CS380: Introduction to Computer Graphics Linear ...vclab.kaist.ac.kr/cs380/slide03-2-Linear.pdf · 18/03/13 1 Min H. Kim (KAIST) Foundations of 3D Computer Graphics, S. Gortler, MIT

18/03/13

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

CS380:IntroductiontoComputerGraphicsLinearTransformation

Chapter2

MinH.KimKAISTSchoolofComputing


GLSLPIPELINE

RECAP

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GLSLPipeline:VertexShader

•  Verticesarestoredinavertexbuffer.•  Whenadrawcallisissued,eachoftheverticespasses

throughthevertexshader•  Oninputtothevertexshader,eachvertex(black)has

associatedattributes.•  Onoutput,eachvertex(cyan)hasavalueforgl_Positionand

foritsvaryingvariables.3


GLSLPipeline:Rasterization

•  Thedataingl_Positionisusedtoplacethethreeverticesofthetriangleonavirtualscreen.

•  Therasterizerfiguresoutwhichpixels(orange)areinsidethetriangleandinterpolatesthevaryingvariablesfromtheverticestoeachofthesepixels.

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GLSLPipeline:FragmentShader

•  Eachpixel(orange)ispassedthroughthefragmentshader,whichcomputesthefinalcolorofthepixel(pink).

•  Thepixelisthenplacedintheframebufferfordisplay.

5


GLSLPipeline:FragmentShader

•  Bychangingthefragmentshader,wecansimulatelightreflectingofdifferentkindsofmaterials.

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TextureMapping

•  Asimplegeometricobjectdescribedbyasmallnumberoftriangles.

•  Anauxiliaryimagecalledatexture.•  Partsofthetexturearegluedontoeachtriangle

givingamorecomplicatedappearance.7


LINEARTRANSFORMATIONChapter2

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Pointvs.Vector•  RepresentPointsusingcoordinates•  Toperformgeometrictransformationstothese

points•  Vectors:3Dmotionvialineartransformations•  Coordinatevector:thepositionofthepoint

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xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥


Points Vector Coordinatesystem

Coordinatevector

Pointvs.CoordinateVector

1.  Point(geometricobject):notatedas(tildeabovetheletter),non-numericalobject.

2.  Vector(motion):notatedas(arrowabovetheletter),non-numericalobject.

3.  Coordinatesystem:denotedas(bold:columnvector,tmakesittranspose),non-numericalobjectbasisforvector;frameforpoint

4.  Coordinatevector:notedas(boldletter),numericalobject

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p

v

c

f t

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VectorSpace•  Avectorspaceissomesetofelements•  NB:Vector(motion)isNOTjustasetofthree

numbers!!!•  Ifasetofvectorsisnotlinearlydependent,wecall

linearlyindependent.•  Ifarelinearlyindependent,allvectors

ofcanbeexpressedwithcoordinatesofabasisof(asetof).

•  isthedimensionofthebasis/space11

V v

b1...bn

V v

ciV bi

v= cibi

i∑ .

n


VectorSpace•  Freemotioninspace,3dimensionalvector•  Invectoralgebranotation:

•  avector•  rowbasisvectors•  columncoordinatevector

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c

v= cibi

i∑ = b1

b2

b3⎡

⎣⎤⎦

c1c2c3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

v

b t

v= b tc.

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LinearTransformation•  Lineartransformationfollowsthesetwo

properties:

•  Vectortransformation(suchthatthebasisislinearlyindependent):

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L(v+ u) = L(v

)+ L(u

)

L(α v) =αL(v

) .

v!⇒ L(v

!) = L cib

!i

i∑⎛⎝⎜

⎞⎠⎟= ciL(b

!i )

i∑ .


3-by-3Transformation•  Rewritethelineartransform

•  isactuallyalinearcombinationoftheoriginalbasisvectors.

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b1

b2

b3⎡

⎣⎤⎦

c1c2c3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥⇒ L(b1

) L(b2

) L(b3

)⎡

⎣⎤⎦

c1c2c3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

L(b1)

L(b1) = b1

b2

b3⎡

⎣⎤⎦

M1,1

M 2,1

M 3,1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

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3-by-3Transformation•  3-by-3matrix:

•  Puttingalltogether:

•  Amatrixtotransformonevectortoanother:

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L(b1!"!) L(b2

!"!) L(b3

!"!)⎡

⎣⎢⎤⎦⎥= b1!"!

b2!"!

b3!"!⎡

⎣⎢⎤⎦⎥

M1,1 M1,2 M1,3

M2,1 M2,2 M2,3

M3,1 M3,2 M3,3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

.

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= b tc⇒ b tMc


Lineartransformofavector•  Avectorundergoesalineartransformation

•  ThematrixMdependsonthechosenlineartransformation.

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v= b tc⇒ b tMc

v!⇒ L(v

!)

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Inversetransform•  Identitymatrix

•  In3Dgraphics,whilemovingobjectsaroundinspace,itwillseldommakesensetouseannon-invertibletransform.

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I =1 0 00 1 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

MM −1 = M −1M = I .


Lineartransformofabasis•  Abasisundergoesalineartransformation

•  Validtomultiplyamatrixtimesacoordinatevector

•  changeabasisofavectorto

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b t

⇒ b tM

v= b tc = a tM −1c . a

t= b tM ,

b t

a t

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Dotproduct•  Input:twovectors•  Output:arealnumber•  dotproduct=thesquaredlength

•  Theanglebetweenthetwovectors:

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v⋅w

v 2:= v⋅v

cosθ = v

⋅w

vw .

θ ∈[0...π ]


3Dorthogonalbasis•  Orthogonalvectors:•  Aright-handedorthogonalcoordinatesystem.

Thezaxiscomesoutofthescreen(OpenGL).•  Aleft-handedorthogonalcoordinatesystem.

Thezaxisgoesintothescreen(DirectX).

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v⋅w= 0

right-handed

left-handed

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Crossproduct•  Input:twovectors•  Output:avector

•  whereisaunitvectorthatisorthogonaltotheplanespannedbyand

•  formsaright-handedbasis

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[v,w,n]

n

v

w

v×w:= vwsinθn,


Crossproduct•  Inaright-handedorthogonalbasis

•  Wecancomputeacross-productas

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(b tc)× (b

td) =

c2d3 − c3d2c3d1 − c1d3c1d2 − c2d1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

b t

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2DRotation•  Letbea2Dright-handedorthonormal

(orthogonalandunitvectors)basis

•  Rotatedvector

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v= b1

b2⎡

⎣⎤⎦

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥.

b t

x ' = xcosθ − ysinθy ' = xsinθ + ycosθ .

x 'y '

⎡

⎣⎢⎢

⎤

⎦⎥⎥= cosθ −sinθ

sinθ cosθ⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥.


3DRotation•  Everyrotationfixesanaxisofrotationand

rotatesbysomeangleaboutthataxis.•  Rotationaroundthezaxis:

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b1

b2

b3⎡

⎣⎤⎦

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⇒ b1

b2

b3⎡

⎣⎤⎦

cosθ −sinθ 0sinθ cosθ 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

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3DRotation•  Rotationaroundthexaxis

•  Rotationaroundtheyaxis

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1 0 00 cosθ −sinθ0 sinθ cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

cosθ 0 sinθ0 1 0

−sinθ 0 cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥


xyz-Euleranglerotation•  Axisofrotation

•  xyz-Euleranglerotationmatrix

where26

kx2v + c kxkyv − kzs kxkzv + kys

kykxv + kzs ky2v + c kykzv − kxs

kzkxv − kys kzkyv + kxs kz2v + c

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

k= kx ,ky ,kz⎡⎣ ⎤⎦

t

c := cosθ , s := sinθ , v := 1− c.

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Scales•  Scalingoperations

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b1

b2

b3⎡

⎣⎤⎦

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⇒ b1

b2

b3⎡

⎣⎤⎦

α 0 00 β 00 0 γ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

CS380: Introduction to Computer Graphics Linear ...vclab.kaist.ac.kr/cs380/slide03-2-Linear.pdf · 18/03/13 1 Min H. Kim (KAIST) Foundations of 3D Computer Graphics, S. Gortler, MIT

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