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Transformations in Computer Graphics

Transformations are used to move and manipulate 3D objects in computer graphics. The key types of 2D transformations are: 1. Translation moves objects by adding offsets to the x and y coordinates. 2. Scaling enlarges or shrinks objects by multiplying the x and y coordinates by scale factors. 3. Rotation spins objects around the origin by applying trigonometric functions to the coordinates. Transformations can be represented using matrices, allowing multiple transformations to be combined through matrix multiplication. Common 2D transformations like translation, scaling, rotation, and shearing can be encoded concisely using 2×2 transformation matrices.

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Computer Graphics 1 Topic: Transformations Name: Jatender Khatri
2 Transformations. What is a transformation? • P=T(P) What does it do? Transform the coordinates / normal vectors of objects Why use them? • Modelling -Moving the objects to the desired location in the environment -Multiple instances of a prototype shape -Kinematics of linkages/skeletons – character animation • Viewing – Virtual camera: parallel and perspective projections
3 Types of Transformations – Geometric Transformations • Translation • Rotation • scaling • Linear (preserves parallel lines) • Non-uniform scales, shears or skews – Projection (preserves lines) • Perspective projection • Parallel projection – Non-linear (lines become curves) • Twists, bends, warps, morphs,
12/12/2019 Lecture 1 4 Geometric Transformation – Once the models are prepared, we need to place them in the environment – Objects are defined in their own local coordinate system – We need to translate, rotate and scale them to put them into the world coordinate system
5 2D Translations. TPP d d T y x P y x P dyydxx yxP yxPP yx                         Now ,, torscolumn vectheDefine axis.ytoparalleldaxis,xtoparallelddistancea),(Pointtotranslate ),,(asdefinedPoint y x yx P P’
6 2D Scaling from the origin.                               y x . 0 0 y x or Now 0 0 matrixtheDefine .,. axis.ythealongsand axis,xthealongsfactoraby),(Pointto(stretch)scaleaPerform ),,(asdefinedPoint y x y x y x yx s s PSP s s S ysyxsx yxP yxPP P P’
7 2D Rotation about the origin. y x r r P’(x’,y’) P(x,y) 
8 2D Rotation about the origin. y x r r P’(x’,y’) P(x,y)   y   sin. cos. ry rx   x
9 2D Rotation about the origin. y x r r P’(x’,y’) P(x,y)   y   sin. cos. ry rx   x   co.sin.sin.cos.)sin(. sin.sin.cos.cos.)cos(. rrry rrrx  
10 2D Rotation about the origin.   sin. cos. ry rx     cos.sin.sin.cos.)sin(. sin.sin.cos.cos.)cos(. rrry rrrx   Substituting for r : Gives us :   cos.sin. sin.cos. yxy yxx  
11 2D Rotation about the origin.   cos.sin. sin.cos. yxy yxx   Rewriting in matrix form gives us :                      y x y x . cossin sincos   PRPR         , cossin sincos matrixtheDefine  
Transformations. • Translation. – P=T + P • Scale – P=S  P • Rotation – P=R  P • We would like all transformations to be multiplications so we can concatenate them •  express points in homogenous coordinates. 12                      y x y x . cossin sincos  
Example: 2D Geometric Transformation Modeling Coordinates World Coordinates
Example: 2D Scaling Modeling Coordinates World Coordinates Scale(0.3, 0.3)
Example: 2D Rotation Modeling Coordinates Scale(0.3, 0.3) Rotate(-90) World Coordinates
Example: 2D Translation Modeling Coordinates Scale(0.3, 0.3) Rotate(-90) Translate(5, 3) World Coordinates
Example: 2D Geometric Transformation Modeling Coordinates World Coordinates Again?
Example: 2D Geometric Transformation Modeling Coordinates World Coordinates Scale Translate Scale Rotate Translate
Basic 2D Transformations • Translation – – • Scale – – • Rotation – – • Shear – – txxx  tyyy  sxxx  syyy  sinθy-cosθxx  cosθysinθyy  yhxxx  xhyyy 
Basic 2D Transformations • Translation – – • Scale – – • Rotation – – • Shear – – txxx  tyyy  sxxx  syyy  sinθy-cosθxx  cosθysinθyy  yhxxx  xhyyy  Transformations can be combined (with simple algebra)
Basic 2D Transformations • Translation – – • Scale – – • Rotation – – • Shear – – txxx  tyyy  sxxx  syyy  sinθy-cosθxx  cosθysinθyy  yhxxx  xhyyy  syy sxxx   y
Basic 2D Transformations • Translation – – • Scale – – • Rotation – – • Shear – – )cossy)(ysinsx)((xy )sinsy)(ycossx)((xx     txxx  tyyy  sxxx  syyy  sinθy-cosθxx  cosθysinθyy  yhxxx  xhyyy 
Basic 2D Transformations • Translation – – • Scale – – • Rotation – – • Shear – – ty)cossy)(ysinsx)((xy tx)sinsy)(ycossx)((xx     txxx  tyyy  sxxx  syyy  sinθy-cosθxx  cosθysinθyy  yhxxx  xhyyy 
Basic 2D Transformations • Translation – – • Scale – – • Rotation – – • Shear – – txxx  tyyy  sxxx  syyy  sinθy-cosθxx  cosθysinθyy  yhxxx  xhyyy  ty)cossy)(ysinsx)((xy tx)sinsy)(ycossx)((xx    
Matrix Representation • Represent a 2D Transformation by a Matrix • Apply the Transformation to a Point                     y x dc ba y x dycxy byaxx         dc ba Transformation Matrix Point
Matrix Representation • Transformations can be combined by matrix multiplication                                 y x lk ji hg fe dc ba y x Matrices are a convenient and efficient way to represent a sequence of transformations Transformation Matrix
2×2 Matrices • What types of transformations can be represented with a 2×2 matrix? 2D Identity 2D Scaling yy xx   ysyy xsxx                       y x y x 10 01                     y x sy sx y x 0 0
2×2 Matrices • What types of transformations can be represented with a 2×2 matrix? 2D Rotation 2D Shearing                      y x y x   cossin sincos                     y x shy shx y x 1 1 yxy yxx     cossin sincos yxshyy yshxxx  
2×2 Matrices • What types of transformations can be represented with a 2×2 matrix? 2D Mirror over Y axis 2D Mirror over (0,0) yy xx   yy xx                       y x y x 10 01                       y x y x 10 01
2×2 Matrices • What types of transformations can be represented with a 2×2 matrix? 2D Translation txxx  tyyy  NO!! Only linear 2D transformations can be Represented with 2x2 matrix
2D Translation • 2D translation can be represented by a 3×3 matrix – Point represented with homogeneous coordinates txxx  tyyy                                   1100 10 01 1 y x ty tx y x
Homogeneous coordinates • Add an extra coordinate, W, to a point. – P(x,y,W). • Two sets of homogeneous coordinates represent the same point if they are a multiple of each other. – (2,5,3) and (4,10,6) represent the same point. • If W 0 , divide by it to get Cartesian coordinates of point (x/W,y/W,1). • If W=0, point is said to be at infinity. 32
Translations in homogenised coordinates • Transformation matrices for 2D translation are now 3x3. 33                                  1 . 100 10 01 1 y x d d y x y x 11   y x dyy dxx
Concatenation. • When we perform 2 translations on the same point 34 ),(),(),( :expectweSo ),(),(),( ),( ),( 21212211 21212211 22 11 yyxxyxyx yyxxyxyx yx yx ddddTddTddT PddddTPddTddTP PddTP PddTP    
Concatenation. 35 ? 100 10 01 . 100 10 01 :is),(),(productmatrixThe 2 2 1 1 2211                       y x y x yxyx d d d d ddTddT Matrix product is variously referred to as compounding, concatenation, or composition
Concatenation. 36 Matrix product is variously referred to as compounding, concatenation, or composition.                                   100 10 01 100 10 01 . 100 10 01 :is),(),(productmatrixThe 21 21 2 2 1 1 2211 yy xx y x y x yxyx dd dd d d d d ddTddT
Properties of translations. 37 ),(),(T4. ),(),(),(),(3. ),(),(),(2. )0,0(1. 1- yxyx yxyxyxyx yyxxyxyx ssTss ssTttTttTssT tstsTttTssT IT     Note : 3. translation matrices are commutative.
Homogeneous form of scale. 38        y x yx s s ssS 0 0 ),( Recall the (x,y) form of Scale :            100 00 00 ),( y x yx s s ssS In homogeneous coordinates :
Concatenation of scales. 39 !multeasy to-matrixin theelementsdiagonalOnly 100 0ss0 00ss 100 0s0 00s . 100 0s0 00s :is),(),(productmatrixThe y2y1 x2x1 y2 x2 y1 x1 2211                                   yxyx ssSssS
Homogeneous form of rotation. 40                                   1 . 100 0cossin 0sincos 1 y x y x   )()( :i.e,orthogonalarematricesRotation ).()( matrices,rotationFor 1 1   T RR RR    
Orthogonality of rotation matrices. 41                        100 0cossin 0sincos )(, 100 0cossin 0sincos )(      T RR                         100 0cossin 0sincos 100 0cossin 0sincos )(     R
Other properties of rotation. 42 carefulmorebetoneed,rotations3DFor sametheis rotationofaxisthebecauseonlyisBut this )()()()( and )()()( )0(   RRRR RRR IR   
How are transforms combined? 02/10/09 Lecture 4 43 (0,0) (1,1) (2,2) (0,0) (5,3) (3,1) Scale(2,2) Translate(3,1) TS = 2 0 0 2 0 0 1 0 0 1 3 1 2 0 0 2 3 1= Scale then Translate Use matrix multiplication: p' = T ( S p ) = TS p Caution: matrix multiplication is NOT commutative! 0 0 1 0 0 1 0 0 1
Non-commutative Composition 02/10/09 Lecture 4 44 Scale then Translate: p' = T ( S p ) = TS p Translate then Scale: p' = S ( T p ) = ST p (0,0) (1,1) (4,2) (3,1) (8,4) (6,2) (0,0) (1,1) (2,2) (0,0) (5,3) (3,1) Scale(2,2) Translate(3,1) Translate(3,1) Scale(2,2)
Non-commutative Composition 02/10/09 Lecture 4 45 TS = 2 0 0 0 2 0 0 0 1 1 0 0 0 1 0 3 1 1 ST = 2 0 0 2 0 0 1 0 0 1 3 1 Scale then Translate: p' = T ( S p ) = TS p 2 0 0 0 2 0 3 1 1 2 0 0 2 6 2 = = Translate then Scale: p' = S ( T p ) = ST p 0 0 1 0 0 1 0 0 1
How are transforms combined? 02/10/09 Lecture 4 46 (0,0) (1,1) (0,sqrt(2)) (0,0) (3,sqrt(2)) (3,0) Rotate 45 deg Translate(3,0) Rotate then Translate Caution: matrix multiplication is NOT commutative! Translate then Rotate (0,0) (1,1) (0,0) (3,0) Rotate 45 deg Translate(3,0) (3/sqrt(2),3/sqrt(2))
Non-commutative Composition 02/10/09 Lecture 4 47 TR = 0 1 0 -1 0 0 0 0 1 1 0 0 0 1 0 3 1 1 RT = 1 0 0 1 3 1 Rotate then Translate: p' = T ( R p ) = TR p 0 0 0 -1 0 0 3 1 1 0 1 0 -1 0 0 -1 3 1 = = Translate then Rotate: p' = R ( T p ) = RT p 0 0 1 0 1 0 -1 0 0 0 0 1 0 1 0 -1 0 0 0 0 1 0 1 0 -1 0 0 0 0 1 0 1 0 -1 0 0 0 0 1 0 1 0 -1 0 0 0 0 1
Types of transformations. • Rotation and translation – Angles and distances are preserved – Unit cube is always unit cube – Rigid-Body transformations. • Rotation, translation and scale. – Angles & distances not preserved. – But parallel lines are. 48
3D Transformations. • Use homogeneous coordinates, just as in 2D case. • Transformations are now 4x4 matrices. • We will use a right-handed (world) coordinate system - ( z out of page ). Lecture 4 49
Translation in 3D. 50              1000 100 010 001 ),,( z y x zyx d d d dddT Simple extension to the 3D case:
Scale in 3D. 51              1000 000 000 000 ),,( z y x zyx s s s sssS Simple extension to the 3D case:
Rotation in 3D • Need to specify which axis the rotation is about. • z-axis rotation is the same as the 2D case. 52               1000 0100 00cossin 00sincos )(   zR
Rotating About the x-axis Rx() 53                                                11000 0θcosθsin0 0θsinθcos0 0001 1 z y x z y x
Rotating About the y-axis Ry() 54                                                11000 0θcos0θsin 0010 0θsin0θcos 1 z y x z y x
Rotation About the z-axis Rz() 55                                               11000 0100 00θcosθsin 00θsinθcos 1 z y x z y x
Rotation • Not commutative if the axis of rotation are not parallel )()()()(  xyyx RRRR 
Calculating the world coordinates of all vertices • For each object, there is a local-to-global transformation matrix • So we apply the transformations to all the vertices of each object • We now know the world coordinates of all the points in the scene 57
Normal Vectors 58 • We also need to know the direction of the normal vectors in the world coordinate system • This is going to be used at the shading operation • We only want to rotate the normal vector • Do not want to translate it
Normal Vectors - (2) 59 • We need to set elements of the translation part to zero                          1000 0 0 0 1000 111111 111111 111111 111111 111111 111111 rrr rrr rrr trrr trrr trrr z y x
Summary. • Transformations: translation, rotation and scaling • Using homogeneous transformation, 2D (3D) transformations can be represented by multiplication of a 3x3 (4x4) matrix. • Multiplication from left-to-right can be considered as the transformation of the coordinate system 60
That’s All! Thank You For Listening…

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Transformations in Computer Graphics

  • 1.
  • 2.
    2 Transformations. What is atransformation? • P=T(P) What does it do? Transform the coordinates / normal vectors of objects Why use them? • Modelling -Moving the objects to the desired location in the environment -Multiple instances of a prototype shape -Kinematics of linkages/skeletons – character animation • Viewing – Virtual camera: parallel and perspective projections
  • 3.
    3 Types of Transformations –Geometric Transformations • Translation • Rotation • scaling • Linear (preserves parallel lines) • Non-uniform scales, shears or skews – Projection (preserves lines) • Perspective projection • Parallel projection – Non-linear (lines become curves) • Twists, bends, warps, morphs,
  • 4.
    12/12/2019 Lecture 14 Geometric Transformation – Once the models are prepared, we need to place them in the environment – Objects are defined in their own local coordinate system – We need to translate, rotate and scale them to put them into the world coordinate system
  • 5.
  • 6.
    6 2D Scaling fromthe origin.                               y x . 0 0 y x or Now 0 0 matrixtheDefine .,. axis.ythealongsand axis,xthealongsfactoraby),(Pointto(stretch)scaleaPerform ),,(asdefinedPoint y x y x y x yx s s PSP s s S ysyxsx yxP yxPP P P’
  • 7.
    7 2D Rotation aboutthe origin. y x r r P’(x’,y’) P(x,y) 
  • 8.
    8 2D Rotation aboutthe origin. y x r r P’(x’,y’) P(x,y)   y   sin. cos. ry rx   x
  • 9.
    9 2D Rotation aboutthe origin. y x r r P’(x’,y’) P(x,y)   y   sin. cos. ry rx   x   co.sin.sin.cos.)sin(. sin.sin.cos.cos.)cos(. rrry rrrx  
  • 10.
    10 2D Rotation aboutthe origin.   sin. cos. ry rx     cos.sin.sin.cos.)sin(. sin.sin.cos.cos.)cos(. rrry rrrx   Substituting for r : Gives us :   cos.sin. sin.cos. yxy yxx  
  • 11.
    11 2D Rotation aboutthe origin.   cos.sin. sin.cos. yxy yxx   Rewriting in matrix form gives us :                      y x y x . cossin sincos   PRPR         , cossin sincos matrixtheDefine  
  • 12.
    Transformations. • Translation. – P=T+ P • Scale – P=S  P • Rotation – P=R  P • We would like all transformations to be multiplications so we can concatenate them •  express points in homogenous coordinates. 12                      y x y x . cossin sincos  
  • 13.
  • 14.
  • 15.
    Example: 2D Rotation Modeling Coordinates Scale(0.3,0.3) Rotate(-90) World Coordinates
  • 16.
    Example: 2D Translation Modeling Coordinates Scale(0.3,0.3) Rotate(-90) Translate(5, 3) World Coordinates
  • 17.
  • 18.
    Example: 2D Geometric Transformation Modeling Coordinates WorldCoordinates Scale Translate Scale Rotate Translate
  • 19.
    Basic 2D Transformations •Translation – – • Scale – – • Rotation – – • Shear – – txxx  tyyy  sxxx  syyy  sinθy-cosθxx  cosθysinθyy  yhxxx  xhyyy 
  • 20.
    Basic 2D Transformations •Translation – – • Scale – – • Rotation – – • Shear – – txxx  tyyy  sxxx  syyy  sinθy-cosθxx  cosθysinθyy  yhxxx  xhyyy  Transformations can be combined (with simple algebra)
  • 21.
    Basic 2D Transformations •Translation – – • Scale – – • Rotation – – • Shear – – txxx  tyyy  sxxx  syyy  sinθy-cosθxx  cosθysinθyy  yhxxx  xhyyy  syy sxxx   y
  • 22.
    Basic 2D Transformations •Translation – – • Scale – – • Rotation – – • Shear – – )cossy)(ysinsx)((xy )sinsy)(ycossx)((xx     txxx  tyyy  sxxx  syyy  sinθy-cosθxx  cosθysinθyy  yhxxx  xhyyy 
  • 23.
    Basic 2D Transformations •Translation – – • Scale – – • Rotation – – • Shear – – ty)cossy)(ysinsx)((xy tx)sinsy)(ycossx)((xx     txxx  tyyy  sxxx  syyy  sinθy-cosθxx  cosθysinθyy  yhxxx  xhyyy 
  • 24.
    Basic 2D Transformations •Translation – – • Scale – – • Rotation – – • Shear – – txxx  tyyy  sxxx  syyy  sinθy-cosθxx  cosθysinθyy  yhxxx  xhyyy  ty)cossy)(ysinsx)((xy tx)sinsy)(ycossx)((xx    
  • 25.
    Matrix Representation • Representa 2D Transformation by a Matrix • Apply the Transformation to a Point                     y x dc ba y x dycxy byaxx         dc ba Transformation Matrix Point
  • 26.
    Matrix Representation • Transformationscan be combined by matrix multiplication                                 y x lk ji hg fe dc ba y x Matrices are a convenient and efficient way to represent a sequence of transformations Transformation Matrix
  • 27.
    2×2 Matrices • Whattypes of transformations can be represented with a 2×2 matrix? 2D Identity 2D Scaling yy xx   ysyy xsxx                       y x y x 10 01                     y x sy sx y x 0 0
  • 28.
    2×2 Matrices • Whattypes of transformations can be represented with a 2×2 matrix? 2D Rotation 2D Shearing                      y x y x   cossin sincos                     y x shy shx y x 1 1 yxy yxx     cossin sincos yxshyy yshxxx  
  • 29.
    2×2 Matrices • Whattypes of transformations can be represented with a 2×2 matrix? 2D Mirror over Y axis 2D Mirror over (0,0) yy xx   yy xx                       y x y x 10 01                       y x y x 10 01
  • 30.
    2×2 Matrices • Whattypes of transformations can be represented with a 2×2 matrix? 2D Translation txxx  tyyy  NO!! Only linear 2D transformations can be Represented with 2x2 matrix
  • 31.
    2D Translation • 2Dtranslation can be represented by a 3×3 matrix – Point represented with homogeneous coordinates txxx  tyyy                                   1100 10 01 1 y x ty tx y x
  • 32.
    Homogeneous coordinates • Addan extra coordinate, W, to a point. – P(x,y,W). • Two sets of homogeneous coordinates represent the same point if they are a multiple of each other. – (2,5,3) and (4,10,6) represent the same point. • If W 0 , divide by it to get Cartesian coordinates of point (x/W,y/W,1). • If W=0, point is said to be at infinity. 32
  • 33.
    Translations in homogenised coordinates •Transformation matrices for 2D translation are now 3x3. 33                                  1 . 100 10 01 1 y x d d y x y x 11   y x dyy dxx
  • 34.
    Concatenation. • When weperform 2 translations on the same point 34 ),(),(),( :expectweSo ),(),(),( ),( ),( 21212211 21212211 22 11 yyxxyxyx yyxxyxyx yx yx ddddTddTddT PddddTPddTddTP PddTP PddTP    
  • 35.
  • 36.
    Concatenation. 36 Matrix product isvariously referred to as compounding, concatenation, or composition.                                   100 10 01 100 10 01 . 100 10 01 :is),(),(productmatrixThe 21 21 2 2 1 1 2211 yy xx y x y x yxyx dd dd d d d d ddTddT
  • 37.
  • 38.
    Homogeneous form ofscale. 38        y x yx s s ssS 0 0 ),( Recall the (x,y) form of Scale :            100 00 00 ),( y x yx s s ssS In homogeneous coordinates :
  • 39.
    Concatenation of scales. 39 !multeasyto-matrixin theelementsdiagonalOnly 100 0ss0 00ss 100 0s0 00s . 100 0s0 00s :is),(),(productmatrixThe y2y1 x2x1 y2 x2 y1 x1 2211                                   yxyx ssSssS
  • 40.
    Homogeneous form ofrotation. 40                                   1 . 100 0cossin 0sincos 1 y x y x   )()( :i.e,orthogonalarematricesRotation ).()( matrices,rotationFor 1 1   T RR RR    
  • 41.
    Orthogonality of rotation matrices. 41                       100 0cossin 0sincos )(, 100 0cossin 0sincos )(      T RR                         100 0cossin 0sincos 100 0cossin 0sincos )(     R
  • 42.
    Other properties ofrotation. 42 carefulmorebetoneed,rotations3DFor sametheis rotationofaxisthebecauseonlyisBut this )()()()( and )()()( )0(   RRRR RRR IR   
  • 43.
    How are transformscombined? 02/10/09 Lecture 4 43 (0,0) (1,1) (2,2) (0,0) (5,3) (3,1) Scale(2,2) Translate(3,1) TS = 2 0 0 2 0 0 1 0 0 1 3 1 2 0 0 2 3 1= Scale then Translate Use matrix multiplication: p' = T ( S p ) = TS p Caution: matrix multiplication is NOT commutative! 0 0 1 0 0 1 0 0 1
  • 44.
    Non-commutative Composition 02/10/09 Lecture4 44 Scale then Translate: p' = T ( S p ) = TS p Translate then Scale: p' = S ( T p ) = ST p (0,0) (1,1) (4,2) (3,1) (8,4) (6,2) (0,0) (1,1) (2,2) (0,0) (5,3) (3,1) Scale(2,2) Translate(3,1) Translate(3,1) Scale(2,2)
  • 45.
    Non-commutative Composition 02/10/09 Lecture4 45 TS = 2 0 0 0 2 0 0 0 1 1 0 0 0 1 0 3 1 1 ST = 2 0 0 2 0 0 1 0 0 1 3 1 Scale then Translate: p' = T ( S p ) = TS p 2 0 0 0 2 0 3 1 1 2 0 0 2 6 2 = = Translate then Scale: p' = S ( T p ) = ST p 0 0 1 0 0 1 0 0 1
  • 46.
    How are transformscombined? 02/10/09 Lecture 4 46 (0,0) (1,1) (0,sqrt(2)) (0,0) (3,sqrt(2)) (3,0) Rotate 45 deg Translate(3,0) Rotate then Translate Caution: matrix multiplication is NOT commutative! Translate then Rotate (0,0) (1,1) (0,0) (3,0) Rotate 45 deg Translate(3,0) (3/sqrt(2),3/sqrt(2))
  • 47.
    Non-commutative Composition 02/10/09 Lecture4 47 TR = 0 1 0 -1 0 0 0 0 1 1 0 0 0 1 0 3 1 1 RT = 1 0 0 1 3 1 Rotate then Translate: p' = T ( R p ) = TR p 0 0 0 -1 0 0 3 1 1 0 1 0 -1 0 0 -1 3 1 = = Translate then Rotate: p' = R ( T p ) = RT p 0 0 1 0 1 0 -1 0 0 0 0 1 0 1 0 -1 0 0 0 0 1 0 1 0 -1 0 0 0 0 1 0 1 0 -1 0 0 0 0 1 0 1 0 -1 0 0 0 0 1
  • 48.
    Types of transformations. •Rotation and translation – Angles and distances are preserved – Unit cube is always unit cube – Rigid-Body transformations. • Rotation, translation and scale. – Angles & distances not preserved. – But parallel lines are. 48
  • 49.
    3D Transformations. • Usehomogeneous coordinates, just as in 2D case. • Transformations are now 4x4 matrices. • We will use a right-handed (world) coordinate system - ( z out of page ). Lecture 4 49
  • 50.
  • 51.
  • 52.
    Rotation in 3D •Need to specify which axis the rotation is about. • z-axis rotation is the same as the 2D case. 52               1000 0100 00cossin 00sincos )(   zR
  • 53.
    Rotating About thex-axis Rx() 53                                                11000 0θcosθsin0 0θsinθcos0 0001 1 z y x z y x
  • 54.
    Rotating About they-axis Ry() 54                                                11000 0θcos0θsin 0010 0θsin0θcos 1 z y x z y x
  • 55.
    Rotation About thez-axis Rz() 55                                               11000 0100 00θcosθsin 00θsinθcos 1 z y x z y x
  • 56.
    Rotation • Not commutativeif the axis of rotation are not parallel )()()()(  xyyx RRRR 
  • 57.
    Calculating the world coordinatesof all vertices • For each object, there is a local-to-global transformation matrix • So we apply the transformations to all the vertices of each object • We now know the world coordinates of all the points in the scene 57
  • 58.
    Normal Vectors 58 • Wealso need to know the direction of the normal vectors in the world coordinate system • This is going to be used at the shading operation • We only want to rotate the normal vector • Do not want to translate it
  • 59.
    Normal Vectors -(2) 59 • We need to set elements of the translation part to zero                          1000 0 0 0 1000 111111 111111 111111 111111 111111 111111 rrr rrr rrr trrr trrr trrr z y x
  • 60.
    Summary. • Transformations: translation,rotation and scaling • Using homogeneous transformation, 2D (3D) transformations can be represented by multiplication of a 3x3 (4x4) matrix. • Multiplication from left-to-right can be considered as the transformation of the coordinate system 60
  • 61.