cs380: introduction to computer graphics projection chapter 10...

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MinH.Kim(KAIST) Foundationsof3DComputerGraphics,S.Gortler,MITPress,2012

CS380:IntroductiontoComputerGraphicsProjectionChapter10

MinH.KimKAISTSchoolofComputing


SUMMARYSmoothInterpolation

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CubicBezierSpline•  Toevaluatethefunctionatanyvalueof,weperformthefollowingsequenceoflinearinterpolations:

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c(t) t

f = (1− t)c0 + td0g = (1− t)d0 + te0h = (1− t)e0 + tc0m = (1− t) f + tgn = (1− t)g + thc(t) = (1− t)m + tn

when t = 0.3


CubicBezierSpline•  Let’stakethefirst-orderderivatives:

•  Weseethat•  Weseeindeedthattheslopeofmatchestheslopeofthecontrolpolygonat0and1.

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c(t)= c0(1−t)3+3d0t(1−t)2+3e0t2(1−t)+c1t3

′c (t) = 3c1t2 − 3e0t

2 − 3c0 (t −1)2 + 3d0 (t −1)

2 − 6e0t(t −1)+ 3d0t(2t − 2)

symsabcdtf(t)=a*(1-t)^3+3*b*t*(1-t)^2+3*c*t^2*(1-t)+d*t^3df=diff(f)df(0)=3*b-3*adf(1)=3*d-3*c

′c (0) = 3(d0 − c0 ), ′c (1) = 3(c1 − e0 )c(t)

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Catmull-RomSplines(CRS)•  Thisalsocanbestatedas

•  Sinceand,intheBezierrepresentation,thistellsusthatweneedtoset:

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′c (t) i+1 =12ci+2 − ci( )

′c (i) = 3(di − ci ) ′c (i +1) = 3(ci+1 − ei )

di =

16 ci+1 −ci−1( )+ci

ei =−16 ci+2−ci( )+ci+1

′c (t) i =12ci+1 − ci−1( )

cidi

ei ci+1

ci−1

ci+2c(t)= ci(1−t)

3+3dit(1−t)2+3eit2(1−t)+ci+1t3


Curves•  Wecancleanlyapplythescalartheorytodescribecurvesintheplaneorspace.

•  Thesplinecurveiscontrolledbyasetofcontrolpointsin2Dor3D.

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ciBeziercurve(2D) Catmull-Romcurve(2D)

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QuaternionSplining•  Bezierevaluationstepsoftheform:

•  Become:•  Andtheequationvaluesaredefinedas

•  Inordertointerpolate“theshortway”,wedoconditionalnegationonbeforeapplyingthepoweroperator.

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r = (1− t)p + tq

r = slerp(p, q, t)di and ei

ei = ((ci+2ci−1)−1/6 )ci+1

di = ((ci+1ci−1−1 )1/6 )ci

(ci+1ci−1−1 )

⇐ di =16ci+1 − ci−1( ) + ci

⇐ ei =−16

ci+2 − ci( ) + ci+1


PROJECTIONChapter10

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Chapters1—10Object&Animation

Chapters10—13Camera

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RECAP:Modelviewmatrix•  Modelviewmatrix(MVM)

– Describestheorientationandpositionoftheviewandtheorientationandpositionoftheobjectwithrespecttotheeyeframe

–  Thevertexshaderwilltakethesevertexdataandperformthemultiplication,producingtheeyecoordinatesusedinrendering

–  uniformmat4uModelViewMatrix;

•  normalMatrix()producestheinversetransposeofthelinearfactortogetuniformnormalNMVM–  uniformmat4uNormalMatrix;

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E−1OE−1

O

p =otc = wtOc = etE−1Oc

et

E−1Oc


RECAP:Stagesofvertextransformation

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Modelviewmatrix

Projectionmatrix

Perspectivedivision

Viewporttransforma

tion

eyecoordinates

http://www.glprogramming.com/red/chapter03.html

clipcoordinates

normalizeddevicecoordinates

windowcoordinates

vertex

xeyeze1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

= E−1O

xoyozo1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

xoyozo1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

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RECAP:Cameraprojection(overview)•  Intrinsicpropertiesofthecamera(theeye)– Afieldofview– Windowaspectratio– Near/farfieldinZ(clipping)– Representedascamerafrustum

•  sendProjectionMatrix:sendthecameramatrixtothevertexshader

•  uniformmat4uProjMatrix;11


RECAP:Vertexshader•  Takestheobjectcoordinatesofeveryvertexpositionandturnsthemintoeyecoordinates,aswellasthevertex’snormalcoordinates

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#version130uniformMatrix4uModelViewMatrix;uniformMatrix4uNormalMatrix;uniformMatrix4uProjMatrix;invec3aColor;invec4aNormal;invec4aVertex;outvec3vColor;outvec3vNormal;outvec3vPosition;

voidmain(){vColor=aColor;vPosition=uModelViewMatrix*aVertex;vec4normal=vec4(aNormal.x,aNormal.y,aNormal.z,0.0);vNormal=vec3(uNormalMatrix*normal);gl_Position=uProjMatrix*vPosition;}

E−1Oc

PE−1OcForcamera(3Dàà2D)

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Cameratransforms•  Untilnowwehaveconsideredallofourgeometryina3Dspace

•  Ultimatelyeverythingendedupineyecoordinateswithcoordinates

•  Wesaidthatthecameraisplacedattheoriginoftheeyeframe,andthatitislookingdowntheeye’snegativez-axis.

•  Thissomehowproducesa2Dimage.•  Wehadamagicmatrixwhichcreatedgl_Position•  Nowwewillstudythisstep

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[xe, ye, ze,1]t

et


Pinholecameramodel

•  Aslighttravelstowardsthefilmplane,mostisblockedbyanopaquesurfaceplacedattheplane.

•  Butweplaceaverysmallholeinthecenterofthesurface,atthepointwitheyecoordinates

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ze = 0

[0,0,0,1]t

[0,0,0,1]t

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Pinholecameramodel

•  Onlyraysoflightthatpassthroughthispointreachthefilmplaneandhavetheirintensityrecordedonfilm.Theimageisrecordedatafilmplaneplacedat,say,

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ze = 1

[0,0,0,1]t

ze = 1


Pinholecameramodel

– Aphysicalcameraneedsafiniteapertureandalens,butwewillignorethis.

•  Toavoidtheimageflip,wecanmathematicallymodelthiswiththefilmplaneinfrontofthepinhole,sayatthe

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ze = −1

[0,0,0,1]t

ze = −1

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Pinholecameramodel

•  Ifweholdupthephotographattheplane,andobserveitwithourowneye,placedattheorigin,itwilllooktousjustliketheoriginscenewouldhave.

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ze = −1

[0,0,0,1]t

ze = −1


Basicmathematicalmodel

•  Letususenormalizedcoordinatestospecifypointsonourfilmplane.–  Fornow,letthemmatcheyecoordinatesonthisfilmplane.

•  Wheredoestherayfromtotheoriginhitsthefilmplane?

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[xn , yn ]t

p

[0,0,0,1]t

p

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•  Allpointsontherayhitthesamepixel.•  Allpointsontherayareallscales•  Sopointsonrayare:

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[xe , ye ,ze ]t =α[xn , yn ,−1]t

[0,0,0,1]t

p

α



•  So•  So

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[xe, ye, ze ]t = −ze[xn , yn ,−1]

t

xn = − xeze, yn = − ye

ze

[0,0,0,1]t

−ze

ye p

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Projectionmatrix

•  Wecanmodelthisexpressionasamatrixoperationasfollows.

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1 0 0 00 1 0 0− − − −0 0 −1 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

xeyeze1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

xnwn

ynwn

−wn

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

xcyc−wc

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

[0,0,0,1]t

p


Inmatrixform

•  Therawoutputofthematrixmultiply,arecalledtheclipcoordinatesof.

•  isanewvariablecalledthew-coordinate.–  Insuchclipcoordinates,thefourthentryofthecoordinate4-vectorisnotnecessarilyazerooraone.

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[0,0,0,1]t

[xc , yc ,−,wc ]t

pwn = wc

p

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Dividebyw

•  Wesaythatand.Ifwewanttoextractalone,wemustperformthedivision

•  Thisrecoversourcameramodel

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[0,0,0,1]t

xnwn = xcxn

ynwn = yc

p

1 0 0 00 1 0 0− − − −0 0 −1 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

xeyeze1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

xnwn

ynwn

−wn

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

xcyc−wc

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

xn =

xcwn

=xnwn

wn


Dividebyw

•  Ouroutputcoordinates,withsubscripts‘n’,arecallednormalizeddevicecoordinates(NDC)becausetheyaddresspointsontheimageinabstractunitswithoutspecificreferencetonumbersofpixels.

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[0,0,0,1]t

p

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Dividebyw

•  Wekeepalloftheimagedatainthecanonicalsquare,andultimatelymapthisontoawindowonthescreen.–  Dataoutsideofthissquaredoesnotberecordedordisplayed.

–  Thisisexactlythemodelweusedtodescribe2DOpenGL25

[0,0,0,1]t

−1≤ xn ≤ +1, −1≤ yn ≤ +1,

p


Scales

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ze = n n

[0,0,0,1]t

yn =1

yn =−1

•  Bychangingtheentriesintheprojectionmatrix,wecanslightlyaltergeometryofthecameratransformation.

•  Wecouldpushthefilmplaneoutto,whereissomenegativenumber(zoomlens)

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Scales

•  Sopointsonrayare:•  So•  So

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[xe , ye ,ze ]t =α[xn , yn ,zn]t

[xe , ye ,ze ]t =

zen[xn , yn ,zn]t

xn =xenze, yn =

yenze

[0,0,0,1]t

yn =1

yn =−1


Inmatrixform

•  Inmatrixform,thisbecomes:

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[0,0,0,1]t

xnwn

ynwn

−wn

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

−n 0 0 00 −n 0 0− − − −0 0 −1 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

xeyeze1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

(supposingnissomenegativenumber)

yn =1

yn =−1

yn =1

yn =−1

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Inmatrixform

•  Notethismatrixisthesameas

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[0,0,0,1]t

−n 0 0 00 −n 0 0− − − −0 0 0 1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

1 0 0 00 1 0 0− − − −0 0 −1 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

yn =1

yn =−1

yn =1

yn =−1


Inmatrixform

•  Thishasthesameeffectasstartingwithouroriginalcamera,scalingby,andcroppingtothecanonicalsquare.

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[0,0,0,1]t

−n

yn =1

yn =−1

yn =1

yn =−1

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fovY

•  Scalecanbedeterminedbyverticalangularfieldofviewofthedesiredcamera.

•  Ifwewantourcameratohaveafieldofviewofdegrees,thenwecansetgivingus

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[0,0,0,1]t

θ−n = 1

tan θ2

⎛⎝⎜

⎞⎠⎟


fovY

•  Verifythatanypointwho’srayfromtheoriginformsaverticalangleofwiththenegativeaxismapstotheboundaryofthecanonicalsquare

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[0,0,0,1]t

1

tan θ2

⎛⎝⎜

⎞⎠⎟

0 0 0

0 1

tan θ2

⎛⎝⎜

⎞⎠⎟

0 0

− − − −0 0 −1 0

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

θ / 2 z

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fovY

•  Thepointwitheyecoordinates:mapstonormalizeddevicecoordinates

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[0,0,0,1]t

[0, tan θ2

⎛⎝⎜

⎞⎠⎟ ,−1,1]

t

[0,1]t

1

tan θ2

⎛⎝⎜

⎞⎠⎟

0 0 0

0 1

tan θ2

⎛⎝⎜

⎞⎠⎟

0 0

− − − −0 0 −1 0

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥


Dealingwithaspectratio•  Supposethewindowiswiderthanitishigh.Inourcameratransform,weneedtosquishthingshorizontallysoawiderhorizontalfieldofviewfitsintoourretainedcanonicalsquare.

•  Whenthedataislatermappedtothewindow,itwillbestretchedoutcorrespondinglyandwillnotappeardistorted.

•  Define,theaspectratioofawindow,tobeitswidthdividedbyitsheight(measuredsayinpixels).

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a

a = width px( )height px( )

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Dealingwithaspectratio•  Wecanthensetourprojectionmatrixtobe:

•  Sowhenthewindowiswide,wewillkeepmorehorizontalFOV,andwhenthewindowistall,wewillkeeplesshorizontalFOV.

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1atan θ

2⎛

⎝⎜⎞

⎠⎟

0 0 0

0 1tan θ

2⎛

⎝⎜⎞

⎠⎟

0 0

− − − −0 0 −1 0

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥


Dealingwithaspectratio•  Asanalternative,wecouldhaveanfovMin,andwhenthewindowistall,wewouldneedtocalculateanappropriatelargerfovYandthenbuildthematrix.

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FOVissues•  Tobea“window”ontotheworld,theFOVshouldmatchtheangularextentsofthewindowintheviewersfield.

•  Thismightgiveatoolimitedviewontotheworld.

•  Sowecanincreaseittoseemore.•  Butthismightgiveasomewhatunnaturallook.

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Shifts

•  Sometimes,wewishtocroptheimagenon-centrally.

•  Thiscanbemodeledastranslatingthenormalizeddevicecoordinates(NDC)’sandthencroppingcentrally.

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Shifts

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xnwn

ynwn

−wn

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

1 0 0 cx0 1 0 c y− − − −0 0 0 1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

1 0 0 00 1 0 0− − − −0 0 −1 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

xeyeze1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

1 0 −cx 00 1 −c y 0− − − −0 0 −1 0

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

xeyeze1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥


Shifts

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•  Usefulfortileddisplays,stereoviewing,andcertainkindsofimages.

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Frustum•  Shiftsareoftenspecifiedbyfirstspecifyinganear

plane.•  Onthisplane,arectangleisspecifiedwiththeeye

coordinatesofanaxisalignedrectangle.(fornon-distortedoutput,theaspectratioofthisrectangleshouldmatchthatofthefinalwindow.)–  Using

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ze = n

l, r, t, b.

− 2nr − l

0 r + lr − l

0

0 − 2nt − b

t + bt − b

0

− − − −0 0 −1 0

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥


Frustum•  Example1:

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−2nr − l

0 r+ lr − l

0

0 −2nt −b

t +bt −b

0

− − − −0 0 −1 0

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

•  Example2:

xy

z !etl =−3

r =3t =2

b=−2(0,0)

n=−2

2×23+3 0 0

3+3 0

0 2×22+2

02+2 0

− − − −0 0 −1 0

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

xy

z !et

l =2

r =4t =4

b=2(3,3)

n=−2(0,0)

2×24−2 0 4+2

4−2 0

0 2×24−2

4+24−2 0

− − − −0 0 −1 0

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

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Context•  Projectioncouldbeappliedtoeverypointinthescene.

•  InCG,wewillapplyittotheverticestopositionatriangleonthescreen.

•  Therestofthetrianglewillthengetfilledinonthescreenasweshallsee.

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Summary:Projectionfrom3Dto2D

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•  3Dà2D

X

Y

−Zx

yX

p

x

cameracenter

imageplane

principalaxis

C +Z

xnyn1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

xcycwc

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

−n 0 −cx0 −n −cy0 0 −1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xeyeze

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

u

v

cx

cy p x

y

xnyn1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

xcwn

ycwn

wc

wn

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

=

n xeze+ cx

n yeze+ cy

1

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

wc =wn = −zen < 0normalization

NBThecameraframeoriginlookstheoppositedirectionincomputervision

cs380: introduction to computer graphics projection chapter 10...

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