2D Geometrical Transformations
•
Translation
- Moves points to new locations by adding translation amounts to the coordinates of the points
.
P’ (x’,y’)
T
.
P(x,y)
 x′   x   dx 
x′ = x + dx, y′ = y + dy or   =   +  
 y′   y   dy 
P′ = P + T
- To translate an object, translate every point of the object by the same amount
(translate only the endpoints of line segments - redrawing is required)
-2•
Scaling
- Changes the size of the object by multiplying the coordinates of the points by
scaling factors
 x′   s x
x′ = x s x , y′ = y s y or   = 
 y′   0
0 x
  , s x ,s y > 0
sy   y 
P′ = S P
- Scale factors affect size as following:
* If s x = s y uniform scaling
* If s x ≠ s y nonuniform scaling
* If s x , s y < 1, size is reduced, object moves closer to origin
* If s x , s y > 1, size is increased, object moves further from origin
* If s x = s y = 1, size does not change
-3- Control the location of a scaled object by choosing the location of a point
(fixed point) with respect to which the scaling is performed
P2
P1
.
P3
(Xf,Yf)
x′ = x f + (x − x f )s x or x′ = xs x + x f (1 − s x )
y′ = y f + (y − y f )s y or y′ = ys y + y f (1 − s y )
 x′   s x
 =
 y′   0
0   x   x f (1 − s x ) 

 +
s y   y   y f (1 − s y ) 
P′ = S P + T f
-4•
Rotation
- Rotates points by an angle
about origin ( >0: counterclockwise rotation)
- From ABP triangle:
cos( ) = x/r or x = rcos( )
sin( ) = y/r or y = rsin( )
- From ACP′ triangle:
cos( + ) = x′/r or x′ = rcos( + ) = rcos( )cos( ) − rsin( )sin( )
sin( + ) = y′/r or y′ = rsin( + ) = rcos( )sin( ) + rsin( )cos( )
- From the above equations we have:
x′ = xcos( ) − ysin( ), y′ = xsin( ) + ycos( ) or
 x′   cos( ) −sin( )   x 
 =
 
y′
sin(
)
cos(
)
  
 y
P′ = R P
- To rotate an object, rotate every point of the object by the same amount
-5(rotate only the endpoints of line segments - redrawing is required)
- Performing rotation about an arbitrary point
.
P’(x’,y’)
.
r
.
P(x,y)
r
Pr (Xr,Yr)
x′ = x r + (x − x r )cos( ) − (y − y r )sin( )
y′ = y r + (x − x r )sin( ) + (y − y r )cos( )
 x′   cos( ) −sin( )   x − x r   x r 
+ 
 =

y
−
y
y′
sin(
)
cos(
)
r
  yr 
  

P′ = R(P − P r ) + P r
•
Summary of transformations
Translation: P′ = P + T (addition causes problems !!)
Scale: P′ = S P
Rotation: P′ = R P
- Idea: use homogeneous coordinates to express translation as matrix multiplication
-6•
Homogeneous coordinates (projective space)
- Idea: add a third coordinate: (x, y) --> (x h , y h , w)
- Homogenize (x h , y h , w):
x=
yh
xh
, y=
,w≠0
w
w
- In general: (x, y) --> (xw, yw, w) (i.e., x h =xw, y h =yw)
- w can assume any value (w ≠ 0), for example, w = 1:
(x, y) − − > (x, y, 1) (no division is required when you homogenize !!)
(x, y) − − > (2x, 2y, 2) (division is required when you homogenize !!)
- (x, y) can be represented by an infinite number of points in homogeneous
coordinates
If w = 6, (1/3, 1/2) − − > (2, 3, 6)
If w = 12, (1/3, 1/2) − − > (4, 6, 12)
...
- All these points lie on a line in the space of homogeneous coordinates !!
-7•
Translation using homogeneous coordinates
 x′   1
 y′  =  0
  
 1  0
0
1
0
dx  x
 
dy   y 

1  1
(Verification: x′ = 1x + 0y + 1dx = x + dx)
(Verification: y′ = 0x + 1y + 1dy = y + dy)
(Homogenize: divide by 1 !!)
P′ = T (dx, dy) P
- Successive translations
.
P’’ (x’’,y’’)
.
P’ (x’,y’)
T
.
P(x,y)
P′ = T (dx 1 , dy 1 ) P,
P′′ = T (dx 2 , dy 2 ) P′
Thus, P′′ = T (dx 2 , dy 2 )T (dx 1 , dy 1 ) P = T (dx 1 + dx 2 , dy 1 + dy 2 ) P
1
0

0
0
1
0
dx 2   1
dy 2   0

1  0
0
1
0
dx 1   1
dy 1  =  0
 
1  0
0
1
0
dx 1 + dx 2 
dy 1 + dy 2 

1

-8•
Scaling using homogeneous coordinates
 x′   s x
 y′  =  0
  
1 0
0
sy
0
0 x
 
0  y

1 1
(Verify: x′ = s x x + 0y + 01 = s x x)
(Verify: y′ = 0x + s y y + 01 = s y y)
(homogenize: divide by 1 !!)
P′ = S(s x , s y ) P
- Successive scalings
P′′ = S(s x 2 , s y2 ) P′
P′ = S(s x 1 , s y1 ) P,
Thus, P′′ = S(s x 2 , s y2 )S(s x 1 , s y1 ) P = S(s x 1 s x 2 , s y1 s y2 ) P
 s x2
 0

 0
0
s y2
0
0   s x1
0  0

1  0
0   s x2 s x1
0 =  0
 
1  0
0
s y1
0
0
s y2 s y1
0
- Scaling about a fixed point
 x′   s x
 y′  =  0
  
1 0
x f (1 − s x )  x
 
y f (1 − s y )   y 
 
1
 1
0
sy
0
- Decomposing the above transformation:
1
0

0
0
1
0
xf
yf
1
  sx
 0

 0
0
sy
0
0 1
0 0

1 0
0
1
0
−x f
−y f
1




0
0

1
-9-
•
Rotation using homogeneous coordinates
 x′   cos( ) −sin( ) 0   x 
 y′  =  sin( ) cos( ) 0   y 

  
0
1 1
1  0
P′ = R( ) P
- Successive rotations
P′ = R( 1 ) P, P′′ = R( 2 ) P′
Thus, P′′ = R( 1 )R( 2 ) P = R(
1
+
2)
P
- Rotation about an arbitrary point
 x′   cos( ) −sin( )
 y′  =  sin( ) cos( )
  
0
1  0
x r (1 − cos( )) + y r sin( )  x
 
y r (1 − cos( )) − x r sin( )   y 
 
1
 1
- Decomposing the above transformation:
1
0

0
0
1
0
x r   cos( ) −sin( ) 0   1
y r   sin( ) cos( ) 0   0


0
1 0
1  0
0
1
0
−x r 
−y r 

1 
-10-
•
Composition of a series of transformations
- The transformation matrices of a series of transformations can be concatenated
into a single transformation matrix
- Example
* Translate P 1 to origin
* Perform scaling and rotation
* Translate to P 2
M = T (x 2 , y 2 )R( )S(s x , s y )T (−x 1 , − y 1 )
- It is important to reserve the order in which a sequence of transformations is
performed !!
-11-
•
General form of transformation matrix
 x′   a11
 y′  =  a
   21
1  0
a12
a22
0
a13  x
 
a23   y 

1  1
- Representing a sequence of transformations as a single transformation matrix
is more efficient
x′ = a11 x + a12 y + a13
y′ = a21 x + a22 y + a23
only 4 multiplications and 4 additions
•
Similarity transformations
- Involve rotation, translation, scaling
•
Rigid-body transformations
- Involve only translations and rotations
- Preserve angles and lengths
- General form of a rigid-body transformation matrix
 x′   r 11
 y′  =  r
   21
1  0
r 12
r 22
0
tx  x
 
ty   y 

1  1
-12- Properties
* The upper 2x2 submatrix is ortonormal
u1 = (r 11 , r 12 ), u2 = (r 21 , r 22 )
u1 . u1 = ||u1 ||2 = r 11 2 + r 12 2 = 1
u2 . u2 = ||u2 ||2 = r 21 2 + r 22 2 = 1
u1 . u2 = r 11 r 21 + r 12 r 22 = 0
- Example:
 cos( ) −sin( ) 0 
 sin( ) cos( ) 0 


0
1
 0
u1 . u1 = cos( )2 + sin(− )2 = 1
u2 . u2 = cos( )2 + sin( )2 = 1
u1 . u2 = cos( )sin( ) − sin( )cos( ) = 0
•
Affine transformations
- Involve translations, rotations, scale, and shear
- Preserve parallelism of lines but not lengths and angles
-13•
Shear transformations
- Changes the shape of the object.
- Shear along the x-axis:
x′ = x + ay, y′ = y
1
SH x =  0

0
a
1
0
0
0

1
- Shear along the y-axis:
x′ = x, y′ = bx + y
1
SH y =  b

0
0
1
0
0
0

1