CMSC427TransformationsI

Credit:slides9+fromProf.Zwicker

• Typesoftransformations• Specific:translation,rotation,scaling,shearing• Classes:rigid,affine,projective

• Representingtransformations• Unifyingrepresentationwithhomogeneouscoordinates• Transformationsrepresentedasmatrices

• Composingtransformations• Sequencingmatrices• SequencingusingOpenGLstackmodel

• Transformationexamples• Rotatingorscalingaboutapoint• Rotatingtoanewcoordinateframe

• Applications• Modelingtransformations(NOW)• Viewingtransformations(LATER)

Transformations: outline

• Createinstanceofobjectinobjectcoordinatespace• Createcircleatorigin

• Transformobjecttoworldcoordinatespace• Scaleby1.5• Movedownby2unit

• Dosoforotherobjects• Tworects makehat• Threecirclesmakebody• Twolinesmakearms

• Objectcoordinatespace

• Worldcoordinatespace

Modeling with transformations

• Rigid• Translate,rotate,uniformscale• Nodistortiontoobject

• Affine• Translate,rotate,scale(non-uniform),shear,reflect• Limiteddistortions• Preserveparallellines

Classes of transformations

• Affine• Preservesparallellines

• Projective• Foreshortens• Linesconverge• Forviewing/rendering

Classes of transformations

• Affine• Reshape,sizeobject

• Rigid• Place,moveobject

• Projective• Viewobject• Later…

• Non-linear,arbitrary• Twists,pinches,pulls• Notinthisunit

Classes of transformations: summary

• ScaleapointpbysandtranslatebyT• Vectormultiplicationandaddition• Repeatandweget

• Getsunwieldy• Instead– unifynotationwithhomogeneouscoordinatesandmatrices

First try: scale and rotate vertices in vector notation

𝑞 = 𝑠 ∗ 𝑝 + 𝑇

T

p

q

𝑞 = 𝑠( 𝑠 ∗ 𝑝 + 𝑇 + 𝑇(

𝑞 = 2 ∗ 2,3 +< 2,2 >𝑞 = (6,8)

Matrix practice

𝑀 = 2 00 2

𝑅 = 1 10 3

𝑃 = 23

𝑀𝑅 = 2 00 2 1 1

0 3 =

𝑅𝑀 = 1 10 3

2 00 2 =

𝑀𝑃 = 2 00 2 23 =

Matrix practice

𝑀 = 2 01 2

𝑅 = 1 10 3

𝑃 = 23

𝑀𝑅 = 2 01 2 1 1

0 3 = 2 71 6

𝑅𝑀 = 1 10 3

2 01 2 = 3 2

3 6

𝑀𝑃 = 2 01 2 23 = 4

8

Matrix transpose and column vectors

𝑃 = 23 = 2 3 9

𝑅9 = 1 10 3

9=

𝐻9 = 2 1 34 1 5

9=

Matrix transpose and column vectors

𝑃 = 23 = 2 3 9

𝑅9 = 1 10 3

9= 1 0

1 3

𝐻9 = 2 1 34 1 5

9=

2 41 13 5

Matrices

Abstractpointofview• Mathematicalobjectswithsetofoperations• Addition,subtraction,multiplication,multiplicativeinverse,etc.

• Similartointegers,realnumbers,etc.But• Propertiesofoperationsaredifferent• E.g.,multiplicationisnotcommutative

• Representdifferentintuitiveconcepts• Scalarnumbersrepresentdistances• Matrices canrepresentcoordinatesystems,rigidmotions,in3Dandhigherdimensions,etc.

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Matrices

Practicalpointofview• Rectangulararrayofnumbers

• Squarematrixif• Ingraphicsoften

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Matrix addition

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Multiplication with scalar

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Matrix multiplication

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Matrix multiplication

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Matrix multiplication

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Matrix multiplication

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Matrix multiplication

Specialcase:matrix-vectormultiplication

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Linearity

• Distributivelawholds

i.e.,matrixmultiplicationislinearhttp://en.wikipedia.org/wiki/Linear_map

• Butmultiplicationisnotcommutative,

ingeneral

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Identity matrix

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Matrix inverse

DefinitionIfasquarematrixisnon-singular,thereexistsauniqueinverse suchthat

• Note

• Computation• Gaussianelimination,Cramer’srule(OctaveOnline)• Reviewinyourlinearalgebrabook,orquicksummary

http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html

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Java vs. OpenGL matrices

• OpenGL(underlying3DgraphicsAPIusedintheJavacode,morelater)http://en.wikipedia.org/wiki/OpenGL

• Matrixelementsstoredinarrayoffloatsfloat M[16];• “Columnmajor”ordering

• Javabasecode• “Rowmajor”indexing• ConversionfromJavatoOpenGLconventionhiddensomewhereinbasecode!

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Today

Transformations&matrices• Introduction• Matrices• Homogeneouscoordinates• Affinetransformations• Concatenatingtransformations• Changeofcoordinates• Commoncoordinatesystems

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Vectors & coordinate systems

• Vectorsdefinedbyorientation,length• Describeusingthreebasisvectors

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Points in 3D

• Howdowerepresent3Dpoints?• Arethreebasisvectorsenoughtodefinethelocationofapoint?

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Points in 3D

• Describeusingthreebasisvectorsand referencepoint,origin

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Vectors vs. points

• Vectors

• Points

• Representationofvectorsandpointsusing4th coordinateiscalledhomogeneouscoordinates

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Homogeneous coordinates

• Representanaffinespacehttp://en.wikipedia.org/wiki/Affine_space

• Intuitivedefinition• Affinespacesconsistofavectorspaceandasetofpoints• Thereisasubtraction operationthattakestwopointsandreturnsavector• AxiomI:foranypointa andvectorv,thereexistspoint

b,suchthat(b-a) = v• AxiomII:foranypointsa, b, c wehave

(b-a)+(c-b) = c-a

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Affine space

Vectorspace,http://en.wikipedia.org/wiki/Vector_space

• [xyz]coordinates• representsvectors

Affinespacehttp://en.wikipedia.org/wiki/Affine_space

• [xyz1],[xyz0]homogeneouscoordinates• distinguishespointsandvectors

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Homogeneous coordinates

• Subtractionoftwopointsyieldsavector

• Usinghomogeneouscoordinates

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Today

Transformations&matrices• Introduction• Matrices• Homogeneouscoordinates• Affinetransformations• Concatenatingtransformations• Changeofcoordinates• Commoncoordinatesystems

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Affine transformations

• Transformation,ormapping:functionthatmapseach3Dpointtoanew3Dpoint„f: R3 -> R3“• Affinetransformations:classoftransformationstoposition3Dobjectsinspace• Affinetransformationsinclude• Rigidtransformations

• Rotation• Translation

• Non-rigidtransformations• Scaling• Shearing

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Affine transformations

• Definition:mappings that preserve colinearity andratios of distanceshttp://en.wikipedia.org/wiki/Affine_transformation

• Straightlines are preserved• Parallellines are preserved

• Lineartransformations +translation• Nice:Alldesired transformations (translation,rotation)implemented using homogeneouscoordinates and matrix-vector multiplication

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Translation

Point Vector

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Matrix formulation

Point Vector

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Matrix formulation

• Inverse translation

• Verify that

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Note

• Whathappenswhenyoutranslateavector?

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Rotation

First:rotatingavectorin2D• Convention:positiveanglerotatescounterclockwise• Expressusingrotationmatrix

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Rotating a vector in 2D

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Rotating a vector in 2D

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Rotating a vector in 2D

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Rotation in 3D

Rotationaroundz-axis• z-coordinatedoesnotchange

• Whatisthematrixfor?

v0= R z(µ)v

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Other coordinate axes

• Samematrixtorotatepointsandvectors• Pointsarerotatedaroundorigin

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Rotation in 3D

• Concatenaterotationsaroundx,y,z axestoobtainrotationaroundarbitraryaxesthroughorigin

• arecalledEulerangleshttp://en.wikipedia.org/wiki/Euler_angles

• Disadvantage:resultdependsonorder!

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Gimbalhttps://en.wikipedia.org/wiki/Gimbal

Rotation around arbitrary axis

• Still:origindoesnotchange• Counterclockwiserotation• Angle,unitaxis•

• Intuitivederivationseehttp://mathworld.wolfram.com/RotationFormula.html

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Summary

• Differentwaystodescriberotationsmathematically• Sequenceofrotationsaroundthreeaxes(Eulerangles)• Rotationaroundarbitraryangles(axis-anglerepresentation)• Otheroptionsexist(quaternions,etc.)

• Rotationspreserve• Angles• Lengths• Handednessofcoordinatesystem

• Rigidtransforms• Rotationsandtranslations

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Rotation matrices

• Orthonormal• Rows,columnsareunitlengthandorthogonal

• Inverseofrotationmatrix?

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Rotation matrices

• Orthonormal• Rows,columnsareunitlengthandorthogonal

• Inverseofrotationmatrix?• Itstranspose

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Rotations

• Givenarotationmatrix• Howdoweobtain ?

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Rotations

• Givenarotationmatrix• Howdoweobtain ?

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Rotations

• Givenarotationmatrix• Howdoweobtain ?

• Howdoweobtain …?

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Rotations

• Givenarotationmatrix• Howdoweobtain ?

• Howdoweobtain …?

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Scaling

• Origin does not change

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Scaling

• Inversescaling?

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Scaling

• Inversescaling?

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Shear

• Pureshearifonlyoneparameterisnon-zero• Cartoon-likeeffects

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Summary affine transformations

• Lineartransformations(rotation,scale,shear,reflection)+translation

Vectorspace,http://en.wikipedia.org/wiki/Vector_space

• vectorsas[xyz]coordinates• representsvectors• lineartransformations

Affinespacehttp://en.wikipedia.org/wiki/Affine_space

• pointsandvectorsas[xyz1],[xyz0]homogeneouscoordinates• distinguishespointsandvectors• lineartranforms andtranslation

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Summary affine transformations

• Implementedusing4x4matrices,homogeneouscoordinates• Lastrowof4x4matrixisalways[0 0 0 1]

• Anysuchmatrixrepresentsanaffinetransformationin3D• Factorizationintoscale,shear,rotation,etc.isalwayspossible,butnon-trivial• Polardecomposition

http://en.wikipedia.org/wiki/Polar_decomposition

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Today

Transformations&matrices• Introduction• Matrices• Homogeneouscoordinates• Affinetransformations• Concatenatingtransformations• Changeofcoordinates• Commoncoordinatesystems

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Concatenating transformations

• Build“chains”oftransformations

• Applyfollowedbyfollowedby

• Overalltransformationisanaffinetransformation

• Multiplicationontheleft

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Concatenating transformations

• Resultdependsonorderbecausematrixmultiplicationnotcommutative• Thoughtexperiment• Translationfollowedbyrotationvs.rotationfollowedbytranslation

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Rotating with pivot

Rotation around origin

Rotation with pivot

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1. Translation 2. Rotation 3. Translation

Rotating with pivot

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Rotating with pivot

1. Translation 2. Rotation 3. Translation

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Concatenating transformations

• Arbitrarysequenceoftransformations

• Note:associativity

SoeitherisvalidT=M3.multiply(M2); Mtotal=T.multiply(M1)orT=M2.multiply(M1); Mtotal=M3.multiply(T)

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• Transformationsareusedformodeling• Classesoftransformation:rigidandaffine• Whyweusehomo.coordinatesandmatrices• Howtodomatrixmults,inversion,transpose• Homogenouscoordinates,vectorsvs.points• Propertiesofaffinetransformations• Transforms:translation,scale,rotation,shear• Onlystartingwith3Drotations– don’tbeconcerned

• Orderoftransformations• Theydon’tcommute,butareassociative• Translatetooriginforscaling,rotation

Transformation: summary