cs380: introduction to computer graphics affine transform...

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

CS380:IntroductiontoComputerGraphicsAffineTransformChapter3

MinH.KimKAISTSchoolofComputing


RECAPLinearTransformation

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Geometricdatatypes•  Point:–  representsplace

•  Vector:–  representsmotion/offsetbetweenpoints

•  Coordinatevector:•  Coordinatesystem:– “basis”forvectors– “frame”forpoints

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p

v

c

st


Vectorspace•  Avectorspaceissomesetofelements•  Coordinatevector:•  ifarelinearlyindependent,allvectorsofcanbeexpressedwithcoordinatesofabasisof(asetof).

•  isthedimensionofthebasis/space

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b1...bn

V v

ciV bi

n

v= cibi

i∑ = b1

b2

b3⎡

⎣⎤⎦

c1c2c3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥. v

= b tc.

V v

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Lineartransformation•  3-by-3matrix

•  Lineartransformofavector

•  Lineartransformofabasis

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b1

b2

b3⎡

⎣⎤⎦

c1c2c3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥⇒ b1

b2

b3⎡

⎣⎤⎦

M1,1 M1,2 M1,3

M 2,1 M 2,2 M 2,3

M 3,1 M 3,2 M 3,3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

c1c2c3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

v!⇒ L(v

!) v

!= b"! tc⇒ b"! tMc

v= b tc = a tM −1c .


Productofvectors;rotation•  Dotproduct:

•  Crossproduct:

•  3Drotationmatrices:aroundzaxis, xaxis, yaxis

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v⋅w:= vwcosθ .

v×w:= vwsinθn,

cosθ −sinθ 0sinθ cosθ 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1 0 00 cosθ −sinθ0 sinθ cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

cosθ 0 sinθ0 1 0

−sinθ 0 cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

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AFFINETRANSFORMATIONChapter3

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Pointsvs.vectors•  Vector:=motionbetweenpointsinspace–  livesinaspacewecall– hasthestructureofalinear/vectorspace.– additionandscalarmultiplicationhavemeaning– zerovectorisnomotion– cannotreallytranslatemotion

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v

R3

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Pointsvs.vectors•  Point:=apositioninspace–  livesinaspacewemightcall– hasthestructureofaso-calledaffinespace.– additionandscalarmultiplicationdon’tmakesense– zerodoesn’tmakesense– subtractiondoesmakesense,givesusavector

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pA3


Pointsandframes•  Subtractionofpoints:

•  Movingapointwithavector:

•  Basisisthreevectors

•  Howcouldwepresenttranslations?– Affinetransform(4-by-4matrix)–  usage:transformationofobjectsandcameraprojection(3Dà2D)

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p − q =v

v = cibi

i∑

q +v = p

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Affineframe•  Affinespace:marriageofpointandvector

•  foraffinespacewewilluseaframe– startwithachosenoriginpoint– addtoitalinearcombinationofvectorsusingcoordinatestogettoanydesiredpoint

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o

ci p

p = o+ v.


Affineframe•  Movementofapoint(originalàapoint)

•  Affineframe(madeofthreevectorsin3Dandapointin3D):

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p = o+ cibi

i∑ = b1

b2

b3o⎡

⎣⎤⎦

c1c2c31

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

. p = o+ v

.

o p

b1

b2

b3o⎡

⎣⎤⎦ = f t

p = f tc .

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Defininganaffinematrix•  pointisspecifiedwitha4-coordinatesvector–  fournumbers–  lastoneisalways1– …or0.(andwegetavector)

•  let’sdefineanaffinematrixas4-by-4matrix

•  wearetransformingapointtoanotherwithanaffineframe:

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

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

p = f tc .

c1 c2 c3 1⎡⎣

⎤⎦t


Transformingapoint•  affinetransform

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b1

b2

b3o⎡

⎣⎤⎦

c1c2c31

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⇒

b1

b2

b3o⎡

⎣⎤⎦


⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

c1c2c31

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

.

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Transformingapoint•  forshort

•  transformingcoordinatevectors(4withaoneasthefourthentry)

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p = f tc⇒ f tAc .

b1

b2

b3o⎡

⎣⎤⎦ = f t

c '1c '2c '31

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=


⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

c1c2c31

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

.

where


transformingapoint•  Alternatively,transformingthebasisvectors

•  Thistransformationistoapplytheaffinetransformtoaframeas

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b '1

b '2

b '3

o '⎡⎣

⎤⎦ = b1

b2

b3o⎡

⎣⎤⎦


⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

.

b1

b2

b3o⎡

⎣⎤⎦⇒ b1

b2

b3o⎡

⎣⎤⎦


⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

.

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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

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

.


Lineartransformation•  affinetransformation:

where,Lisa4-by-4matrix;lisa3-by-3matrix.•  Alineartransformisappliedtoapoint.Thisisaccomplishedbyapplyingthelineartransformtoitsoffsetvector.

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L = l 00 1

⎡

⎣⎢

⎤

⎦⎥

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Translationtransform•  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

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

.


Translationtransform•  translationtransformationtopoints

•  translationmatrix

•  where,Tisa4-by-4matrix;iisa3-by-3identitymatrix,tis3-by-1matrixfortranslation.

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c1 ⇒ c1 + txc2 ⇒ c2 + tyc3 ⇒ c3 + tz

T = i t0 1

⎡

⎣⎢

⎤

⎦⎥

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Affinetransformmatrix•  Anaffinematrixcanbefactoredintoalinearpartandatranslationalpart:

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⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

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


Affinetransformmatrix•  NBasmatrixmultiplicationisnotcommutative,theorderofthemultiplicationTLmatters!!!

•  Sincethesematriceshavethesamesize(4-by-4),itisdifficulttodebugwhenyoumesseduptheorder.Payextraattentiononitwhileyouarecoding…

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TL ≠ LT

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Rigidbodytransformation•  Whenthelineartransformisarotation,wecallthisasrigidbodytransformation(rotation+translationonly).

•  Arigidbodytransformationpreservesdotproductbetweenvectors,handednessofabasis,anddistancebetweenpoints.

•  Itsgeometrictopologyismaintainedwhiletransformingit.

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A = TR


Affinetransformactingonvector•  Iffourthcoordinateofciszero,thisjusttransformsavectortoavector.– notethatthefourthcolumnisirrelevant– avectorcannotbetranslated

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

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n ⋅( p1 − p2 ) = 0


Normals•  Weusenormalsforshading•  howdotheytransform•  supposeirotateforward– normalgetsrotatedforward

•  supposesquashintheydirection

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•  Surfacenormalsoftwogeometriesaredifferentbefore/aftertransformation

•  Squashingaspheremakesitsnormalsstretchalongtheyaxisinsteadofsquashing.

•  normalgetshigherintheydirection

•  whatistherule?

Changingashape

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nxnynz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥≠

nx 'ny 'nz '

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.


Transformingnormals•  Sincethenormalandveryclosepointsandareonasurface:

•  AfterapplyinganaffinetransformA,

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nx ny nz ∗⎡⎣

⎤⎦

x1y1z11

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

−

x0y0z01

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

= 0 .

n p1 p2

nx ny nz ∗⎡⎣

⎤⎦A

−1( ) A

x1y1z11

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

−

x0y0z01

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

= 0 .

thenormalofthetransformedgeometry

n ⋅( p1 − p2 ) = 0

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Transformingnormals•  Transformednormals:

•  Transposingthisexpression:

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nx ' ny ' nz '⎡⎣

⎤⎦ = nx ny nz⎡

⎣⎤⎦ l

−1 .

nx 'ny 'nz '

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= l− t

nxnynz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.


Transformingnormals•  Rememberlisarotationmatrix(orthonormal),thusitsinversetransposeisthesameastheoriginal:

•  inversetranspose– soinversetranspose/transposeinverseistherule–  forrotation,transpose=inverse–  forscale,transpose=nothing–  inthecodenextweek,wewillsendAandl-ttothevertexshader.

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l− t = l .

LLt = I (Lt = L−1), det L = 1

cosθ −sinθsinθ cosθ

⎡

⎣⎢

⎤

⎦⎥

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Transformingnormals•  Renormalizetocorrectunitnormalsofsquashedshape:

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nx 'ny 'nz '

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= l− t

nxnynz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

nxnynz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥≠

nx 'ny 'nz '

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.


Normaltransformation

•  Forrotation,wecansimplyapplyltonormals,sameasapplyingltovertices– Asincaseofrotation

•  Fordeformation,wemustapplytonormalsinsteadof.

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l−t = l

nx 'ny 'nz '

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= l− t

nxnynz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

l−tl

cs380: introduction to computer graphics affine transform...

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