Transcript 슬라이드 1

Graphics
2D Geometric
Transformations
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Graphics Lab @ Korea University
Contents
CGVR

Definition & Motivation
 2D Geometric Transformation







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
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Geometric Transformation

Definition


CGVR
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
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Example: 2D Geometric
Transformation
CGVR
Modeling
Coordinates
World Coordinates
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Example: 2D Scaling
CGVR
Modeling
Coordinates
Scale(0.3, 0.3)
World Coordinates
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Example: 2D Rotation
CGVR
Modeling
Coordinates
Scale(0.3, 0.3)
Rotate(-90)
World Coordinates
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Example: 2D Translation
CGVR
Modeling
Coordinates
Scale(0.3, 0.3)
Rotate(-90)
Translate(5, 3)
World Coordinates
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Example: 2D Geometric
Transformation
CGVR
Modeling
Coordinates
Again?
World Coordinates
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Example: 2D Geometric
Transformation
Modeling
Coordinates
CGVR
Scale
Translate
Scale
Rotate
Translate
World Coordinates
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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
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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
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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
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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
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x  ((x sx)  cos  (y  sy) sin )
y  ((x sx)  sin  (y  sy) cos )
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
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x  ((x sx)  cos  (y  sy) sin )  tx
y  ((x sx)  sin  (y  sy) cos )  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
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x  ((x sx)  cos  (y  sy) sin )  tx
y  ((x sx)  sin  (y  sy) cos )  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
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Point
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Matrix Representation

CGVR
Transformations can be combined by matrix
multiplication
 x  a b  e f  i j   x 
 y  c d   g h k l   y 
  


 
Transformation
Matrix
Matrices are a convenient and efficient way
to represent a sequence of transformations
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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
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 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  cos  x  sin   y
y  sin   x  cos  y
 x  cos  sin    x 
 y  sin  cos   y 
  
 
2D Shearing
x  x  shx  y
y  shy  x  y
 x  1 shx  x 
 y  shy 1  y 
  
 
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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
  
 
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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
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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
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 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  cos  sin  0   x 
 y  sin  cos
 y
0
  
 
1  0
0
1  1 
 x  1 shx 0   x 
 y    shy 1 0  y 
  
 
1  0 0 1 1 
Rotate
Shear
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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
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Matrix Composition

CGVR
Transformations can be combined by matrix
multiplication
 x   1 0 tx  cosθ - 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( )
S(sx, sy)
p
Efficiency with premultiplication

Matrix multiplication is associative
p  (T  (R  (S  p)))
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p  (T  R  S)  p
Graphics Lab @ Korea University
Matrix Composition

CGVR
Rotate by  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)
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Pivot-Point Rotation
(xr,yr)
Translate
(xr,yr)
Rotate
CGVR
(xr,yr)
(xr,yr)
Translate
T xr , yr  R  T  xr , yr   Rxr , yr , 
1 0 x r  cos  sin  0 1 0  x r  cos  sin  x r (1  cos )  y r sin  
0 1 y    sin  cos 0  0 1  y    sin  cos y (1  cos )  x sin  
r 
r
r
r

 


0 0 1   0

0
1 0 0 1   0
0
1
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General Fixed-Point Scaling
(xf,yf)
(xf,yf)
Scale
Translate
CGVR
(xf,yf)
(xf,yf)
Translate
T x f , y f  S sx , sy  T  x f , y f   S x f , y f , sx , sy 
1 0
0 1

0 0
x f  s x
y f    0
1   0
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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
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1
2
3
2’
3’
1’
•
y
2
x
y
3
xy
 1 0 0
 0  1 0


 0
0 1
y
1’
3’
3’
2
x
1’
3
1
2
x
2
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’
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Graphics Lab @ Korea University
Shear

CGVR
Converted to a parallelogram
1 sh x 0
0 1 0 


0 0 1
x’ = x + shx · y, y’ = y
y
(0,1)
(1,1)
(0,0)
(1,0)
y
(2,1)
x
(0,0)
(1,0)
(3,1)
x
(Shx=2)

Transformed to a shifted parallelogram
(Y = Yref)
1 sh x
0 1

0 0
 sh x  y ref 

0


1
x’ = x + shx · (y-yref),
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y’ = y
y
(0,1)
y
(1,1)
(2,1)
(1,1)
(1/2,0)
(0,0)
(1,0)
x
(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)
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