CS 4300
Computer Graphics
Prof. Harriet Fell
Fall 2012
Lecture 11 – September 27, 2012
April 13, 2015
©College of Computer and Information Science, Northeastern University
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Today’s Topics
• Linear Algebra Review
 Matrices
 Transformations
• New Linear Algebra
 Homogeneous Coordinates
April 13, 2015
©College of Computer and Information Science, Northeastern University
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Matrices
 a11 a12 
A 

a
a
22 
 21
 b11 b12
B  b21 b22

 b31 b32
b13 
b23 

b33 
 c11 c12
c
c22
21
C
 c31 c32

 c41 c42
c13
c23
c33
c43
c14 
c24 

c34 

c44 
• We use 2x2, 3x3, and 4x4 matrices in computer
graphics.
• We’ll start with a review of 2D matrices and
transformations.
April 13, 2015
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Basic 2D Linear Transforms
 a11
a
 21
a12   x   a11 x  a12 y 





a22   y   a21 x  a22 y 
 a11 a12  1  a11 
 
a



 21 a22  0 a21 
April 13, 2015
 a11 a12  0  a12 
 
a



 21 a22  1  a22 
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Scale by .5
scale .5,.5 
.5 0 
 0 .5


(1, 0)
(0.5, 0)
(0, 1)
(0, 0.5)
April 13, 2015
©College of Computer and Information Science, Northeastern University
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Scaling by .5
y
y
0.5 0 
 0 0.5


x
April 13, 2015
©College of Computer and Information Science, Northeastern University
x
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General Scaling
y
y
 s xs 0 0 1   s x0
 0 x s   0    
  0 ysy  01   0sy 


 
x
April 13, 2015
©College of Computer and Information Science, Northeastern University
x
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General Scaling
y
y
scale  sx , s y  
1
 sx
0

x
1
April 13, 2015
sy
0
s y 
x
sx
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Rotation

cos    sin   

sin   cos   

rot  




φ sin(φ)
cos(φ)
-sin(φ)
φ cos(φ)
April 13, 2015
©College of Computer and Information Science, Northeastern University
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Rotation

cos    sin   

sin   cos   

rot  
y




y
φ
x
April 13, 2015
©College of Computer and Information Science, Northeastern University
x
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Reflection in y-axis
reflect-y 
 1 0
 0 1


April 13, 2015
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Reflection in y-axis
y
reflect-y 
y
 1 0
 0 1


x
April 13, 2015
©College of Computer and Information Science, Northeastern University
x
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Reflection in x-axis
reflect-x 
1 0 
0 1


April 13, 2015
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Reflection in x-axis
y
y
reflect-x 
1 0 
0 1


x
April 13, 2015
©College of Computer and Information Science, Northeastern University
x
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Shear-x
shear-x  s  
1 s 
 0 1


s
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Shear x
y
y
1 s 
 0 1


x
April 13, 2015
©College of Computer and Information Science, Northeastern University
x
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Shear-y
shear-y  s  
1 0 
 s 1


s
April 13, 2015
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Shear y
y
y
1 0 
 s 1


x
April 13, 2015
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x
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Linear Transformations
• 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)
April 13, 2015
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Composing Linear
Transformations
• If T1 and T2 are transformations
 T2 T1(v) =def T2( T1(v))
• If T1 and T2 are linear and are represented
by matrices M1 and M2
 T2 T1 is represented by M2 M1
 T2 T1(v) = T2( T1(v)) = (M2 M1)(v)
April 13, 2015
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Reflection About an
Arbitrary Line (through the origin)
y
y
x
April 13, 2015
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x
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Reflection as a Composition
y
x
April 13, 2015
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Decomposing
Linear Transformations
• Any 2D Linear Transformation can be
decomposed into the product of a rotation,
a scale, and a rotation if the scale can
have negative numbers.
• M = R1SR2
April 13, 2015
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Rotation about
an Arbitrary Point
y
y
φ
φ
x
x
This is not a linear transformation. The origin moves.
April 13, 2015
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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.
April 13, 2015
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Homogeneous Coordinates
y
y
Embed the xy-plane in R3 at z = 1.
(x, y)  (x, y, 1)
x
x
z
April 13, 2015
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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




 21 a22   y   a21 x  a22 y 
or a 3x3 matrix
 a11
a
 21
 0
April 13, 2015
a12
a22
0
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 
This is a 3D shear that acts as a
translation on the plane z = 1.
April 13, 2015
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Translation as a Shear
y
y
x
x
z
April 13, 2015
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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).
• With homogeneous coordinates, we can
represent all 2D affine transformations as 3D
linear transformations.
• We can then use matrix multiplication to
transform objects.
April 13, 2015
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Rotation about
an Arbitrary Point
y
y
φ
φ
x
April 13, 2015
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x
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Rotation about
an Arbitrary Point
y
T(cx, cy) R(φ)T(-cx, -cy)
φφ
φφ
April 13, 2015
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)
April 13, 2015
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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
April 13, 2015
 a11 a12
a13 
a
a22
21


a23 

 a31 a32
a33 

0
0
a13
a22
a33
0
0
0

0

1
©College of Computer and Information Science, Northeastern University
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3D Affine Transformations
 sx
0
scale  sx , s y , sz   
0

0
0
sy
0
0
0
0
sz
0
0
0

0

1
1
0
translate  t x , t y , tz   
0

0
April 13, 2015
0
1
0
0
0 tx 
0 ty 

1 tz 

0 1
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3D Rotations
0
0
1
0 cos    sin  
rotate x    
0 sin   cos  

0
0
0
rotate z   
April 13, 2015
0
0

0

1
 cos  

0

rotate y   
  sin  

cos    sin   0 0
0



sin

cos

0
0









0
0
0
0
0 sin  
1
0
0 cos  
0
0
0

0
0

1
1 0

0 1
©College of Computer and Information Science, Northeastern University
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