CS380: Introduction to Computer Graphics Respect Chapter 4 ...vclab.kaist.ac.kr/cs380/slide05-respect.pdf18/03/20 1 Min H. Kim (KAIST) Foundations of 3D Computer Graphics, S. Gortler,

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

CS380:IntroductiontoComputerGraphicsRespectChapter4

MinH.KimKAISTSchoolofComputing


SUMMARYAffineTransformation

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Lineartransformation•  3-by-3transformmatrixà4-by-4affinetransform

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b1

b2

b3o⎡

⎣⎤⎦

c1c2c31

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⇒

b1

b2

b3o⎡

⎣⎤⎦

a b c 0e f g 0i j k 00 0 0 1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

c1c2c31

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

.


Translationtransformation•  translationtransformationtopoints

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b1

b2

b3o⎡

⎣⎤⎦

c1c2c31

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⇒ b1

b2

b3o⎡

⎣⎤⎦

1 0 0 tx0 1 0 ty0 0 1 tz0 0 0 1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

c1c2c31

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

.

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Affinetransformmatrix•  Anaffinematrixcanbefactoredintoalinearpartandatranslationalpart:

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a b c de f g hi j k l0 0 0 1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

1 0 0 d0 1 0 h0 0 1 l0 0 0 1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

a b c 0e f g 0i j k 00 0 0 1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

.

l t0 1

⎡

⎣⎢

⎤

⎦⎥ =

i t0 1

⎡

⎣⎢

⎤

⎦⎥

l 00 1

⎡

⎣⎢

⎤

⎦⎥

A = TL


Normals•  Normal:avectorthatisorthogonaltothetangentplaneofthesurfacesatthatpoint.–  thetangentplaneisaplaneofvectorsthataredefinedbysubtracting(infinitesimally)nearbysurfacepoints:

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n ⋅( p1 − p2 ) = 0

nx 'ny 'nz '

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= l− t

nxnynz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

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RESPECTChapter4

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Scalingapointoverframe•  Wearetransformingapointinaframe•  Withamatrix

•  Performingatransform:

•  Supposeanotherframe:8

S =

2 0 0 00 1 0 00 0 1 00 0 0 1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

p f t

p =f tc

f tc⇒

f tSc

at =f t A

thestretchesbyfactoroftwoinfirstaxisof

f t

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Scalingapointoverframe•  Wecouldexpressthepointwithanewcoordinatevector

•  Nowtransformsthepointwithrespectto

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p =f tc = atd

at =f t A

f tc =

f t Ad

1A−=d c

atS

atd⇒ atSd

p

f t = at A−1


Left-ofrule•  Pointistransformedwithrespecttothetheframethatappearsimmediatelytotheleftofthetransformationmatrixintheexpression.

•  Weread

•  Weread

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tt S⇒f fr r

Sistransformedbywithrespectto tfr

f t

istransformedbywithrespectto

1 1tt t A SA− −= ⇒f aar rr

S tar f t

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Scalingapointoverframe

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t tp S= ⇒f c cfrr

%Sistransformedbywithrespectto p

f t



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p =at A−1c⇒ atSA−1c

Sistransformedbywithrespectto p at

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Rotatingapointoverframe•  Thesamereasoningtotransformationsofframesthemselves:

•  Inanotherframe:

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tt R⇒f fr r

R f t tf

r


1 1tt t A RA− −= ⇒f aar rr

R f t

tar

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Rotatingapointoverframe

15istransformedbywithrespectto

t tp R= ⇒f c cfrr

%R p

f t



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istransformedbywithrespectto1 1t tp A RA− −= ⇒a c a crr%

R p at

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AuxiliaryFrame•  Youwanttobuildthesolarsystem– TheMoonrotatesaroundtheEarth’sframe– TheEarthrotatesaroundtheSun’sframe

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TheSun

TheEarth

TheMoon

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TransformsusinganAuxiliaryFrame•  Sometimesweneedtotransformaframeinsomespecificway,representedbyamatrix,withrespecttosomeauxiliaryframe

•  Thetransformframecanthenbeexpressedas

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f tM

at

at ⇒

f t A

f t

=at A−1

⇒atMA−1

=f t AMA−1


MultipleTransformations•  Rotationandtranslationwithframe

•  NBingeneral,matrixmultiplicationisnotcommutative!!!

•  Therearetwodifferentwaystoapplymultipletransformations– Localtransformation– Globaltransformation

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f t ⇒

f tTR

f tTR ≠

f t RT

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LocalTransformations•  Localtransformations

•  Inthefirststep,

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f t ⇒

f tTR

f t ⇒

f tT =

f 't

istransformedbywithrespecttoT f t

f t

astheresultingframe: f 't


LocalTransformations•  Localtransformations

•  Inthesecondstep,

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f t ⇒

f tTR

f t ⇒

f tTR,' .t t R⇒ ff

r r

istransformedbywithrespecttoR f t

f 't

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GlobalTransformations•  Globaltransformations

•  Inthefirststep(inthereverseorder)

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f t ⇒

f t R =

f t

istransformedbywithrespecttoR f t

f t

astheresultingframe: f t

f t ⇒

f tTR


GlobalTransformations•  Globaltransformations

•  Inthesecondstep

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t t tR TR= ⇒ ff for rr

istransformedbywithrespecttoT f t

f t

f t ⇒

f tTR

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Twointerpretationsoftransformations

•  Twodifferentwaysformultipletransformations:1.  (Localtransformations)Translatewithrespectto

thenrotatewithrespecttotheintermediateframe

2.  (Globaltransformations)Rotatewithrespecttothentranslatewithrespecttotheoriginalframe

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f t

f 't

f t

f t

Localtransformations Globaltransformations f t ⇒

f tTR

CS380: Introduction to Computer Graphics Respect Chapter 4 ...vclab.kaist.ac.kr/cs380/slide05-respect.pdf18/03/20 1 Min H. Kim (KAIST) Foundations of 3D Computer Graphics, S. Gortler,

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