Download Linear and affine transformations

Linear and affine transformations
•
Linear Algebra Review
 Matrices
 Transformations
Affine transformations in Euclidean space
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
The linear transformation given by a matrix
Let A be an mn matrix. The function T defined by
T ( v)  Av
is a linear transformation from Rn into Rm.

Note:
R n vector
 a11
a
Av   21


 am 1
a12
a22
am 2
R m vector
a1n   v1   a11v1  a12v2 
a2 n   v2   a21v1  a22v2 

  
  
amn   vn   am 1v1  am 2v2 
 a1nvn 
 a2 nv n 


 amn vn 
T ( v)  Av
T : Rn 
 R m
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
Two representations of the linear transformation
T:R3→R3 :
(1)T ( x1 , x2 , x3 )  (2 x1  x2  x3 , x1  3x2  2 x3 ,3x2  4 x3 )
 2 1  1  x1 
(2)T (x)  Ax   1 3  2  x2 

 
 0 3 4   x3 

Three reasons for matrix representation of a linear
transformation:

It is simpler to write.

It is simpler to read.

It is more easily adapted for computer use.
Geometric transformations
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•
•
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Geometric transformations map points in one space to points in
another: (x',y',z') = f(x,y,z), i.e. in vector form X’ = f(X)
These transformations can be very simple, such as scaling each
coordinate, or complex, such as non-linear twists and bends.
We'll focus on transformations that can be represented easily with
matrix operations.
We'll start in 2D...
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Representation of 2D linear map
• We can represent a 2-D transformation M by a matrix
a b
M  

c d
• If x is a column vector, M goes on the left: x '  Mx
 x  a b   x 
 y    c d   y 
  
  

• If x is a row vector, MT goes on the right: x  xM
 x
y    x
• We will use column vectors.
T
a c 
y 

b d 
Basic 2D Linear Transforms
Base vectors map to columns of matrix.
x '  Mx
x  M x
 a b   x   ax  by 
M x
   

 c d   y   cx  dy 
 a b  1   a 
 c d  0   c 

   
 x  a b   x 
 y    c d   y 
  
  
 a b   0  b 
 c d  1   d 

   
(1, 0)
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3
x  M x
Scaling by 0.5
y
y
M
0.5 0 
 0 0.5


x
x
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General Scaling
x  scale(sx , s y ) x
y
y
scale  s x , s y  
1
 sx
0

x
1
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sy
0
s y 
x
sx
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Rotation x  rot ( ) x
rot   
y
y
cos    sin   


 sin   cos   

x
x
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Reflection in y-axis x  ref y x
y
ref y 
y
 1 0 
 0 1


x
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x
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Reflection in x-axis x  ref x x
y
y
ref x 
1 0 
0 1


x
x
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Shear-x
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x  M x
y
y
1 s 
 0 1


x
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x
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Shear-y
x  M x
y
y
1 0 
 s 1


x
x
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Composing Linear Transformations
T1 ( v)  M1  v
• If T1 and T2 are transformations
T2 ( v)  M 2  v
 T2 T1(v) =def T2( T1(v))
• If T1 and T2 are represented by matrices M1 and M2
 T2 T1 is represented by M2 M1
 T2 T1(v) = T2( T1(v)) = (M2 M1)(v)
• Order is important!
reflect(x) (rot(O,𝛂)): A → A’ → A’’
rot(O,𝛂) (reflect(x)): A → A → A’
A '  Rot  A
A ''  Ref  A '
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Composing Linear Transformations
• Order is important!
reflect(x) (rot(O,𝛂)): A → A’ → A’’
rot(O,𝛂) (reflect(x)): A → A → A’
A '  Rot  A
A ''  Ref  A '
 cos   sin  
1 0 
Rot  
;
Ref




 sin  cos  
 0 1
 cos  sin  
Rot*Ref  

 sin   cos  
 cos   sin  
Ref*Rot  

  sin   cos  
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Decomposing Linear Transformations
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Any 2D Linear Transformation can be decomposed into the product
of a rotation, a scale (or line reflection), and a rotation
M = R1SR2 .
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Any 2D Linear Transformation can be decomposed into the product
of 3 line reflection at the most.
Isometry (congruent transformation)
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Isometry preserves length, whereas direct isometry preserves
orientation and opposite does not preserve orientation
Direct Isometry |R| = 1 (Rotation)
Opposite Isometry |R| = -1 (Line Reflection)
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Reflection About an
Arbitrary Line (through the origin)
y
y
x
x
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Reflection as a Composition
xF  ref  x F new base xB = B  x  x = B -1  x B
y
B-1 xB = ref  B-1 xB
xB = B  ref  B-1 xB
B  e1 , e2
y
F  f1 ,f 2
f1
x
f2
x
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Linear Transformations
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Scale, Reflection, Rotation, and Shear are all linear transformations
They satisfy: T(au + bv) = aT(u) + bT(v)
 u and v are vectors
 a and b are scalars
If T is a linear transformation
 T((0, 0)) = (0, 0)
Q: What important operation does that leave out?
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Rotation about
an Arbitrary Point
y
y


x
x
This is not a linear transformation. The origin moves.
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Translation
(x, y)(x+a,y+b)
y
y
(a, b)
x
x
This is not a linear transformation. The origin moves.
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2D Linear Transformations
as 3D Matrices
Any 2D linear transformation can be
represented by a 2x2 matrix
 a11 a12   x   a11 x  a12 y 
a
  y   a x  a y 
a
22 
 21 22     21
or a 3x3 matrix
 a11 a12
a
a
 21 22
0
 0
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0  x   a11 x  a12 y 
0  y    a21 x  a22 y 
  

1  1  
1

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2D Linear Translations
as 3D Matrices
Any 2D translation can be represented by a 3x3 matrix.
1 0 a   x   x  a 
0 1 b   y    y  b 

  

0 0 1   1   1 
With homogeneous coordinates, we can represent all 2D
affine transformations as 3D linear transformations.
We can then use matrix multiplication to transform objects.
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2D Affine Transformations
• An affine transformation is any transformation
that preserves co-linearity (i.e., all points lying on
a line initially still lie on a line after
transformation) and ratios of distances (e.g., the
midpoint of a line segment remains the midpoint
after transformation).
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Rotation about
an Arbitrary Point
T(cx, cy) R()T(-cx, -cy)
y
y


x
x
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Windowing Transforms
(A,B)
(a,b)
translate
(A-a,B-b)
scale
(C,D)
(C-c,D-d)
translate (c,d)
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3D Transformations
Remember:
 x
 x
 
 y   y
 
z
 z 
 
1
A 3D linear transformation can be represented by a
3x3 matrix.
 a11 a12
a
a
 21 22
 a31 a32
 a11 a12
a13 
a
a

a23   21 22

 a31 a32
a33 

0
0
a13
a22
a33
0
0
0

0

1
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3D Affine Transformations
 sx
0
scale  sx , s y , sz   
0

0
1
0
translate  t x , t y , tz   
0

0
0
0
sy
0
0
sz
0
0
0
0

0

1
0 0 tx 
1 0 ty 

0 1 tz 

0 0 1
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3D Rotations
0
0
1
0 cos    sin  
rotate x    
0 sin   cos  

0
0
0
cos    sin  

sin   cos  
rotatez    
 0
0

0
 0
0
0

0

1
 cos  

0
rotate y    
  sin  

 0
0 0

0 0
1 0

0 1
0 sin  
1
0
0 cos  
0
0
0

0
0

1
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