交互式计算机图形学17章课后题答案.docx

交互式计算机图形学17章课后题答案.docx

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交互式计算机图形学17章课后题答案

交互式计算机图形学（第五版）1-7章课后题答案（总13页）

Angel:

InteractiveComputerGraphics,FifthEdition

Chapter1Solutions

Themainadvantageofthepipelineisthateachprimitivecanbe

processedindependently.Notonlydoesthisarchitectureleadtofast

performance,itreducesmemoryrequirementsbecauseweneednotkeepall

objectsavailable.Themaindisadvantageisthatwecannothandlemost

globaleffectssuchasshadows,reflections,andblendinginaphysically

correctmanner.

WederivethisalgorithmlaterinChapter6.First,wecanformthe

tetrahedronbyfindingfourequallyspacedpointsonaunitspherecentered

attheorigin.Oneapproachistostartwithonepointonthezaxis

（0,0,1）.Wethencanplacetheotherthreepointsinaplaneofconstantz.

Oneofthesethreepointscanbeplacedontheyaxis.Tosatisfythe

requirementthatthepointsbeequidistant,thepointmustbeat

（0,2p2/3,−1/3）.Theothertwocanbefoundbysymmetrytobeat

（−

p6/3,−

p2/3,−1/3）and（p6/3,−

p2/3,−1/3）.

Wecansubdivideeachfaceofthetetrahedronintofourequilateral

trianglesbybisectingthesidesandconnectingthebisectors.However,the

bisectorsofthesidesarenotontheunitcirclesowemustpushthese

pointsouttotheunitcirclebyscalingthevalues.Wecancontinuethis

processrecursivelyoneachofthetrianglescreatedbythebisectionprocess.

InExercise,wesawthatwecouldintersectthelineofwhichthe

linesegmentispartindependentlyagainsteachofthesidesofthewindow.

Wecoulddothisprocessiteratively,eachtimeshorteningthelinesegment

ifitintersectsonesideofthewindow.

Inaone–pointperspective,twofacesofthecubeisparalleltothe

projectionplane,whileinatwo–pointperspectiveonlytheedgesofthe

cubeinonedirectionareparalleltotheprojection.Inthegeneralcaseofa

three–pointperspectivetherearethreevanishingpointsandnoneofthe

edgesofthecubeareparalleltotheprojectionplane.

Eachframefora480x640pixelvideodisplaycontainsonlyabout

300kpixelswhereasthe2000x3000pixelmovieframehas6Mpixels,or

about18timesasmanyasthevideodisplay.Thus,itcantake18timesas

muchtimetorendereachframeifthereisalotofpixel-levelcalculations.

TherearesinglebeamCRTs.Oneschemeistoarrangethephosphors

inverticalstripes（red,green,blue,red,green,....）.Themajordifficultyis

thatthebeammustchangeveryrapidly,approximatelythreetimesasfast

aeachbeaminathreebeamsystem.Theelectronicsinsuchasystemthe

electroniccomponentsmustalsobemuchfaster（andmoreexpensive）.

Chapter2Solutions

Wecansolvethisproblemseparatelyinthexandydirections.The

transformationislinear,thatisxs=ax+b,ys=cy+d.Wemust

maintainproportions,sothatxsinthesamerelativepositioninthe

viewportasxisinthewindow,hence

x−xmin

xmax−xmin

=

xs−u

w

xs=u+w

x−xmin

xmax−xmin

.

Likewise

ys=v+h

x−xmin

ymax−ymin

.

Mostpracticaltestsworkonalinebylinebasis.Usuallyweuse

scanlines,eachofwhichcorrespondstoarowofpixelsintheframebuffer.

Ifwecomputetheintersectionsoftheedgesofthepolygonwithaline

passingthroughit,theseintersectionscanbeordered.Thefirst

intersectionbeginsasetofpointsinsidethepolygon.Thesecond

intersectionleavesthepolygon,thethirdreentersandsoon.

Therearetwofundamentalapproaches:

vertexlistsandedgelists.

Withvertexlistswestorethevertexlocationsinanarray.Themeshis

representedasalistofinteriorpolygons（thosepolygonswithnoother

polygonsinsidethem）.Eachinteriorpolygonisrepresentedasanarrayof

pointersintothevertexarray.Todrawthemesh,wetraversethelistof

interiorpolygons,drawingeachpolygon.

Onedisadvantageofthevertexlististhatifwewishtodrawtheedgesin

themesh,byrenderingeachpolygonsharededgesaredrawntwice.We

canavoidthisproblembyforminganedgelistoredgearray,eachelement

isapairofpointerstoverticesinthevertexarray.Thus,wecandraweach

edgeoncebysimplytraversingtheedgelist.However,thesimpleedgelist

hasnoinformationonpolygonsandthusifwewanttorenderthemeshin

someotherwaysuchasbyfillinginteriorpolygonswemustaddsomething

tothisdatastructurethatgivesinformationastowhichedgesformeach

polygon.

Aflexiblemeshrepresentationwouldconsistofanedgelist,avertexlist

andapolygonlistwithpointerssowecouldknowwhichedgesbelongto

whichpolygonsandwhichpolygonsshareagivenvertex.

TheMaxwelltrianglecorrespondstothetrianglethatconnectsthe

red,green,andblueverticesinthecolorcube.

Considerthelinesdefinedbythesidesofthepolygon.Wecanassign

adirectionforeachoftheselinesbytraversingtheverticesina

counter-clockwiseorder.Oneverysimpletestisobtainedbynotingthat

anypointinsidetheobjectisontheleftofeachoftheselines.Thus,ifwe

substitutethepointintotheequationforeachofthelines（ax+by+c）,we

shouldalwaysgetthesamesign.

Thereareeightverticesandthus256=28possibleblack/white

colorings.Ifweremovesymmetries（black/whiteandrotational）thereare

14uniquecases.SeeAngel,InteractiveComputerGraphics（Third

Edition）orthepaperbyLorensenandKlineinthereferences.

Chapter3Solutions

Thegeneralproblemishowtodescribeasetofcharactersthatmight

havethickness,curvature,andholes（suchasinthelettersaandq）.

Supposethatweconsiderasimpleexamplewhereeachcharactercanbe

approximatedbyasequenceoflinesegments.Onepossibilityistousea

move/linesystemwhere0isamoveand1aline.Thenacharactercanbe

describedbyasequenceoftheform（x0,y0,b0）,（x1,y1,b1）,（x2,y2,b2）,.....

wherebiisa0or1.ThisapproachisusedintheexampleintheOpenGL

ProgrammingGuide.Amoreelaboratefontcanbedevelopedbyusing

polygonsinsteadoflinesegments.

Thereareacoupleofpotentialproblems.Oneisthattheapplication

programcanmapdifferentpointsinobjectcoordinatestothesamepoint

inscreencoordinates.Second,agivenpositiononthescreenwhen

transformedbackintoobjectcoordinatesmaylieoutsidetheuser’s

window.

Eachscanisallocated1/60second.Foragivenscanwehavetotake

10%ofthetimefortheverticalretracewhichmeansthatwestarttodraw

scanlinenat.9n/（60*1024）secondsfromthebeginningoftherefresh.

Butallocating10%ofthistimeforthehorizontalretraceweareatpixelm

onthislineattime.81nm/（60*1024）.

Whenthedisplayischanging,primitivesthatmoveorareremoved

fromthedisplaywillleaveatraceormotionbluronthedisplayasthe

phosphorspersist.Longpersistencephosphorshavebeenusedintextonly

displayswheremotionblurislessofaproblemandthelongpersistence

givesaverystableflicker-freeimage.

Chapter4Solutions

Ifthescalingmatrixisuniformthen

RS=RS（α,α,α）=αR=SR

ConsiderRx（θ）,ifwemultiplyandusethestandardtrigonometric

identitiesforthesineandcosineofthesumoftwoangles,wefind

Rx（θ）Rx（φ）=Rx（θ+φ）

Bysimplymultiplyingthematriceswefind

T（x1,y1,z1）T（x2,y2,z2）=T（x1+x2,y1+y2,z1+z2）

Thereare12degreesoffreedominthethree–dimensionalaffine

transformation.Considerapointp=[x,y,z,1]Tthatistransformedto

p_=[x_y_,z_,1]TbythematrixM.Hencewehavetherelationship

p_=MpwhereMhas12unknowncoefficientsbutpandp_areknown.

Thuswehave3equationsin12unknowns（thefourthequationissimply

theidentity1=1）.Ifwehave4suchpairsofpointswewillhave12

equationsin12unknownswhichcouldbesolvedfortheelementsofM.

Thusifweknowhowaquadrilateralistransformedwecandeterminethe

affinetransformation.

Intwodimensions,thereare6degreesoffreedominMbutpandp_have

onlyxandycomponents.Henceifweknow3pointsbothbeforeandafter

transformation,wewillhave6equationsin6unknownsandthusintwo

dimensionsifweknowhowatriangleistransformedwecandeterminethe

affinetransformation.

Itiseasytoshowbysimplymultiplyingthematricesthatthe

concatenationoftworotationsyieldsarotationandthattheconcatenation

oftwotranslationsyieldsatranslation.Ifwelookattheproductofa

rotationandatranslation,wefindthattheleftthreecolumnsofRTare

theleftthreecolumnsofRandtherightcolumnofRTistheright

columnofthetranslationmatrix.IfwenowconsiderRTR_whereR_isa

rotationmatrix,theleftthreecolumnsareexactlythesameastheleft

threecolumnsofRR_andtheandrightcolumnstillhas1asitsbottom

element.Thus,theformisthesameasRTwithanalteredrotation（which

istheconcatenationofthetworotations）andanalteredtranslation.

Inductively,wecanseethatanyfurtherconcatenationswithrotationsand

translationsdonotalterthisform.

Ifwedoatranslationby-hweconverttheproblemtoreflectionabout

alinepassingthroughtheorigin.Frommwecanfindananglebywhich

wecanrotatesothelineisalignedwitheitherthexoryaxis.Nowreflect

aboutthexoryaxis.Finallyweundotherotationandtranslationsothe

sequenceisoftheformT−1R−1SRT.

Themostsensibleplacetoputtheshearissecondsothattheinstance

transformationbecomesI=TRHS.Wecanseethatthisordermakes

senseifweconsideracubecenteredattheoriginwhosesidesarealigned

withtheaxes.Thescalegivesusthedesiredsizeandproportions.The

shearthenconvertstherightparallelepipedtoageneralparallelepiped.

Finallywecanorientth