Matrices can be used to represent 2D geometric transformations such as translation, scaling, and rotation. Translations are represented by adding a translation vector to coordinates. Scaling is represented by multiplying coordinates by scaling factors. Rotations are represented by premultiplying coordinates with a rotation matrix. Multiple transformations can be combined by multiplying their respective matrices. Using homogeneous coordinates allows all transformations to be uniformly represented as matrix multiplications.

1. 2D Transformations with Matrices

2. Matrices
 a1,1

A = a2,1
 a3,1

a1, 2
a2 , 2
a3, 2
a1,3 

a2 , 3 
a3,3 

•
A matrix is a rectangular array of numbers.
•
A general matrix will be represented by an upper-case italicised
letter.
•
The element on the ith row and jth column is denoted by ai,j. Note
that we start indexing at 1, whereas C indexes arrays from 0.

3. Matrices – Addition
• Given two matrices A and B if we want to add B to A
(that is form A+B) then if A is (n×m), B must be (n×m),
Otherwise, A+B is not defined.
• The addition produces a result, C = A+B, with elements:
Ci , j = Ai , j + Bi , j
1 2 5 6 1 + 5 2 + 6  6 8 
3 4 + 7 8 = 3 + 7 4 + 8 = 10 12

 
 
 


4. Matrices – Multiplication
• Given two matrices A and B if we want to multiply B by A (that
is form AB) then if A is (n×m), B must be (m×p), i.e., the
number of columns in A must be equal to the number of rows
in B. Otherwise, AB is not defined.
• The multiplication produces a result, C = AB, with elements:
m
(Basically we multiply Ci , j = ∑ aik bkj of A with the first column of
the first row
k =1
B and put this in the c1,1 element of C. And so on…).

5. Matrices – Multiplication (Examples)
2×6+ 6×3+ 7×2=44
2 6 7 6 8 44 76
4 5 8 × 3 3 = 55 95

 
 

9 2 3 2 6 66 96

 
 

6 8
 2 6 
×  3 3
 4 5


  2 6


Undefined!
2x2 x 3x2 2!=3
2x2 x 2x4 x 4x4 is allowed. Result is 2x4 matrix

6. Matrices -- Basics
• Unlike scalar multiplication, AB ≠ BA
• Matrix multiplication distributes over addition:
A(B+C) = AB + AC
•
Identity matrix for multiplication is defined as I.
• The transpose of a matrix, A, is either denoted AT or A’ is
obtained by swapping the rows and columns of A:
 a1,1
A=
a2,1
a1, 2
a2 , 2
 a1,1
a1,3 

⇒ A' = a1, 2
a2 , 3 

 a1,3

a2,1 

a2 , 2 
a2 , 3 


7. 2D Geometrical Transformations
Translate
Rotate
Shear
Scale

8. Translate Points
Recall.. We can translate points in the (x, y) plane to new positions
by adding translation amounts to the coordinates of the points. For
each point P(x, y) to be moved by dx units parallel to the x axis and by dy
units parallel to the y axis, to the new point P’(x’, y’ ). The translation has
the following form:
x' = x + d x
P’(x’,y’)
dy
dx
P(x,y)
y' = y + d y
In matrix format:
 x '   x  d x 
 y ' =  y  + d 
     y
d x 
If we define the translation matrix T =   , then we have P’ =P + T.
d y 

9. Scale Points
Points can be scaled (stretched) by sx along the x axis and by sy
along the y axis into the new points by the multiplications:
We can specify how much bigger or smaller by means of a “scale factor”
To double the size of an object we use a scale factor of 2, to half the size of
an obejct we use a scale factor of 0.5
x' = s x x
P(x,y)
y
s y• y
P’(x’,y’)
s x• x
s x
If we define S = 
0
x
y' = s y y
 x'  s x
 y ' =  0
  
0
, then we have P’ =SP
sy 

0 x 
sy  y
 

10. Rotate Points (cont.)
Points can be rotated through an angle θ about the origin:
| OP' |=| OP |= l
x' =| OP' | cos(α + θ ) = l cos(α + θ )
= l cos α cos θ − l sin α sin θ
= x cos θ − y sin θ
y ' =| OP' | sin(α + θ ) = l sin(α + θ )
= l cos α sin θ + l sin α cos θ
= x sin θ + y cos θ
P’(x’,y’)
y
y’
O
θ
P(x,y)
l
α
x’
 x'  cos θ
 y ' =  sin θ
  
x
− sin θ   x 
cos θ   y 
 
P’ =RP

11. Review…
• Translate:
• Scale:
• Rotate:
P’ = P+T
P’ = SP
P’ = RP
• Spot the odd one out…
– Multiplying versus adding matrix…
– Ideally, all transformations would be the same..
• easier to code
• Solution: Homogeneous Coordinates

12. Homogeneous Coordinates
For a given 2D coordinates (x, y), we introduce a third dimension:
[x, y, 1]
In general, a homogeneous coordinates for a 2D point has the form:
[x, y, W]
Two homogeneous coordinates [x, y, W] and [x’, y’, W’] are said to be of the
same (or equivalent) if
x = kx’
y = ky’
W = kW’
for some k ≠ 0
eg: [2, 3, 6] = [4, 6, 12]
where k=2
Therefore any [x, y, W] can be normalised by dividing each element by W:
[x/W, y/W, 1]

13. Homogeneous Transformations
Now, redefine the translation by using homogeneous coordinates:
x '  1
 y ' = 0
  
1  0
  
x '  x  d x 
 y ' =  y  + d 
     y
0
1
0
d x x 
d y  y 
 
1 1 
 
P' = T × P
Similarly, we have:
Scaling
x'  s x
 y ' =  0
  
1   0
  
P’
=
Rotation
0
0 x 
0  y 
 
11 
 
S
×
0
sy
P
x'  cos θ
 y ' = sin θ
  
1   0
  
P’
=
− sin θ
cos θ
0 x 
0  y 
 
11 
 
0
R
×
P

14. Composition of 2D Transformations
1. Additivity of successive translations
We want to translate a point P to P’ by T(dx1, dy1) and then to P’’ by
another T(dx2, dy2)
P ' ' = T (d x 2 , d y 2 ) P ' = T (d x 2 , d y 2 )[T (d x1 , d y1 ) P ]
On the other hand, we can define T21= T(dx1, dy1) T(dx2, dy2) first, then
apply T21 to P:
P ' ' = T21 P
where
T21 = T ( d x 2 , d y 2 )T (d x1 , d y1 )
1 0 d x 2  1 0 d x1 
=  0 1 d y 2   0 1 d y1 



0 0 1  0 0 1 



1 0 d x1 + d x 2 
=  0 1 d y1 + d y 2 


0 0

1



15. Examples of Composite 2D Transformations
T(-1,2)
(1,3)
T(1,-1)
(2,1)
1
T21 = 0

0

1
= 0

0

0
1
0
0
1
0
1  1 0 − 1
− 1 0 1 2 


1  0 0 1 


0
1

1

(2,2)

16. Composition of 2D Transformations (cont.)
2. Multiplicativity of successive scalings
P ' ' = S ( s x 2 , s y 2 )[ S ( s x1 , s y1 ) P]
= [ S ( s x 2 , s y 2 ) S ( s x1 , s y1 )]P
= S 21 P
where
S 21 = S ( s x 2 , s y 2 ) S ( s x1 , s y1 )
s x 2
= 0

0

0
sy2
0
s x 2 * s x1
= 0

 0

0s x1
0  0

1  0

0
s y 2 * s y1
0
0
s y1
0
0
0

1

0
0

1


17. Composition of 2D Transformations (cont.)
3. Additivity of successive rotations
P ' ' = R (θ 2 )[ R (θ1 ) P ]
= [ R (θ 2 ) R (θ1 )]P
= R21 P
where
R21 = R (θ2 ) R (θ1 )
cos θ2
= sin θ2

 0

− sin θ2
cos θ2
0
cos(θ2 +θ1 )
= sin(θ2 +θ1 )


0

0cos θ1
0sin θ1

1  0

− sin(θ2 +θ1 )
cos(θ2 +θ1 )
0
− sin θ1
cos θ1
0
0
0

1

0
0

1


18. Composition of 2D Transformations (cont.)
4. Different types of elementary transformations discussed above
can be concatenated as well.
P ' = R (θ )[T (d x , d y ) P ]
= [ R (θ )T (d x , d y )]P
= MP
where
M = R (θ )T ( d x , d y )

19. Consider the following two questions:
1) translate a line segment P1 P2, say, by -1 units in the x direction
and -2 units in the y direction.
P2(3,3)
P’2
P1(1,2)
P’1
2). Rotate a line segment P1 P2, say by θ degrees counter clockwise,
about P1.
P’2
P2(3,3)
P1(1,2)
θ
P’1

20. Other Than Point Transformations…
Translate Lines:
translate both endpoints, then join them.
Scale or Rotate Lines: More complex. For example, consider to
rotate an arbitrary line about a point P1,
three steps are needed:
1). Translate such that P1 is at the origin;
2). Rotate;
3). Translate such that the point at the origin
returns to P1.
P2
P2(3,3)
T(-1,-2)
P2(2,1)
P1(1,2)
P1
P2
R(θ)
θ
P1
T(1,2)
P1

21. Another Example.
Translate
Scale
Translate
Rotate

22. Order Matters!
As we said, the order for composition of 2D geometrical transformations matters,
because, in general, matrix multiplication is not commutative. However, it is easy
to show that, in the following four cases, commutativity holds:
1).
Translation + Translation
2).
Scaling + Scaling
3).
Rotation + Rotation
4).
Scaling (with sx = sy) + Rotation
just to verify case 4:
M 1 = S ( s x , s y ) R (θ )
M 2 = R (θ ) S ( s x , s y )
0cos θ − sin θ
s y 0sin θ
cos θ

0 1  0
0

s x * cos θ − s x * sin θ 0
=  s y * sin θ
s y * cos θ 0



0
0
1


s x
= 0

0

0
0
0

1

if sx = sy, M1 = M2.
cos θ
= sin θ

 0

− sin θ
0s x
cos θ 0 0

0
1  0

s x * cos θ − s y * sin θ
=  s x * sin θ
s y * cos θ


0
0

0
sy
0
0
0

1

0
0

1


23. Rigid-Body vs. Affine Transformations
A transformation matrix of the form
r11
r
 21
0

r12
r22
0
tx 
ty 

1

where the upper 2×2 sub-matrix is orthogonal, preserves angles and
lengths. Such transforms are called rigid-body transformations,
because the body or object being transformed is not distorted in any
way. An arbitrary sequence of rotation and translation matrices
creates a matrix of this form.
The product of an arbitrary sequence of rotation, translations, and
scale matrices will cause an affine transformation, which have the
property of preserving parallelism of lines, but not of lengths and
angles.

24. Rigid-Body vs. Affine Transformations (cont.)
Rigid- body
Transformation
Affine
Transformation
45º
Unit cube
Scale in x, not in y
Shear transformation is also affine.
Shear in the x direction
1
SH x = 0

0

a
1
0
0
0

1

1
SH y = b

0

Shear in the y direction
0
1
0
0
0

1
