2D Transformations
2D Transformations
•
•
•
•
•
•
•
World Coordinates
Translate
Rotate
Scale
Viewport Transforms
Hierarchical Model Transforms
Putting it all together
Transformations
• Rigid Body Transformations - transformations that do
not change the object.
• Translate
– If you translate a rectangle, it is still a rectangle
• Scale
– If you scale a rectangle, it is still a rectangle
• Rotate
– If you rotate a rectangle, it is still a rectangle
Vertices
• We have always represented vertices as
(x,y)
• An alternate method is:
 x
( x, y )   
 y
• Example:
 2.1
(2.1,4.8)   
4.8
Matrix * Vector
 x' a b   x 
 y '   c d   y 
  
 
x'  ax  by
 x'  a b c   x 
 y '   d e f   y 
  
 
 z '   g h i   z 
x'  ax  by  cz
y '  cx  dy
y '  dx  ey  fz
z '  gx  hy  iz
1 0
I 

0
1


1 0 0 


I  0 1 0 
0 0 1
Matrix * Matrix
x y 
a b 
,B  
A


w
z
d
c




ax  bz ay  bw
A* B  

dw

cy
dz

cx


 a b  1 0 
?
 0 1 
d
c



Does A*B = B*A? NO
What does the identity do?
AI=A
Translation
• Translation - repositioning an object along a
straight-line path (the translation distance)
from one coordinate location to another.
(x’,y’)
(tx,ty)
(x,y)
Translation
• Given:
• We want:
• Matrix form:
P  ( x, y )
T  (t x , t y )
x'  x  t x
y'  y  t y
P  (3.7,4.1)
T  (7.1,8.2)
x '  3 . 7  7 .1
y '  4.1  8.2
 x'  3.7   7.1
 x'  x  t x   y '    4.1  8.2
  
 y '   y   t    
     y  x '  3.4
P'  P  T
y '  4 .1
Translation Examples
• P=(2,4), T=(-1,14), P’=(?,?)
• P=(8.6,-1), T=(0.4,-0.2), P’=(?,?)
• P=(0,0), T=(1,0), P’=(?,?)
Recall
• A point is a position specified with
coordinate values in some reference frame.
• We usually label a point in this reference
point as the origin.
• All points in the reference frame are given
with respect to the origin.
Applying to Triangles
(tx,ty)
What do we have here?
• You know how to:
Scale
• Scale - Alters the size of an object.
• Scales about a fixed point
(x’,y’)
(x,y)
Scale
• Given:
• We want:
• Matrix
P  ( x, y )
P  (1.4,2.2)
S  (sx , s y )
S  (3,3)
x'  s x x
x '  3 * 1 .4
y'  s y y
y '  3 * 2 .2
 x'  s x
 y '   0
form:   
P'  S  P
 x' 3 0 1.4 
 y '  0 3 2.2
0  x   
 
s y   y  x'  4.2
y '  6 .6
Non-Uniform Scale
(x’,y’)
(x,y)
S=(1,2)
Rotation
• Rotation - repositions an object along a
circular path.
• Rotation requires an  and a pivot point

Rotation
p  ( x, y )
R  ( )
x  r cos 
y  r sin 
x'  r cos(  )
y '  r sin(   )
x'  r cos  cos   r sin  sin 
y '  r cos  sin   r sin  cos 
x'  x cos   y sin 
y '  x sin   y cos 
 x' cos 
 y '   sin 
  
p '  Rp
 sin    x 
cos    y 
Example
• P=(4,4)
• =45 degrees
Rotations
V(-0.6,0) V(0,-0.6) V(0.6,0.6)
Rotate -30 degrees
V(0,0.6) V(0.3,0.9) V(0,1.2)
Combining Transformations
Q: How do we
specify each
transformation?
Specifying 2D Transformations
• Translation
– T(tx, ty)
– Translation distances
• Scale
– S(sx,sy)
– Scale factors
• Rotation
– R()
– Rotation angle
Combining Transformations
• Using translate, rotation, and scale, how do
we get:
Combining Transformations
• Note there are two ways to combine rotation
and translation. Why?
Let’s look at the equations
P ( x, y )
T (t x , t y )
R( )
P'  P  T
P' '  R  P'
x'  x  t x
y'  y  t y
x"  x' cos   y ' sin 
y"  x' sin   y ' cos 
x"   x  t x  cos    y  t y ' sin 
y"   x  t x sin    y  t y ' cos 
P'  R  P
P"  P 'T
x'  x cos   y sin 
y '  x sin   y cos 
x"   x cos   y sin    t x
y"   x sin   y cos    t y
Combining them
• We must do each step in turn. First we
rotate the points, then we translate, etc.
• Since we can represent the transformations
by matrices, why don’t we just combine
them?
P'  P  T
P'  R  P
P'  S  P
2x2 -> 3x3 Matrices
• We can combine transformations by
expanding from 2x2 to 3x3 matrices.
1 0 t x 
 x  t x 
T t x , t y         0 1 t y 
 y  t y  0 0 1 


 s x 0 0
s
0
 x
 

S s x , s y   

0
s
0
 
y

0
s
y

  0 0 1


cos   sin 
cos   sin   
R   
  sin  cos 

 sin  cos    0
0

0
0
1
Homogenous Coordinates
• We need to do something to the vertices
• By increasing the dimensionality of the
problem we can transform the addition
component of Translation into
multiplication.
xh 
6


x

3


h
2
4
6 
 xh  


 x  
3    14 
y h   4  
P      yh    y   Ex.   2, Ex.   14  7  
7
h   2
2

 y  h  

1
 2   2 
   h 
 h 
 2 
Homogenous Coordinates
• Homogenous Coordinates - term used in
mathematics to refer to the effect of this
representation on Cartesian equations. Converting
a pt(x,y) and f(x,y)=0 -> (xh,yh,h) then in
homogenous equations mean (v*xh,v*yh,v*h) can
be factored out.
• What you should get: By expressing the
transformations with homogenous equations and
coordinates, all transformations can be expressed
as matrix multiplications.
Final Transformations Compare Equations
 x' 1 0 t x   x 
 x '  t x   x    


T t x , t y            y '  0 1 t y   y 
 y ' t y   y   1  0 0 1   1 
  
 
P  T t x , t y   P  P  T t x , t y  P
 x'  s x
 x'  s x 0   x    
S s x , s y      
  y '   0



 y '  0 s y   y   1   0
  
P  S s x , s y  P  P  S s x , s y  P
0
sy
0
 x' cos 
 x' cos   sin    x    
R      
  y '   sin 



 y '  sin  cos    y   1   0
  
P  R   P  P  R   P
0  x 
0  y 
1  1 
 sin 
cos 
0
0  x 
0  y 
1  1 
Combining Transformations
P'  A  P, P"  B  P'  P"  A  B  P
P(3,4), T (0.4641,2), R(60)
 x' 1 0 t x   x 
 y '  0 1 t   y 
y  
  
 1  0 0 1   1 
 x" cos   sin 
 y"   sin  cos 
  
 1   0
0
 x" cos 
 y"   sin 
  
 1   0
 sin 
cos 
0
0  x ' 
0  y '
1  1 
0 1 0 t x   x'
0 0 1 t y   y '
1 0 0 1   1 
 x" cos   sin  t x cos   t y sin    x 
 y"   sin  cos  t sin   t cos   y 
x
y
  
 
 1   0
  1 
0
1
 x" 1 0 t x  cos   sin  0  x 
 y"  0 1 t   sin  cos 0  y 
y 
  
 
 1  0 0 1   0
0
1  1 
 x" cos   sin  t x   x 
 y"   sin  cos  t   y 
y  
  
 1   0
0
1   1 
How would we get:
How would we get:
Coordinate Systems
• Object Coordinates
• World Coordinates
• Eye Coordinates
Object Coordinates
World Coordinates
Screen Coordinates
Coordinate Hierarchy
Screen Coordinates
Transformation
World->Screen
World Coordinates
Transformation
Object #1 ->
World
Transformation
Object #2 ->
World
Transformation
Object #3 ->
World
Object #1
Object Coordinates
Object #2
Object Coordinates
Object #3
Object Coordinates
Let’s reexamine assignment 1
Transformation Hierarchies
• (See chapter 10 for details)
• For example, a robot arm
Transformation Hierarchies
• Let’s examine:
Transformation Hierarchies
• What is a better way?
Transformation Hierarchies
• What is a better way?
World
Coordinates
Transformation Hierarchies
• We can have transformations be
in relation to each other
Transformation:
Upper Arm -> World
Upper Arm
Coordinates
Transformation:
Lower -> Upper
Lower Arm
Coordinates
Transformation:
Hand-> Lower
Hand
Coordinates
Rotation about a Fixed Point
Start with identity matrix: C  I
Move fixed point to origin: C  CT
Rotate: C  CR
Move fixed point back: C  CT -1
Result: C = TR T –1 which is backwards – Cp
This result is a consequence of doing postmultiplications.
Let’s try again.
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Angel: Interactive Computer
Graphics 5E © Addison-Wesley 2009
Reversing the Order
We want C = T –1 R T
so we must do the operations in the following order
CI
C  CT -1
C  CR
C  CT
Each operation corresponds to one function call in the
program.
Note that the last operation specified is the first executed in
the program
Angel: Interactive Computer
45
Graphics 5E © Addison-Wesley 2009
OpenGL Example
• Rotation about z axis by 30 degrees with a fixed
point of (1.0, 2.0, 3.0)
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
glTranslatef(1.0, 2.0, 3.0);
glRotatef(30.0, 0.0, 0.0, 1.0);
glTranslatef(-1.0, -2.0, -3.0);
glBegin(GL_TRIANGLES);
...
• Remember that last transform specified in the
program is the first applied
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Angel: Interactive Computer
Graphics 5E © Addison-Wesley 2009
Matrix Stacks
• In many situations we want to save
transformation matrices for use later
– Traversing hierarchical data structures (Chapter 10)
– Avoiding state changes when executing display lists
• OpenGL maintains stacks for each type of
matrix
– Access present type (as set by glMatrixMode)
by glPushMatrix()
glPopMatrix()
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Angel: Interactive Computer
Graphics 5E © Addison-Wesley 2009