Transformations
CS 445/645
Introduction to Computer Graphics
David Luebke, Spring 2003
Admin
● Call roll
● Assn 1
■ Remember, due Monday 10 AM (ie, don’t skip class)
■ Several people have turned in already – nice job
David Luebke
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Recap: Translations
● For convenience we usually describe objects in
relation to their own coordinate system
● We can translate or move points to a new position
by adding offsets to their coordinates:
 x'  x  tx 
 y '   y   ty 
     
 z '   z  tz 
■ Note that this translates all points uniformly
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Recap: Scaling
● Scaling operation:
● Or, in matrix form:
 x' ax 
 y '  by 
   
 z '   cz 
 x '   a 0 0  x 
 y'  0 b 0  y 
  
 
 z '  0 0 c   z 
scaling matrix
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Recap: 3-D Rotation
● 3x3 matrix for rotation about the z-axis:
 x' cos()  sin( ) 0  x 
 y '   sin( ) cos() 0  y 
  
 
 z '   0
0
1  z 
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Recap: 3-D Rotation
● 3x3 matrix for rotation about the y-axis:
 x'  cos() 0 sin( )   x 
 y '   0



1
0
y
  
 
 z '   sin( ) 0 cos()  z 
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Recap: 3-D Rotation
● 3x3 matrix for rotation about the x-axis:
0
0  x
 x' 1
 y '  0 cos()  sin( )  y 
  
 
 z '  0 sin( ) cos()   z 
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Recap:
Composing Canonical Rotations
● Goal: rotate about arbitrary vector A by 
■ Idea: we know how to rotate about X,Y,Z
So, rotate about Y by  until A lies in the YZ plane
Then rotate about X by  until A coincides with +Z
Then rotate about Z by 
Then reverse the rotation about X (by -)
Then reverse the rotation about Y (by -)
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Composing Canonical
Rotations
● First: rotating about Y by  until A lies in YZ
■ Draw it…
● How exactly do we calculate ?
■ Project A onto XZ plane
■ Find angle  to X:
 = -(90° - ) =  - 90 °
● Second: rotating about X by  until A lies on Z
● How do we calculate ?
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Composing Canonical
Rotations
● Why are we slogging through all this tedium?
● A: Because you’ll have to do it on the test
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3-D Rotation Matrices
● So an arbitrary rotation about A composites several
●
●
●
●
canonical rotations together
We can express each rotation as a matrix
Compositing transforms == multiplying matrices
Thus we can express the final rotation as the product
of canonical rotation matrices
Thus we can express the final rotation with a single
matrix!
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Compositing Matrices
● So we have the following matrices:
p: The point to be rotated about A by 
Ry : Rotate about Y by 
Rx  : Rotate about X by 
Rz : Rotate about Z by 
Rx  -1: Undo rotation about X by 
Ry-1 : Undo rotation about Y by 
● In what order should we multiply them?
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Compositing Matrices
● Remember: the transformations, in order, are written
from right to left
■ In other words, the first matrix to affect the vector goes
next to the vector, the second next to the first, etc.
■ This is the rule with column vectors (OpenGL); row
vectors would be the opposite
● So in our case:
p’ = Ry-1 Rx  -1 Rz Rx  Ry p
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Rotation Matrices
● Notice these two matrices:
Rx  : Rotate about X by 
Rx  -1: Undo rotation about X by 
● How can we calculate Rx  -1?
■ Obvious answer: calculate Rx (-)
■ Clever answer: exploit fact that rotation matrices are
orthogonal
● What is an orthogonal matrix?
● What property are we talking about?
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Rotation Matrices
● Rotation matrix is orthogonal
■ Columns/rows linearly independent
■ Columns/rows sum to 1
● The inverse of an orthogonal matrix is just its
transpose:
a b
d e

 h i
David Luebke
1
c
a b


f   d e
 h i
j 
15
T
c
a


f   b
 c
j 
d
e
f
h

i
j 
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Translation Matrices?
● We can composite scale matrices just as we did
rotation matrices
● But how to represent translation as a matrix?
● Answer: with homogeneous coordinates
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Homogeneous Coordinates
● Homogeneous coordinates: represent coordinates in 3
dimensions with a 4-vector
 x / w  x 
 y / w  y 
 
( x, y , z )  
 z / w  z 

  
 1   w
(Note that typically w = 1 in object coordinates)
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Homogeneous Coordinates
● Homogeneous coordinates seem unintuitive, but they
make graphics operations much easier
● Our transformation matrices are now 4x4:
0
0
1
0 cos()  sin( )
Rx  
0 sin( ) cos()

0
0
0
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0
0
0

1
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Homogeneous Coordinates
● Homogeneous coordinates seem unintuitive, but they
make graphics operations much easier
● Our transformation matrices are now 4x4:
 cos()
 0
Ry  
 sin( )

 0
David Luebke
0 sin( ) 0
1
0
0
0 cos() 0

0
0
1
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Homogeneous Coordinates
● Homogeneous coordinates seem unintuitive, but they
make graphics operations much easier
● Our transformation matrices are now 4x4:
cos()  sin( )
 sin( ) cos()
Rz  
 0
0

0
 0
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0 0
0 0
1 0

0 1
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Homogeneous Coordinates
● Homogeneous coordinates seem unintuitive, but they
make graphics operations much easier
● Our transformation matrices are now 4x4:
 Sx 0
 0 Sy
S
0 0

0 0
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0 0
0 0
Sz 0 

0 1
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Homogeneous Coordinates
● How can we represent translation as a
4x4 matrix?
● A: Using the rightmost column:
1
0
T
0

0
David Luebke
0 0 Tx 
1 0 Ty 
0 1 Tz 

0 0 1
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Translation Matrices
● Now that we can represent translation as a matrix, we
can composite it with other transformations
● Ex: rotate 90° about X, then 10 units down Z:
 x'  1
 y '  0
 
 z '  0
  
 w' 0
David Luebke
0 0 0  1
0
0
1 0 0  0 cos(90)  sin( 90)
0 1 10 0 sin( 90) cos(90)

0 0 1  0
0
0
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0  x 
0  y 
0  z 
 
1   w
3/15/2016
Translation Matrices
● Now that we can represent translation as a matrix, we
can composite it with other transformations
● Ex: rotate 90° about X, then 10 units down Z:
 x'  1
 y '  0
 
 z '  0
  
 w' 0
David Luebke
0 0 0  1
1 0 0  0
0 1 10 0

0 0 1  0
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0 0 0  x 
0  1 0  y 
1 0 0  z 
 
0 0 1   w
3/15/2016
Translation Matrices
● Now that we can represent translation as a matrix, we
can composite it with other transformations
● Ex: rotate 90° about X, then 10 units down Z:
 x'  1
 y '  0
 
 z '  0
  
 w' 0
David Luebke
0 0 0  x 
0  1 0   y 
1 0 10  z 
 
0 0 1   w
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Translation Matrices
● Now that we can represent translation as a matrix, we
can composite it with other transformations
● Ex: rotate 90° about X, then 10 units down Z:
 x'   x 
 y'   z 
 

 z '   y  10
  

 w'  w 
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Transformation Commutativity
● Is matrix multiplication, in general, commutative?
Does AB = BA?
● What about rotation, scaling, and translation
matrices?
■ Does RxRy = RyRx?
■ Does RAS = SRA ?
■ Does RAT = TRA ?
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