CS380: Introduction to Computer Graphics Frames in ...vclab.kaist.ac.kr/cs380/slide06-frames.pdf · 18/03/22 1 Min H. Kim (KAIST) Foundations of 3D Computer Graphics, S. Gortler,

18/03/22

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

CS380:IntroductiontoComputerGraphicsFramesinGraphics

Chapter5

MinH.KimKAISTSchoolofComputing


SUMMARYRespect

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Left-ofrule•  Pointistransformedwithrespecttotheframethatappearsimmediatelytotheleftofthetransformationmatrixintheexpression.

•  Weread

•  Weread

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f t ⇒

f tS

Sistransformedbywithrespectto f t

f t

istransformedbywithrespectto f t = at A−1 ⇒ atSA−1

S at

f t

f t

at

SA−1A

at =f t A


Scalingapointoverframe

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!f t = !at A−1

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Rotatingapointoverframe

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!f t = !at A−1


Twointerpretationsoftransformations•  Twodifferentwaysformultipletransformations:

1.  (Localtransformations)Translatewithrespecttothenrotatewithrespecttotheintermediateframe

2.  (Globaltransformations)Rotatewithrespecttothentranslatewithrespecttotheoriginalframe

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f t

f 't

f t

f t

Localtransformations Globaltransformations f t ⇒

f tTR

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FRAMESINGRAPHICSChapter5

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World,objectandeyeframes•  Worldframe(worldcoordinates)– abasicright-handedorthonormalframe– weneveralterthisframe– otherframescanbedescribedwrttheworldframe

•  Objectframe(objectcoordinates)– modelthegeometryoftheobjectusingvertexcoordinates

– notneedtobeawareoftheglobalplacement– aright-handedorthonormalframeofobject

•  Eyeframe(cameracoordinates):lateron8

wt

ot

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Worldvs.objectframe•  Therelationshipbetweentheworldframeandobjectframe:– affine4-by-4matrix(rigidbodytransformation:rotation+translationonly)

•  Themeaningofistherelationshipbetweentheworldframetotheobject’scoordinatesystem.

•  Tomovethepositionoftheobjectframeitself,wechangethematrix.

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ot = wtO

O

O

ot

O


Theeye’sview•  Theworldframeisinred•  Theobjectframeisingreen•  Theeyeframeisinblue– Theeyeislookingdownitsnegativeztowardtheobject.

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-Z

objectframe

worldframe

eyeframe

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Theeyeframe•  Eyeframe(cameracoordinates)– aright-handedorthonormalframe–  theeyelooksdownitsnegativezaxistomakeapicture

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et

et = wtE

-Z

objectframe

worldframe

eyeframe


Extrinsictransformationoftheeye•  weexplicitlystorethematrix

– Objectcoordinates:– Worldcoordinates:– Eyecoordinates:– Calculatingtheeyecoordinatesofeveryvertexes:

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E

p =otc

cOc

E−1Oc

xeyeze1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

= E−1O

xoyozo1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=wtOc =

etE−1Oc

!ot = !wtO,!et = !wtE

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MovinganObject

•  Wewanttheobjecttorotatearounditsowncenterabouttheviewer’syaxis,whenwemovethemousetotheright.

•  Howwecoulddothis?13


MovinganObject•  Basicidea:setaframe

•  MovingtransformationisM.•  Whatisthebestauxiliaryframetodothis?

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at = wt A

ot

=wtO

=at A−1O

⇒atMA−1O

=wt AMA−1O.

at

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MovinganObject•  Whatifwechoose•  wetransformthisobjectwithrespecttoratherthanwithrespecttoourobservationthroughthewindow.

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ot

ot


MovinganObject•  Whatifwetransformwithrespectto•  wewillrotateabouttheoriginoftheeye’sframe(itappearstoorbitaroundtheeye).

•  Thenwhatframeitshouldbe?

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ot

et

et

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MovinganObject

•  Weactuallywanttwodifferentoperations1.  totransform(rotate)theobjectatitsorigin2.  buttherotationaxisshouldbetheyaxisofthe

eye.17


HowtomoveanObject•  RecallingtheAffinetransform.:•  Theobject’sAffinetransform.:

(wewanttheobject’srotationabouttheobject’sorigin)•  Theeye’sAffinetransform.:

(wewanttheobject’srotationabouttheeye’syaxis)•  Thedesiredauxiliaryframe

(imagineinainverseway):

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A = TRO = (O)T (O)R

E = (E)T (E)R

at = wt (O)T (E)RA = (O)T (E)R

!at

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Summary•  Fromtheleft,wetranslatetheworldframetotheoriginOoftheobject’sframe,andthenrotatingtheobject’sframebyMaboutthatpointtoalignwiththedirectionsoftheeyeE.

19 at = wt (O)T (E)R

O← AMA−1OA = (O)T (E)R

=etE−1Oc p =

otc =wtOc

!ot = !wtO,!et = !wtE


Movingtheeye•  Weusethesameauxiliarycoordinatesystem.•  Butinthiscase,theeyewouldorbitaroundthecenteroftheobject.

•  Applyanaffinetransformdirectlytotheeye’sownframe(turningone’shead,first-personmotion)

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et = wtE,E← EM

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Theeyematrix(cameratransform)•  Specifyingtheeyematrixby:–  theeyepoint–  theviewpoint(wheretheeyelooksat)–  theupvector

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z = normalize(p − q)x = normalize(u× z)y = z × xnormalize(c) =

c / c12 + c2

2 + c32

E =

x1 y1 z1 p1x2 y2 z2 p2x3 y3 z3 p30 0 0 1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

p

q

u

et = wtE

x

y=u

z

pq

NBmatrixsenttothevertexshaderisE^{-1}*O

NBP.39containserrors(seeerrata)!!!


Theviewmatrix(gluLookAt)•  Specifyingtheviewmatrix–  theeyepoint–  theviewpoint(wheretheeyelooksat)–  theupvector

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z = normalize(q − p)x = normalize(u× z)y = x × z

p

q

u

V = E−1

xeyeze1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

= E−1O

xoyozo1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

x y

zu

p

q

V =E−1

modelmatrix:Oviewmatrix:V =E−1

movelviewmatrix:VO=E−1O

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SceneGraph:HierarchyFrames•  Anobjectcanbetreatedasbeingassembledbysomefixeandmovablesubobjects.

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ot = wtO o 't = otO ' at = ot A bt = at B b 't =

bt B '

ct =btC

dt = ctD d 't =

dtD '

f t = otF


SceneGraph:HierarchyFrames•  WejustupdateitsOmatrixtotheobjectframe,insteadofrelatingittotheworldframe

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ot = wtO

ot = wtOat = wtOAbt = wtOABb 't = wtOABB 'ct = wtOABCdt = wtOABCDd 't = wtOABCDD '

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Matrixstack•  Matrixstackdatastructurecanbeusedtokeeptrackofthematrix

•  push(M)–  createsanew‘topmost’matrix–  acopyoftheprevioustopmostmatrix– M.multipliesthisnewtopmatrix

•  pop()–  removesthetopmostlayerofthestack

•  descending–  descenddowntoasubobject,whenapushoperationisdone

–  thismatrixispoppedoffthestackwhenreturningfromthisdescenttotheparent

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Movinglimbs

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Movinglimbs

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Scenegraphpseudocode

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…matrixStack.initialize(inv(E));matrixStack.push(O);

matrixStack.push(O’); draw(matrixStack.top(),cube);\\bodymatrixStack.pop();\\O’

matrixStack.push(A);\\grouping matrixStack.push(B); matrixStack.push(B’); draw(matrixStack.top(),sphere);\\upperarm matrixstack.pop();\\B’

matrixStack.push(C); matrixStack.push(C’); draw(matrixStack.top(),sphere);\\lowerarm matrixStack.pop();\\C’ matrixStack.pop();\\C matrixStack.pop();\\BmatrixStack.pop();\\A

\\currenttopmatrixisinv(E)*O\\wecannowdrawanotherarm

matrixStack.push(F);…

!p="etE−1OO'c →draw()

!p="etE−1OABB 'c →draw()

!p="etE−1OABCC 'c →draw()

!p="etE−1Oc

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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

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

CS380: Introduction to Computer Graphics Frames in ...vclab.kaist.ac.kr/cs380/slide06-frames.pdf · 18/03/22 1 Min H. Kim (KAIST) Foundations of 3D Computer Graphics, S. Gortler,

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