsidertheproblemintwodimensions.Supposethatthefirstmaterialhasavelocityoflightofv1andthesecondmaterialhasalightvelocityofv2.Furthermore,assumethattheaxisy=0separatesthetwomaterials.Placeapointlightsourceat(0,h)whereh>0andaviewerat(x,y)wherey<0.Lightwilltravelinastraightlinefromthesourcetoapoint(t,0)whereitwillleavethefirstmaterialandenterthesecond.Itwillthentravelfromthispointinastraightlineto(x,y).Wemustfindthetthatminimizesthetimetravelled.Usingsomesimpletrigonometry,wefindthelinefromthesourceto(t,0)haslengthl1=√h2+t2andthelinefromtheretotheviewerhaslength1l2=_y2+(x?t)2.Thetotaltimelighttravelsisthusl1v1+l2v2.Minimizingovertgivesdesiredresultwhenwenotethetwodesiredsinesaresinθ1=h√h2+t2andsinθ2=?y√(y2+(x?t)2.Shadingrequiresthatwhenwetransformnormalsandpoints,wemaintaintheanglebetweenthemorequivalentlyhavethedotproductp·v=p_·v_whenp_=Mpandn_=Mp.IfMTMisanidentitymatrixanglesarepreserved.Suchamatrix(M?1=MT)iscalledorthogonal.Rotationsandtranslationsareorthogonalbutscalingandsheararenot.Probablytheeasiestapproachtothisproblemistorotatethegivenplanetoplanez=0androtatethelightsourceandobjectsinthesameway.Nowwehavethesameproblemwehavesolvedandcanrotateeverythingbackattheend.Aglobalrenderingapproachwouldgenerateallshadowscorrectly.Inaglobalrenderer,aseachpointisshaded,acalculationisdonetoseewhichlightsourcesshineonit.Theprojectionapproachassumesthatwecanprojecteachpolygonontoallotherpolygons.Iftheshadowofagivenpolygonprojectsontomultiplepolygons,wecouldnotcomputetheseshadowpolygonsveryeasily.Inaddition,wehavenotaccountedforthedifferentshadeswemightseeiftherewereintersectingshadowsfrommultiplelightsources.Chapter7SolutionsFirst,considertheproblemintwodimensions.Wearelookingforan_and_suchthatbothparametricequationsyieldthesamepoint,thatisx(_)=(1?_)x1+_x2=(1?_)x3+_x4,y(_)=(1?_)y1+_y2=(1?_)y3+_y4.Thesearetwoequationsinthetwounknowns_and_and,aslongasthelinesegmentsarenotparallel(aconditionthatwillleadtoadivisionbyzero),wecansolvefor__.Ifboththesevaluesarebetween0and1,thesegmentsintersect.Iftheequationsarein3D,wecansolvetwoofthemforthe_and_wherexandymeet.Ifwhenweusethesevaluesoftheparametersinthetwoequationsforz,thesegmentsintersectifwegetthesamezfrombothequations.Ifweclipaconvexregionagainstaconvexregion,weproducetheintersectionofthetworegions,thatisthesetofallpointsinbothregions,whichisaconvexsetanddescribesaconvexregion.Toseethis,consideranytwopointsintheintersection.Thelinesegmentconnectingthemmustbeinbothsetsandthereforetheintersectionisconvex.SeeProblem.Nonuniformscalingwillnotpreservetheanglebetweenthenormalandothervectors.NotethatwecoulduseOpenGLto,produceahiddenlineremovedimagebyusingthezbufferanddrawingpolygonswithedgesandinteriorsthesamecolorasthebackground.Butofcourse,thismethodwasnotusedinpre–rastersystems.Hidden–lineremovalalgorithmsworkinobjectspace,usuallywitheitherpolygonsorpolyhedra.Back–facingpolygonscanbeeliminated.Ingeneral,edgesareintersectedwithpolygonstodetermineanyvisibleparts.Goodalgorithms(seeFoleyorRogers)usevariouscoherencestrategiestominimizethenumberofintersections.TheO(k)wasbaseduponcomputingtheintersectionofrayswiththeplanescontainingthekpolygons.Wedidnotconsiderthecostoffillingthepolygons,whichcanbealargepartoftherenderingtime.Ifweconsiderascenewhichisviewedfromagivenpointtherewillbesomepercentageof1theareaofthescreenthatisfilledwithpolygons.Aswemovetheviewerclosertotheobjects,fewerpolygonswillappearonthescreenbuteachwilloccupyalargerareaonthescreen,thusleavingtheareaofthescreenthatisfilledapproximatelythesame.Thustherenderingtimewillbeaboutthesameeventhoughtherearefewerpolygonsdisplayed.ThereareanumberofwayswecanattempttogetO(klogk)performance.Oneistouseabettersortingalgorithmforthedepthsort.Otherstrategiesarebasedondivideandconquersuchabinaryspatialpartitioning.Ifweconsideraraytracerthatonlycastsraystothefirstintersectionanddoesnotcomputeshadowrays,reflectedortransmittedrays,thentheimageproducedusingaPhongmodelatthepointofintersectionwillbethesameimageasproducedbyourpipelinerenderer.ThisapproachissometimescalledraycastingandisusedinvolumerenderingandCSG.However,thedataareprocessedinadifferentorderfromthepipelinerenderer.Theraytracerworksraybyraywhilethepipelinerendererworksobjectbyobject.Consideracirclecenteredattheorigin:x2+y2=r2.Ifweknowthatapoint(x,y)isonthecurvethan,wealsoknow(?x,y),(x,?y),(?x,?y),