CS380: Introduction to Computer Graphics Track-/Arc-ball ...vclab.kaist.ac.kr/cs380/slide09-arcball.pdf · 18/04/06 1 Min H. Kim (KAIST) Foundations of 3D Computer Graphics, S. Gortler,

18/04/06

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

CS380:IntroductiontoComputerGraphicsTrack-/Arc-ball

Chapter8

MinH.KimKAISTSchoolofComputing


SUMMARYQuaternion

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Unitnormquats.==rotations•  Squarednormissumof4squares.•  Anyquaternionoftheformhasaunitform

•  Conversely,anysuchunitnormquaternioncanbeinterpreted(alongwithitsnegation)asauniquerotationmatrix.

3

ωc

⎡

⎣⎢

⎤

⎦⎥

1= ω2+ x2+ y2+z2 , c =[x , y ,z]t

cos θ2⎛

⎝⎜⎞

⎠⎟

sin θ2⎛

⎝⎜⎞

⎠⎟k

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

cos θ2⎛

⎝⎜⎞

⎠⎟

sin θ2⎛

⎝⎜⎞

⎠⎟k x

sin θ2⎛

⎝⎜⎞

⎠⎟k y

sin θ2⎛

⎝⎜⎞

⎠⎟kz

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥ k =1

ω c


Operations•  Quat*quatmultiply

•  Whereandarethedotandcrossproducton3dimensionalcoordinatevectors.

•  Correctlymodels(rotationmatrix)*(rotationmatrix)multiplications

•  Trackballexample:

4

ω1

c1

⎡

⎣⎢⎢

⎤

⎦⎥⎥

ω 2

c2

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

(ω1ω 2 − c1 ⋅ c2 )(ω1c2 +ω 2c1 + c1 × c2 )

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⋅ ×

M=Matrix4::makeXRotation(-dy)*Matrix4::makeYRotation(dx);

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Unitnormquats•  Identityrotationexample

•  Fliprotationexample

•  Unitquaternionmultiplication

5

10

⎡

⎣⎢⎢

⎤

⎦⎥⎥, −1

0

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0k

⎡

⎣⎢⎢

⎤

⎦⎥⎥, 0

− k

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0c1

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0c2

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

−c1 ⋅ c2c1 × c2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

k1 ⋅ k2k1 × k2

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥=

0k2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0−k1

⎡

⎣⎢⎢

⎤

⎦⎥⎥


Operations•  scalar*quatmultiply

– NBQuaternionsqand–qarethesamerotation!!!

•  Unitquaternionmultiplicativeinverse(conjugate)

6

cos θ2⎛

⎝⎜⎞

⎠⎟

sin θ2⎛

⎝⎜⎞

⎠⎟k

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

−1

=

cos θ2⎛

⎝⎜⎞

⎠⎟

−sin θ2⎛

⎝⎜⎞

⎠⎟k

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

, 10

⎡

⎣⎢

⎤

⎦⎥=

cos θ2⎛

⎝⎜⎞

⎠⎟

−sin θ2⎛

⎝⎜⎞

⎠⎟k

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

cos θ2⎛

⎝⎜⎞

⎠⎟

sin θ2⎛

⎝⎜⎞

⎠⎟k

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

−1

α ω

c

⎡

⎣⎢

⎤

⎦⎥= αω

αc

⎡

⎣⎢

⎤

⎦⎥, example)-1 ω

c

⎡

⎣⎢

⎤

⎦⎥= −ω

−c

⎡

⎣⎢

⎤

⎦⎥

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Rotateavectorbyaunitquaternion•  Performthefollowingtriplequaternionmultiplication:

•  Weneedinv(q)additionally,suchthatqandvarenot

orthogonal!•  Ifqandvareorthogonal,v’=qv.•  Ifwejustmultiplyqwithv,qvcannotbecomeavector(thefirst

elementwillbenon-zero).Theadditionalrightmultiplicationofinverseallowsustogetarotatedvector.

•  Butwewillwritethisincodeas:cvec=quat*cvec

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0ˆ ʹc

⎡

⎣⎢

⎤

⎦⎥=

cos θ2⎛

⎝⎜⎞

⎠⎟

sin θ2⎛

⎝⎜⎞

⎠⎟k

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

0c

⎡

⎣⎢

⎤

⎦⎥

cos θ2⎛

⎝⎜⎞

⎠⎟

sin θ2⎛

⎝⎜⎞

⎠⎟k

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

−1

v '=qvq−1v

q v '

ω1

c1

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ω2

c2

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥=

(ω1ω2− c1 ⋅ c2)(ω1c2+ω2c1+ c1× c2)

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥


LERP(LinearInterpolation)•  Aneveneasierhackistodo4DLinear

intERPolation(LERP)andrenormalization

8

p = a+α(b−a)p = a+α!vp = (1−α)a+αb

a bp

α (1-α) !v

(1−α )cos θ0

2⎛⎝⎜

⎞⎠⎟

sin θ02

⎛⎝⎜

⎞⎠⎟ k0

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

+αcos θ1

2⎛⎝⎜

⎞⎠⎟

sin θ12

⎛⎝⎜

⎞⎠⎟ k1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

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Power-basedSLERP•  SphericalLinearInterpolation

9

p = !vq = rq−1q

!v =cos θ

2⎛

⎝⎜⎞

⎠⎟

sin θ2⎛

⎝⎜⎞

⎠⎟k

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

α!v =α θ

2⎛

⎝⎜⎞

⎠⎟k =

αθ2

⎛

⎝⎜

⎞

⎠⎟k

p = !vαq = rq−1( )α q

!vα =cos αθ

2⎛

⎝⎜

⎞

⎠⎟

sin αθ2

⎛

⎝⎜

⎞

⎠⎟k

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

!v = rq−1

q r

pα

!v

p = a+α(b−a)p = a+α!vp = (1−α)a+αb

a bp

α (1-α) !v


Power-basedSLERP•  Inordertoselect“theshortinterpolation”oflessthan180degrees,

•  Whenweinterpolate,beforecallingthepoweroperator,wefirstcheckthesignofthefirstcoordinate,andconditionallynegatethequaternion.

•  Wecallthistheconditionalnegationoperator•  Quaternionsqand–qarethesamerotation!!!•  Finally,weoutput:

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cn(): −1 ω

c

⎡

⎣⎢

⎤

⎦⎥= −ω

−c

⎡

⎣⎢

⎤

⎦⎥

cncos θ1

2⎛⎝⎜

⎞⎠⎟

sin θ12

⎛⎝⎜

⎞⎠⎟ k1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

cos θ02

⎛⎝⎜

⎞⎠⎟

sin θ02

⎛⎝⎜

⎞⎠⎟ k0

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

−1⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

α

cos θ02

⎛⎝⎜

⎞⎠⎟

sin θ02

⎛⎝⎜

⎞⎠⎟ k0

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

[−ω ,−c]

[ω ,c]

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Sphere-basedSLERPing•  Inanydimensionn,atrigonometricargumentcanbeusedtoshowthatsphericallinearinterpolationbetweenanytwounitvectorsin,canbecalculatedas:

11

sin[(1−α)Ω]sin(Ω)

!v0+sin[αΩ]sin(Ω)

!v1

whereΩ= cos−1(!v0 ⋅!v1)

sin[(1−α )Ω]sin(Ω)

cos θ02

⎛⎝⎜

⎞⎠⎟

sin θ02

⎛⎝⎜

⎞⎠⎟ k0

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

+ sin[αΩ]sin(Ω)

cos θ12

⎛⎝⎜

⎞⎠⎟

sin θ12

⎛⎝⎜

⎞⎠⎟ k1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

!n


(RBT*RBT)andinv(RBT)•  Theresultisanewrigidtransformwithtranslationandrotation– Usethistocodeupthe*operator.– MindtheCvec3s(thet’s)andCvec4s(neededforq*v).

12

t1 + r1t2 r1r2

i t1 + r1t20 1

⎡

⎣⎢⎢

⎤

⎦⎥⎥

r1r2 00 1

⎡

⎣⎢⎢

⎤

⎦⎥⎥

4Dvector3Dvector

i t10 1

⎡

⎣⎢⎢

⎤

⎦⎥⎥

r1 00 1

⎡

⎣⎢⎢

⎤

⎦⎥⎥

i t20 1

⎡

⎣⎢⎢

⎤

⎦⎥⎥

r2 00 1

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

i t0 1

⎡

⎣⎢

⎤

⎦⎥

r 00 1

⎡

⎣⎢

⎤

⎦⎥

⎛

⎝⎜⎞

⎠⎟

−1

=i −r−1t0 1

⎡

⎣⎢⎢

⎤

⎦⎥⎥r−1 00 1

⎡

⎣⎢⎢

⎤

⎦⎥⎥

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Morecode•  InGLSL,youwillstilluseitsmatrixdatatype.•  TheonlyMatrix4s(thatwillsurvive)aretheprojMatrix,theMVMandtheNMVM,whichgetsenttoyourshaders.

•  Also,whenweneedtodoobjectscaling,wecannotcapturethisinanRigTform,sothiswillalsobeanMatrix4usedincreatingtheMVM.

•  Tocommunicatewiththevertexshaderusing4-by-4matrices,weneedaproceduremakeTranslation(RigTform)andMatrix4quatToMatrix(RigTform)touseT*R,whichturnsquaternionsintoa4-by-4rotationmatrix.

13


BALLS:TRACKANDARCChapter8

14

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Trackandarcballs•  Howshouldwelinkmousemotiontoobjectrotation

•  Candobetterthanourcurrentsetup•  Wantthefeelingofpushingaspherearound(trackball)

•  Wantpathinvariance(arcball)

•  Reminders:– Affinetransform:– Rigidbodytransform:

15

Aaffine = TLARBT = TR


Trackandarcballs

16

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Setup•  Wearemovinganobjectwithrespecttocube-eye

•  Theuserclicksonthescreenanddragsthemouse.WewishtointerpretthisusermotionassomerotationMthatisappliedtowithrespectto

17

at = wt (O)T (E)R

ot

at


Mentalmodel•  Imagineasphereofsomechosenradiusthatiscenteredat,theoriginof.

•  Userclicksonthescreenatscreenpixeloverthesphereintheimage– Weinterpretthisastheuserselectingsome3Dpointonthesphere.

•  Theuserthenmovesthemousetootherscreenpixeloverthesphere,– Weinterpretasasecondpointonthesphere

18

o ot

s1

p1

s2

p2 s1( !p1)

s2( !p2)

!o

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Mentalmodel•  Definetheunitdirectionvectors:normalizeandnormalizerespectively.

•  Definetheangle

•  Definetheaxis

19

v1,v2

( p1 − o) ( p2 − o)

φ = arccos(v1 ⋅v2 )

k = normalize(v1 ×

v2 ) s1( !p1)

s2( !p2)

!o


Mentalmodel•  Let’strytoformulatetherotationabouttheaxiscanberepresentedbythequaternion.– Wherearethecoordinates3-vectorsrepresentingthevectorswithrespecttotheframe.

20

k

k, v1, v2

k , v1,

v2 at

s1( !p1)

s2( !p2)

!o

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Ballrotationusingquaternions•  Whatwewant

•  Derivation

•  Therefore21

s1( !p1)

s2( !p2)

!v2 =q

!v1 , q=!v2!v1

−1

φ =

θ2 , θ =2φ.

0v2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0−v1

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

v1 ⋅ v2v1× v2

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥=

cos(φ)sin(φ)k

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

q=cos θ2

⎛

⎝⎜⎞

⎠⎟

sin θ2⎛

⎝⎜⎞

⎠⎟k

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=cos(φ)sin(φ)k

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥.

!o

suchthatqand!v1 areorthogonal,and qand

!v2 areorthogonal.


Ballrotationusingquaternions•  Rotationofdegreesabouttheaxiscanberepresentedbythequaternion.

– Wherearethecoordinates3-vectorsrepresentingthevectorswithrespecttotheframe.

22

2φ k

cos(φ)

sin(φ)k

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

v1 ⋅ v2v1 × v2

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

0v2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0−v1

⎡

⎣⎢⎢

⎤

⎦⎥⎥

k, v1, v2

k , v1,

v2 at s1( !p1)

s2( !p2)

!o

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Ballrotationusingquaternions•  Arcball:istherotationofdegreesabouttheaxis.

•  Trackball:istherotationofdegreesabouttheaxis.

•  Couldbeimplementedwithmatricesorquaternions.

•  Arcballisveryeasywithquaternions

23

M

φ kM

2φ

k

s1( !p1)

s2( !p2)

!o


s1( !p1)

s2( !p2)

s3( !p3)

Properties•  Trackballfeelsliketheuserissimplygrabbingaphysicalpointonasphereanddraggingitaround.

•  Butto,followedbytoisdifferentfrommovingdirectlyfromto–  willberotatedto,buttheresultscandifferbysome‘twist’abouttheaxis.

– Thispathdependencealsoexistsinoursimplerotationinterface.

•  Arcball:theobjectappearstospintwicefasterthanexpected.

24

s1 s2 s2 s3s1 s3

p1 p3

o− p3

!o

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ArcballisPathindependent•  Ifwecomposetwoarcballrotations,correspondingtomotionfromtofollowedbymotionfromto,weget

– Readrighttoleft,globalintheunchangingframe

•  whichisexactlywhatwewouldhavegottenhadwemoveddirectlyfromto.

25

p1 p2 p3

v2 ⋅ v3v2 × v3

⎡

⎣⎢⎢

⎤

⎦⎥⎥

v1 ⋅ v2v1 × v2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

at

0v3

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0−v2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0v2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0−v1

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

0v3

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0−v1

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

v1 ⋅ v3v1× v3

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

p1 p3

p2


Implementation•  TrackballandArcballcanbedirectlyimplementedusingeither4-by-4matricesorquaternionstorepresentthetransformationM.– Wewillusequaternions,sincewealreadyhavethem.

•  Theresultingquaterniondependsonlyonvector– Sooriginofframeisirrelevant

•  Wecanworkineyecoordinatesinsteadofcube-eye.

26

v

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Gettingeyecoordinates•  Oneslightlytrickypartiscomputingthecoordinatesofthepointonthespherecorrespondingtoaselectedpixel–  Thisisgeometricraytracing(thisisessentiallyray-tracing,whichwewillcoverlater)

•  Hack:workin“windowcoordinates”–  X-axisisthehorizontalaxisofthescreen,they-axisistheverticalaxisofthescreen,andthez-axisiscomingoutofthescreen.

–  Thinkofthesphere’scenterassimplysittingonthescreen.

27

s1( !p1)

s2( !p2)

!o


Gettingeyecoordinates•  Giventhe(x,y)windowcoordinatesofclick,thezcoordinateonthespherecanbesolvedusing

–  arethewindowcoordinatesofthecenterofthesphere

–  ristheradiusofthespheremeasuredinpixels

–  ifoutsideofthesphere,thenclamptoitsboundary

– Allweneedisnormalized,sojustnormalizedsuchvectors

28

(x − cx )2 + (y − cy )

2 + (z − 0)2 = r2

[cx ,cy ,0]t

v

z = r2 − (x − cx )2 − (y − cy )

2

s1( !p1)

s2( !p2)

!o

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Calculation•  Needthecenterofthesphere•  Sowegiveyoucodethattransformseyecoordinatestoscreencoordinates

•  Wedrawtheballusingobjectcoordinates,soweneedtocalculateitssizeineye/objectcoordinates

•  Soweprovideyouwith

•  Intheballdrawer,yourightmultiplyascalematrixtotheMVM

29

Cvec2getScreenSpaceCoord(constCvec3&p,constMatrix4&projection,doublefrustNear,doublefrustFovY,intscreenWidth,intscreenHeight)

doublegetScreenToEyeScale(doublez,doublefrustFovY,intscreenHeight)


Translation•  NBthedistanceafterperspectiveprojectionshouldbedifferentfromthedistanceintheobjectcoordinates.

•  Intranslation,weinterpretmousedisplacement(measuredinpixels)toobjectdisplacement.

•  MayaswellusethesamescreenToEyeScalefactorsotheobjectmoveswiththemouse

•  Oncetheobjectismoved,orwechangetheeyeweneedtorecalculatethescale– Waitforclickup.

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CS380: Introduction to Computer Graphics Track-/Arc-ball ...vclab.kaist.ac.kr/cs380/slide09-arcball.pdf · 18/04/06 1 Min H. Kim (KAIST) Foundations of 3D Computer Graphics, S. Gortler,

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