Transformations
Aaron Bloomfield
CS 445: Introduction to Graphics
Fall 2006
(Slide set originally by David Luebke)
Graphics coordinate systems



X is red
Y is green
Z is blue
2
Graphics coordinate systems


If you are on the +z axis,
and +y is up, then +x is
to the right
Math fields have +x
going to the left
3
Outline






Scaling
Rotations
Composing Rotations
Homogeneous Coordinates
Translations
Projections
4
Scaling


Scaling a coordinate means multiplying each of its
components by a scalar
Uniform scaling means this scalar is the same for
all components:
2
5
Scaling

Non-uniform
component:
scaling:
different
scalars
per
X  2,
Y  0.5

How can we represent this in matrix form?
6
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
7
Outline






Scaling
Rotations
Composing Rotations
Homogeneous Coordinates
Translations
Projections
8
2-D Rotation
(x’, y’)
(x, y)

x’ = x cos() - y sin()
y’ = x sin() + y cos()
9
2-D Rotation

This is easy to capture in matrix form:
 x' cos  sin   x 
 y '   sin  cos   y 
  
 

3-D is more complicated


Need to specify an axis of rotation
Simple cases: rotation about X, Y, Z axes
10
Rotation example: airplane
11
3-D Rotation

What does the 3-D rotation matrix look like for a
rotation about the Z-axis?

Build it coordinate-by-coordinate
 x' cos()  sin( ) 0  x 
 y '   sin( ) cos() 0  y 
  
 
 z '   0
0
1  z 

2-D rotation from last slide:  x'  cos  sin   x 
 y '  sin  cos   y 
  
 
12
3-D Rotation

What does the 3-D rotation matrix look like for a
rotation about the Y-axis?

Build it coordinate-by-coordinate
 x'  cos() 0 sin( )   x 
 y '   0



1
0
y
  
 
 z '   sin( ) 0 cos()  z 
13
3-D Rotation

What does the 3-D rotation matrix look like for a
rotation about the X-axis?

Build it coordinate-by-coordinate
0
0  x
 x' 1
 y '  0 cos()  sin( )  y 
  
 
 z '  0 sin( ) cos()   z 
14
Rotations about the axes
15
3-D Rotation



General rotations in 3-D require rotating about an
arbitrary axis of rotation
Deriving the rotation matrix for such a rotation
directly is a good exercise in linear algebra
Another approach: express general rotation as
composition of canonical rotations

Rotations about X, Y, Z
16
Outline






Scaling
Rotations
Composing Rotations
Homogeneous Coordinates
Translations
Projections
17
Composing Canonical Rotations

Goal: rotate about arbitrary vector A by 


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 




Idea: we know how to rotate about X,Y,Z
Then reverse the rotation about X (by -)
Then reverse the rotation about Y (by -)
Show video…
18
Composing Canonical Rotations

First: rotating about Y by  until A lies in YZ


How exactly do we calculate ?




Draw it…
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 ?
19
3-D Rotation Matrices
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

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!
20
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?
21
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

22
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
orthonormal
What is an orthonormal matrix?
What property are we talking about?
23
Rotation Matrices

Rotation matrix is orthogonal

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
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
1
c
a b


f   d e
 h i
j 
T
c
a


f   b
 c
j 
d
e
f
h

i
j 
24
Rotation Matrix for Any Axis

Given glRotated (angle, x, y, z)




Let c = cos(angle)
Let s = sin(angle)
And normalize the vector so that ||(x,y,z|| == 1
The produced matrix to rotate something by angle
degrees around the axis (x,y,z) is:
 xx(1  c)  c
 yx(1  c)  zs

 zx(1  c)  ys

0

xy (1  c)  zs
yy (1  c)  c
zy (1  c)  xs
0
xz (1  c)  ys 0 
yz (1  c)  xs 0 
zz (1  c)  c 0 

0
1
25
Outline






Scaling
Rotations
Composing Rotations
Homogeneous Coordinates
Translations
Projections
26
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
27
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
28
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



[x, y, z, 0]T represents a point at infinity (use for vectors)
[0, 0, 0]T is not allowed
Note that typically w = 1 in object coordinates
29
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
0
0
0

1
30
Homogeneous Coordinates


Homogeneous coordinates seem unintuitive, but
they make graphics operations much easier
Our transformation matrices are now 4x4:
 cos()
 0
Ry  
 sin( )

 0
0 sin( )
0
1
0
0
0 cos() 0

0
0
1
31
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
0 0
0 0
1 0

0 1
32
Homogeneous Coordinates



Homogeneous coordinates seem unintuitive, but
they make graphics operations much easier
Our transformation matrices
 Sx 0 0 0 
are now 4x4:


Performing a scale:
1
0

0

0
0 0 0  x 
2 0 0   y 
 x 2y
0 1 0  z 
 
0 0 1  1 
0
S
0

0
Sy
0
0
Sz
0
0
0
0

1
z 1
33
More On Homogeneous Coords

What effect does the following matrix have?
 x'  1
 y '  0
 
 z '  0
  
 w' 0

0  x 
1 0 0   y 
0 1 0  z 
 
0 0 10  w
0 0
Conceptually, the fourth coordinate w is a bit like a
scale factor
34
More On Homogeneous Coords

Intuitively:




The w coordinate of a homogeneous point is
typically 1
Decreasing w makes the point “bigger”, meaning further
from the origin
Homogeneous points with w = 0 are thus “points at
infinity”, meaning infinitely far away in some direction.
(What direction?)
To help illustrate this, imagine subtracting two
homogeneous points
35
Outline






Scaling
Rotations
Composing Rotations
Homogeneous Coordinates
Translations
Projections
36
Homogeneous Coordinates



1
0

0

0
How can we represent translation as a 4x4 matrix?
A: Using the rightmost column:
1 0 0 Tx
Performing a translation:
0   x



1 0 0   y
 x
0 1 10   z 
 
0 0 1  1 
0 0
y


 0 1 0 Ty 

T
 0 0 1 Tz 


0 0 0 1 
z  10 1
37
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
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
0 0
0  x 
0  y 
0  z 
 
1   w
38
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
0  1
1 0 0  0
0 1 10 0

0 0 1  0
0 0
0  x 
0  1 0  y 
1 0 0  z 
 
0 0 1  w
0
0
39
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
0  x 
0  1 0   y 
1 0 10  z 
 
0 0 1   w
0
0
40
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 
41
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 ?
42
Outline






Scaling
Rotations
Composing Rotations
Homogeneous Coordinates
Translations
Projections
43
Perspective Projection


In the real world, objects exhibit perspective
foreshortening: distant objects appear smaller
The basic situation:
44
Perspective Projection

When we do 3-D graphics, we think of the
screen as a 2-D window onto the 3-D world

The view plane
How tall should
this bunny be?
45
Perspective Projection

The geometry of the situation is that of similar triangles.
View from above:
View
plane
X
P (x, y, z)
x’ = ?
(0,0,0)
Z
d

What is x?
46
Perspective Projection

Desired result for a point [x, y, z, 1]T projected onto
the view plane:
x' x
 ,
d z
dx
x
x' 

,
z
z d

y' y

d z
dy
y
y' 

, zd
z
z d
What could a matrix look like to do this?
47
A Perspective Projection Matrix

Answer:
1
0
Mperspective  
0

0
0
0
1
0
0
1
0 1d
0

0
0

0
48
A Perspective Projection Matrix

Example:
 x  1
 y  0


 z  0

 
 z d  0

0
0
1
0
0
1
0 1d
0  x 



0  y 
0  z 
 
0  1 
Or, in 3-D coordinates:
 x

,
z d

y
, d 
zd

49
A Perspective Projection Matrix

OpenGL’s
gluPerspective()
command
generates a slightly more complicated matrix:
 f
 aspect

 0

 0

 0
where

0
0
f
0
 Ζ far  Z near 


Z Z 
far 
 near
1
0
0
0
0
 2  Z far  Z near

 Z Z
near
far

0









 fov y 

f  cot 
 2 
Can you figure out what this matrix does?
50
Projection Matrices


Now that we can express perspective
foreshortening as a matrix, we can composite it
onto our other matrices with the usual matrix
multiplication
End result: can create a single matrix
encapsulating modeling, viewing, and projection
transforms

Though you will recall that in practice OpenGL
separates the modelview from projection matrix
(why?)
51