CS380: Introduction to Computer Graphics Varying Variables ...vclab.kaist.ac.kr/cs380/slide14-varying_variables.pdf · 18/04/26 1 Min H. Kim (KAIST) Foundations of 3D Computer Graphics,

18/04/26

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

CS380:IntroductiontoComputerGraphicsVaryingVariables

Chapter13

MinH.KimKAISTSchoolofComputing


SUMMARYRasterization

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Pathfromvertextopixel

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Clippingcoordinates•  Eyecoordinates(projected)àclipcoordinatesànormalizeddevicecoordinates(NDCs)

•  (reminder)Dividingclipcoordinatesbythecomponent(thefourthcomponentinthehomogeneouscoordinates)yieldsnormalizeddevicecoordinates(NDCs).

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(xc , yc , zc ,wc )wc(wc = wn )

xnwn

ynwn

znwn

wn

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

xcyczcwc

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

sx 0 −cx 00 sy −cy 0

0 0 f + nf − n

− 2 fnf − n

0 0 −1 0

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

xeyeze1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

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Viewportmatrix

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•  Weneedatransformthatmapsthelowerleftcornertoandupperrightcornerto

•  Theappropriatescaleandshiftcanbedoneusingtheviewportmatrix:

[−0.5,−0.5]t

[W − 0.5,H − 0.5]t

xwywzw1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

W / 2 0 0 (W −1) / 20 H / 2 0 (H −1) / 20 0 1/ 2 1 / 20 0 0 1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

xnynzn1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥


TextureViewport•  Theabstractdomainfortexturesisnotthecanonicalsquare,butinsteadistheunitsquare.

•  Inthiscasethecoordinatetransformationmatrixis:

6

xwyw−1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

W 0 0 −1/ 20 H 0 −1/ 2− − − −0 0 0 1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

xtyt−1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

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VARYINGVARIABLESChapter13andAppendixB

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Interpolationofvaryingvariables•  Inbetweenthevertexandfragmentshader,weneedtointerpolatethevaluesofthevaryingvariables.

•  Whatisthedesiredinterpolant,andhowshouldwecomputeit.

•  Thisissurprisinglysubtle.

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Texturemapping

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Wrongrepresentationoftexture

10

Whentexturecoordinatesarelinearlyinterpolatedinwindowcoordinates,anincorrectimageresults.

(xt , yt )

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Correctrepresentationoftexture

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Howcanweinterpolatetexturecoordinatesproperlylikethis?(xt , yt )


Motivation:texturecoordinates•  Letusmapasquarecheckerboardimageontoaquad

•  Webreakupthequadandtheimageeachintotwotriangles.

•  Wewillassociatetexturecoordinatesforeachvertex.

•  Wedesirethatintheinteriorofatriangle,shouldbedeterminedastheuniqueinterpolantfunctionsoverthetrianglethatareaffinein

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[xt , yt ]t

[xt , yt ]t

(xo , yo ,zo)

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Motivation:texturecoordinates•  Evenstepsonthegeometryshouldbeevenstepsinthetexture.

•  Ifweusethisinterpolationandfetchthetexturevalues,weshouldgetanexpectedforeshorteningeffect.

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Linearinterpolation•  Supposewesimplyusedlinearinterpolationonthefunctionsontexture.

•  Then,aswemovebysomefixed2Dvectordisplacementonthescreen,thetexturecoordinateswillbeupdatedbysomefixed2Dvectordisplacementintexturecoordinates.

•  Inthiscase,allofthesquaresofthetexturewillmaptoequalsizeparallelograms.

•  Wewillgetanoddseamwherethetwotrianglesmeet.

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xt and yt

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Affinefunctionsin2D•  Wesayafunctionisanaffinefunctioninthevariablesifitisoftheform:forsomeconstants

•  Thiscanalsobewrittenas:

15

v(x , y)=ax+by+c

v( )

a,b and c

v = a b c⎡⎣

⎤⎦

xy1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

x and y


Affinefunctionsin2D•  Alsocalled‘linear’butnotetheadditiveconstantterm.

•  Anaffinefunctioncanbeevaluatedbyplugginginorbyincrementalevaluationalongalineinspace.

•  Wealreadysawforexamplethatisanaffinefunctionofandsowasa3D“edgefunction”:

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x, yx, y

znxn , yn

zn = axn + byn + c

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Affine/linearinterpolationin2D•  Ifwearegiven,thevaluesofforthree(non-collinear)pointsinthe2Dplane,saytheverticesofatriangle,

•  Thisdeterminesanaffinefunctionovertheentireplane.

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vi v(x, y)

v

vi

x1

x2

x3


Affine/linearinterpolationin2D•  Inthiscase,wesaythatisthelinearinterpolantofthevaluesatthethreevertices.

•  Theprocessofevaluatingthelinearinterpolantofthreeverticesiscalledlinearinterpolation.

•  Giventheonecanusesomesimplematrixoperationstosolvefortheconstants.

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v

vi

[a,b,c]vi

x1

x2

x3

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Barycentric Interpolation How do you interpolate values defined at vertices across the entire triangle?

Barycentric Interpolation How do you interpolate values defined at vertices across the entire triangle?

Solve a simpler problem first:

x 1

x 2

1DInterpolation

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x 2

Barycentric Interpolation How do you interpolate values defined at vertices across the entire triangle? Solve a simpler problem first:

0

x 1 t

Want to define a value for every

0

1

1 t

t ∈ [0,1]

1DInterpolation

Barycentric Interpolation How do we come up with this equation? Look at the picture!

t

1DInterpolation

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Barycentric Interpolation

The further t is from the red point, the more blue we want.

t

1DInterpolation


The further t is from the red point, the more blue we want. The further t is from the blue point, the more red we want.

t

1DInterpolation

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t


The further t is from the red point, the more blue we want. The further t is from the blue point, the more red we want. Percent blue = (length of blue segment)/(total length)

1DInterpolation


The further t is from the red point, the more blue we want. The further t is from the blue point, the more red we want. Percent blue = (length of blue segment)/(total length) Percent red = (length of red segment)/(total length)

t

1DInterpolation

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The further t is from the red point, the more blue we want. The further t is from the blue point, the more red we want. Percent blue = (length of blue segment)/(total length) Percent red = (length of red segment)/(total length)

Value at t = (% blue)(value at blue) + (% red)(value at red)

t

1DInterpolation

The further t is from the red point, the more blue we want. The further t is from the blue point, the more red we want.

Percent blue = t Percent red = 1-t Value at t = tx + (1-t)x

1 2

x 1

x 2

Barycentric Interpolation 0

t 1

1DInterpolation

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Barycentric Interpolation Now what about triangles?

2DInterpolation


Now what about triangles?

Just consider the geometry:

What’s the interpolated value at the point p?

x

p

x 3

1

x 2

2DInterpolation

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Last time (in 1D) we used ratios of lengths.

t

p

x 1

x 2

x 3

2DInterpolation



What about ratios of areas (2D)?

p

2DInterpolation

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p

If we color the areas carefully, the red area (for example) covers more of the triangle as p approaches the red point.

2DInterpolation

p


Just like before: percent red = area of red triangle

total area

percent green = area of green triangle total area

percent blue = area of blue triangle total area

2DInterpolation

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p


Just like before: percent red

Value at p: (% red)(value at red) + (% green)(value at green) + (% blue)(value at blue)

= area of red triangle total area

percent green = area of green triangle total area

percent blue = area of blue triangle total area

2DInterpolation

p

x 3

A 3

A 1

A 1

A 2

x 1

x 2


“barycentric interpolation” a.k.a.“convex combination” a.k.a.“affine linear extension”

Just like before: percent red =

Σ λ =1 i i

Why? Look at the picture!

“barycentric coordinates”

= λ 1

= λ 2

= λ3

Value at p: (A x + A x + A x )/A

1 1 2 2 3 3

A A 2

A A 3

A

percent green =

percent blue =

Now convert this to a bunch of ugly symbols if you want... just don’t think about it that way!

2DInterpolation

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Affinefunctionsin3D•  Wesayafunctionisaffineinvariableifitisoftheform:

•  Suchafunctioncanbeuniquelydeterminedbyitsvaluesatthefourverticesofatetrahedronsittingin3D.

37

vx, y and z

v(x , y ,z)=ax+by+cz+d


Trianglesin3D•  Givenatrianglein3D,supposewespecifythevalueofafunctionatitsthreevertices.

•  Theremaybemanyfunctionsthatareaffineinthatagreewiththesethreevalues.

•  Butallofthesefunctionswillagreewhenrestrictedtotheplaneofthetrianglein2D.

•  Assuch,wecanrefertothisrestrictedfunctionoverthetriangleasthelinearinterplantofthevertexvalues.

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(x, y, z)

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Interpolatingcolorsofanobject•  Whenweassociatecolorswithvertices,itisnaturaltointerpretourdesiredcolorfieldtobetheuniqueinterpolatingfunctionoverthetrianglethatisaffineintheobjectcoordinates,,notinscreencoordinates!

•  Duringtexturemapping,itisnaturaltointerpreteachofthetexturecoordinates,,astheuniqueinterpolatingfunctionsoverthetrianglethataffinein.

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(xo , yo ,zo)

(xt , yt )

(xo, yo, zo )


Interpolatingcolorsofanobject•  Asaratherselfreferentialexample,wecaneventhinkofeachofthethreeobjectcoordinatesofapointonsometrianglein3Dasaffinefunctionsin.Forexample,

•  Forthisreason,thedefaultsemanticsofOpenGListointerpolateallvaryingvariablesasfunctionsovertrianglesthatareaffinein.

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(xo, yo, zo )

v(xo , yo ,zo)= xo (whena=1andb= c =d =0)

(xo, yo, zo )

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Interpolatingcolorsofanobject•  Aswewillsee,thisisequivalenttoafunctionbeingaffineovereyecoordinates,,butitisnotequivalenttoafunctionbeingaffineovernormalizeddevicecoordinates,,orwindowcoordinates.

41

(xe, ye, ze )

(xn , yn , zn )


Goingdown•  Ifwehaveafunctionthatisaffineinwhenrestrictedtoatrianglein3D,

•  Thenwecanusethefactthatthetriangleisflattowriteasafunctionthatisaffineinonlytwovariables.

•  Idea:writeasanaffinefunctionofplugthisintotheaffineexpressionfor.

42

v (x, y, z)

v

z (x, y)v

z =ex+ fy+ gv =ax+by+c(ex+ fy+ g)+dv =hx+ iy+ j

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Goingsideways•  Supposewehavesomematrixexpressionoftheform:forsome4-by-4matrixM(whereMdoesnotevenhavetobeanaffinematrix).

•  Then,justlookingatthefourrowsindependently,weseethatareallaffinefunctionsof.

43

x 'y 'z 'w '

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

= M

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

x ', y ',z ', andw'(x , y ,z)


Moresideways•  Ifwehaveafunctionvwhichisaffinein,then,giventherelationofthepreviousequation,wecanseethatvisalsoaffine.Toseethis,notethat:

44

(x, y, z)

(x ', y ', z ',w ')

v = a b c d⎡⎣

⎤⎦

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

v = a b c d⎡⎣

⎤⎦M

−1

x 'y 'z 'w'

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

∵

x 'y 'z 'w'

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=M

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

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Moresideways

•  Sothepropertyofaffine-nessisinagreement– betweenobjectoreyeorclipcoordinates,andalso– betweennormalizedandwindowcoordinates.

45

v = e f g h⎡⎣

⎤⎦

x 'y 'z 'w'

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


Notsideways•  Theonlytimewehavetobecarefuliswhendivisionisdone.

•  Forexample,giventherelation:

vs.

•  Itwillgenerallynotbethecasethatafunctionv,whichwasaffineinwillbeaffineinor.

•  OurvaryingvariablesarenotaffineinNDCorwindowscoordinates.

46

x 'w 'y 'w 'z 'w 'w '

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

= M

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

(x, y, z)(x ', y ', z ') (x ', y ', z ',w ')

x 'y 'z 'w '

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

= M

xyz1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

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Howtoevaluatethevaryingvariables

•  Recall

•  Invertingourmatrices,thisimpliesthatateachpointonthetriangle,wehavetherelation:

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xnwn

ynwn

znwn

wn

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

= PM

xoyozo1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

xoyozo1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

= M −1P−1

xnwn

ynwn

znwn

wn

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥



•  Nowsupposethatvisanaffinefunctionof.

•  Wealsomakeuseoftheobviousfactthattheconstantfunction1isalsoaffinein

•  Thusforsome,wehave

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(xo, yo, zo )

(xo, yo, zo )(a,b,c,d)

v1

⎡

⎣⎢

⎤

⎦⎥ =

a b c d

0 0 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xoyozo1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

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•  Andtherefore:

•  Forsomeappropriatevaluese…l49

v1

⎡

⎣⎢

⎤

⎦⎥=

a b c d

0 0 0 1

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥M−1P−1

xnwn

ynwn

znwn

wn

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

v1

⎡

⎣⎢

⎤

⎦⎥=

e f g h

i j k l

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xnwn

ynwn

znwn

wn

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

∵

xoyozo1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=M−1P−1

xnwn

ynwn

znwn

wn

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥



•  Nowdividebothsidesbyandweget

•  Conclusion:areaffinefunctionsofnormalizeddevicecoordinates.

50

wn

vwn

1wn

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=e f g h

i j k l

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xnynzn1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

vwn

and 1wn

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Howtoevaluatethevaryingvariables•  “goingsideways”wededucethatareaffinefunctionsofwindowcoordinates.

•  “goingdown”wecanconclude:arebothaffinefunctionof.

•  Meaningthatwecancalculatetheirvaluesateachpixel,justgiventheirvaluesattheverticesandlinearinterpolation.

•  Theabovederivationcannowbethrownaway.51

vwn

and 1wn

(xw , yw ,zw )

vwn

and 1wn

(xw , yw )


VVinOpenGL•  NowwecanseehowOpenGLcanperformthecorrect“rationallinearinterpolation”tocalculatevateachpixel.– Thevertexshaderisrunoneachvertex,calculatingclipcoordinatesandvaryingvariablesforeachvertex.

– Clippingisrunoneachtriangle;thismaycreatenewvertices.Linearinterpolationinclipcoordinatesspaceisruntodeterminetheclipcoordinatesandvaryingvariablevaluesforeachsuchnewvertex.

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VVinOpenGL

– Foreachvertex,andforeachvaryingvariablev,OpenGLcreatesaninternalvariable.Additionally,foreachvertexOpenGLcreatesoneinternalvariable.

53

vwn

1wn


VVinOpenGL

– Foreachvertex,divisionisdonetoobtainthenormalizeddevicecoordinates.

54

xn =xcwc

, yn =ycwc

, zn =zcwc

,

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VVinOpenGL

– Foreachvertex,thenormalizeddevicecoordinatesaretransformedtowindowcoordinates.

– Thecoordinatesareusedtopositionthetriangleonthescreen.

55

[xw , yw ]t


– Foreveryinteriorpixelofthetriangle,linearinterpolationisusedtoobtaintheinterpolatedvaluesof.

– Ateachpixel,theinterpolatedvalueisusedforz-buffering.

56

zw ,

vwn

(forallv)and 1wn

zw

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VVinOpenGL

•  Ateachpixel,andforallvaryingvariables,divisionisdoneontheinterpolatedinternalvariablestoobtainthecorrectanswer:

57v = v

wn

⎛

⎝⎜⎜

⎞

⎠⎟⎟/

1wn


VVinOpenGL

•  Thevaryingvariablesvispassedintothefragmentshader.

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Documents

CS380: Introduction to Computer Graphics Varying Variables ...vclab.kaist.ac.kr/cs380/slide14-varying_variables.pdf · 18/04/26 1 Min H. Kim (KAIST) Foundations of 3D Computer Graphics,