04transformation2d

Graphics

2D Geometric Transformations

cgvr.korea.ac.kr

Graphics Lab @ Korea University

Contents  

Definition & Motivation 2D Geometric Transformation   

   

CGVR

Translation Rotation Scaling

Matrix Representation Homogeneous Coordinates Matrix Composition Composite Transformations    

Pivot-Point Rotation General Fixed-Point Scaling Reflection and Shearing Transformations Between Coordinate Systems Graphics Lab @ Korea University

Geometric Transformation 

CGVR

Definition  



Translation, Rotation, Scaling

Motivation – Why do we need geometric transformations in CG?   

As a viewing aid As a modeling tool As an image manipulation tool

Graphics Lab @ Korea University

Example: 2D Geometric Transformation

CGVR

Modeling Coordinates

World Coordinates Graphics Lab @ Korea University

Example: 2D Scaling

CGVR

Modeling Coordinates

Scale(0.3, 0.3)

World Coordinates Graphics Lab @ Korea University

Example: 2D Rotation

CGVR

Modeling Coordinates

Scale(0.3, 0.3) Rotate(-90) World Coordinates Graphics Lab @ Korea University

Example: 2D Translation

CGVR

Modeling Coordinates

Scale(0.3, 0.3) Rotate(-90) Translate(5, 3) World Coordinates Graphics Lab @ Korea University

Example: 2D Geometric Transformation

CGVR

Modeling Coordinates Again?

World Coordinates Graphics Lab @ Korea University

Example: 2D Geometric Transformation Modeling Coordinates

CGVR

Scale Translate

Scale Rotate Translate World Coordinates Graphics Lab @ Korea University

Basic 2D Transformations 

Translation  





x ′ = x × sx y′ = y × sy

Rotation  



x ′ = x + tx y′ = y + ty

Scale 



CGVR

x ′ = x × cosθ - y × sinθ y′ = y × sinθ + y × cosθ

Shear  

x ′ = x + hx × y y′ = y + hy × x Graphics Lab @ Korea University

Basic 2D Transformations 

Translation  





x ′ = x × sx y′ = y × sy

Rotation  



x ′ = x + tx y′ = y + ty

Scale 



CGVR

x ′ = x × cosθ - y × sinθ y′ = y × sinθ + y × cosθ

Shear  

Transformations can be combined (with simple algebra)

x ′ = x + hx × y y′ = y + hy × x Graphics Lab @ Korea University

Basic 2D Transformations 

Translation  





x ′ = x × sx y′ = y × sy

Rotation  



x ′ = x + tx y′ = y + ty

Scale 



CGVR

x ′ = x × cosθ - y × sinθ y′ = y × sinθ + y × cosθ

Shear  

x ′ = x + hx × y y′ = y + hy × x

x′ = x × sx y′ = y × sy

Graphics Lab @ Korea University

Basic 2D Transformations 

Translation  





x ′ = x × sx y′ = y × sy

Rotation  



x ′ = x + tx y′ = y + ty

Scale 



CGVR

x ′ = x × cosθ - y × sinθ y′ = y × sinθ + y × cosθ

Shear  

x ′ = x + hx × y y′ = y + hy × x

x′ = ((x × sx) × cosθ − (y × sy) × sinθ ) y′ = ((x × sx) × sinθ + (y × sy) × cosθ )

Graphics Lab @ Korea University

Basic 2D Transformations 

Translation  





x ′ = x × sx y′ = y × sy

Rotation  



x ′ = x + tx y′ = y + ty

Scale 



CGVR

x ′ = x × cosθ - y × sinθ y′ = y × sinθ + y × cosθ

Shear  

x ′ = x + hx × y y′ = y + hy × x

x′ = ((x × sx) × cosθ − (y × sy) × sinθ ) + tx y′ = ((x × sx) × sinθ + (y × sy) × cosθ ) + ty

Graphics Lab @ Korea University

Basic 2D Transformations 

Translation  





x ′ = x × sx y′ = y × sy

Rotation  



x ′ = x + tx y′ = y + ty

Scale 



CGVR

x ′ = x × cosθ - y × sinθ y′ = y × sinθ + y × cosθ

Shear  

x ′ = x + hx × y y′ = y + hy × x

x′ = ((x × sx) × cosθ − (y × sy) × sinθ ) + tx y′ = ((x × sx) × sinθ + (y × sy) × cosθ ) + ty

Graphics Lab @ Korea University

Matrix Representation 

CGVR

Represent a 2D Transformation by a Matrix

a b  c d    

Apply the Transformation to a Point

x′ = ax + by y′ = cx + dy

 x′   a b   x   y ′ =  c d   y       Transformation Matrix

Point Graphics Lab @ Korea University

Matrix Representation 

CGVR

Transformations can be combined by matrix multiplication

 x ′   a b  e  y ′ = c d   g    

f  i j   x       h  k l   y 

Transformation Matrix Matrices are a convenient and efficient way to represent a sequence of transformations Graphics Lab @ Korea University

2×2 Matrices 

CGVR

What types of transformations can be represented with a 2×2 matrix?

2D Identity

x′ = x y′ = y

 x′  1 0  x   y′ = 0 1  y      

2D Scaling

x′ = sx × x y′ = sy × y

 x′   sx 0  x   y′ = 0 sy   y      

Graphics Lab @ Korea University

2×2 Matrices 

CGVR

What types of transformations can be represented with a 2×2 matrix?

2D Rotation x′ = cosθ × x − sin θ × y y′ = sin θ × x + cosθ × y

 x′  cosθ − sin θ   x   y′ = sin θ cosθ   y      

2D Shearing x′ = x + shx × y y′ = shy × x + y

 x′  1 shx   x   y′ =  shy 1  y      

Graphics Lab @ Korea University

2×2 Matrices 

CGVR

What types of transformations can be represented with a 2×2 matrix?

2D Mirror over Y axis x′ = − x  x′   − 1 0   x  =    y ′ y′ = y y 0 1      2D Mirror over (0,0) x′ = − x  x′   − 1 0   x  =    y y′ = − y ′ y 0 − 1     

Graphics Lab @ Korea University

2×2 Matrices 

CGVR

What types of transformations can be represented with a 2×2 matrix?

2D Translation

x ′ = x + tx y′ = y + ty

NO!!

Only linear 2D transformations can be Represented with 2x2 matrix

Graphics Lab @ Korea University

2D Translation 

CGVR

2D translation can be represented by a 3×3 matrix 

Point represented with homogeneous coordinates x ′ = x + tx y′ = y + ty

 x′  1 0 tx   x   y′ = 0 1 ty   y       1  0 0 1  1 

Graphics Lab @ Korea University

Basic 2D Transformations 

CGVR

Basic 2D transformations as 3x3 Matrices  x′  1 0 tx   x   y′ = 0 1 ty   y       1  0 0 1  1 

 x′   sx 0 0   x   y′ = 0 sy 0  y       1  0 0 1  1 

Translate

Scale

 x′  cosθ − sin θ 0   x   y′ = sin θ cosθ  y 0      1  0 0 1  1 

 x′  1 shx 0   x   y′ =  shy 1 0  y       1  0 0 1 1 

Rotate

Shear Graphics Lab @ Korea University

Homogeneous Coordinates 

CGVR

Add a 3rd coordinate to every 2D point   

(x, y, w) represents a point at location (x/w, y/w) (x, y, 0) represents a point at infinity (0, 0, 0) Is not allowed y 2

(2, 1, 1) or (4, 2, 2) or (6, 3, 3)

1 1 2

x

Convenient Coordinate System to Represent Many Useful Transformations Graphics Lab @ Korea University

Linear Transformations 



CGVR

Linear transformations are combinations of …  Scale  x′   a b 0   x   Rotation  y ′  = c d 0   y   Shear, and       w′ 0 0 1   w  Mirror Properties of linear transformations  Satisfies: T ( s p + s p ) = s T ( p ) + s T ( p ) 1 1 2 2 1 1 2 2     

Origin maps to origin Lines map to lines Parallel lines remain parallel Ratios are preserved Closed under composition

Graphics Lab @ Korea University

Affine Transformations 

Affine transformations are combinations of  



CGVR

Linear transformations, and Translations x′ a b

c x      y′  = d e f   y        w′ 0 0 1   w Properties of affine transformations     

Origin does not map to origin Lines map to lines Parallel lines remain parallel Ratios are preserved Closed under composition

Graphics Lab @ Korea University

Projective Transformations 

Projective transformations…  



CGVR

Affine transformations, and Projective warps x′ a

   b c x   y′  = d e f   y        w′  g h i   w Properties of projective transformations     

Origin does not map to origin Lines map to lines Parallel lines do not necessarily remain parallel Ratios are not preserved Closed under composition

Graphics Lab @ Korea University

Matrix Composition 

CGVR

Transformations can be combined by matrix multiplication  x′   1 0 tx  cos θ - sinθ 0  sx 0 0   x   y′  =  0 1 ty   sinθ cosθ 0 0 sy 0    y           w′  0 0 1   0 0 1  0 0 1    w 

p′ = 

T(tx, ty)

R(θ )

S(sx, sy)

p

Efficiency with premultiplication 

Matrix multiplication is associative

p′ = (T × (R × (S × p)))

p′ = (T × R × S) × p Graphics Lab @ Korea University

Matrix Composition 

CGVR

Rotate by θ around arbitrary point (a,b) 

M = T(a, b) × R(θ ) × T(-a,-b) (a,b)



Scale by sx, sy around arbitrary point (a,b)  M = T(a, b) × S(sx, sy) × T(-a,-b) (a,b)

Graphics Lab @ Korea University

Pivot-Point Rotation

(xr,yr)

Translate

(xr,yr)

CGVR

(xr,yr)

Rotate

(xr,yr)

Translate

T ( xr , y r ) ⋅ R ( θ ) ⋅ T ( − x r , − y r ) = R ( xr , y r , θ )  1 0 x r   cos θ − sin θ 0  1 0 − x r   cos θ − sin θ  0 1 y  ⋅  sin θ cos θ 0 ⋅  0 1 − y  =  sin θ cos θ r  r      0 0 1   0 0 1  0 0 1   0 0

x r (1 − cos θ ) + y r sin θ  y r (1 − cos θ ) − x r sin θ   1 Graphics Lab @ Korea University

General Fixed-Point Scaling

(xf,yf)

(xf,yf)

Translate

Scale

CGVR

(xf,yf)

(xf,yf)

Translate

T ( x f , y f ) ⋅ S ( s x , s y ) ⋅ T ( − x f ,− y f ) = S ( x f , y f , s x , s y ) 1 0 0 1  0 0

 s x ⋅0   1   0

xf yf

0 sy 0

0 1 0 0 ⋅ 0 1 1 0 0

− x f  s x − y f  =  0 1   0

0 sy 0

x f (1 − s x )  y f (1 − s y )   1

Graphics Lab @ Korea University

Reflection 

CGVR

Reflection with respect to the axis •

•

x

 − 1 0 0  0 1 0    0 0 1

1 0 0  0 − 1 0    0 0 1 y

1

1

2

3

2’

3’

2

y

3

x

y

1’ 3’

xy

(

)

 − 1 0 0  0 − 1 0    0 0 1

2 x

1’ 3’

3

1

2 x

2

1’

x

•

y

y Graphics Lab @ Korea University

Reflection 

CGVR

Reflection with respect to a Line y

0 1 0  1 0 0   0 0 1



x

y=x °

Clockwise rotation of 45 → Reflection about the x axis → Counterclockwise rotation of 45° y

y

x

1

y

2

3

2’

3’

x

x

1’ Graphics Lab @ Korea University

Shear 

CGVR

Converted to a parallelogram 1 sh x 0 1  0 0

x’ = x + shx · y,

0 0 1

y (0,1)

(1,1)

(0,0)

(1,0)

y (2,1) x

(1,0)

x

y’ = y

x



(0,0)

(3,1)

(Shx=2)

Transformed to a shifted parallelogram (Y = Yref) 1 sh x 0 1  0 0 x’ = x + shx · (y-yref),

− sh x ⋅ y ref   0   1 y’ = y

y (0,1)

(1,1)

(0,0)

(1,0)

y

x

(1,1) (1/2,0)

(2,1) (3/2,0)x

(0,-1)

(Shx=1/2, yref=-1) Graphics Lab @ Korea University

Shear 

CGVR

Transformed to a shifted parallelogram (X = Xref)  1  sh  y  0

x’ = x,

0 0  1 − sh y ⋅ x ref   0 1

y

(0,1)

(1,1)

(0,0)

(1,0)

(0,3/2)

(1,2)

(0,1/2)

(1,1)

y

x

(-1,0)

x

y’ = shy · (x-xref) + y (Shy=1/2, xref=-1)

Graphics Lab @ Korea University