3. 2D Translations.
2D Scaling
2D Rotation
2D shearing
Matrix Representation of 2D transformation
Formula for Transformation
Matrix of Transformed object= Matrix of object in 2D * Matrix of transformation
3
4. 4
x y
x
y
Point defined as ( , ),
translate to Point ( , ) a distance d parallel to x axis, d parallel to y axis.
, ,
Now
x y
P P x y
P x y
x x d y y d
dx x
P P T
dy y
P P T
′ ′ ′
′ ′= + = +
′
′= = = ′
′ = +
P
P’
Original Point, Transformed point and Displacement in X and
Y can be written in the form of matrix
5. 5
Component-wise addition
v’ = v + t where
and x’ = x + dx
y’ = y + dy
To move polygons: translate vertices of polygon after adding translation factors
and redraw lines between them.
Translation always Preserves lengths and shape of object (Distortion in object
shape will not occur).
dx = 2
dy = 3
Y
X
0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
=
=
=
dy
dx
t
y
x
v
y
x
v ,
'
'
',
3
3
6
5
7. After applying Scaling factor coordinates of
object is either increase or decrease. Expansion
and compression of object is depend upon the
scaling factor Sx and Sy.
Uniform Scaling
If Sx = Sy < 1 Uniform compression occurs.
If Sx = Sy > 1 Uniform Expansion Will occur.
If Sx=Sy=1 No Change will occur.
Non uniform scaling
If Sx ≠ Sy < 1 Non uniform compression occurs.
If Sx ≠ Sy > 1 Non uniform expansion Will
occur.
7
9. 9
Component-wise scalar multiplication of vectors
v’ = Sv where
and
=
=
'
'
',
y
x
v
y
x
v
=
y
x
s
s
S
0
0
ysy
xsx
y
x
=
=
'
'
Y
X
0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
2
3
=
=
y
x
s
s
1
2
1
3
2
6
2
9
10. Add a 3rd coordinate to every 2D point. A point (a, b) in 2D can
be represented as (a,b,1) in homogeneous coordinate system.
Any point (x, y, w) where w!=0 represents a point at location
(x/w, y/w) in this coordinate system.
Point (2/3,4/3) can be represented in homogeneous coordinate
system (6,12,9)
2D case
3D Case
• Add a 4th coordinate to every 3D point. A point (a, b, c) in 3D can be
represented as (a,b,c,1) in homogeneous coordinate system.
Any point (x, y, z, w) in 3D where w!=0 represents a point at location
(x/w, y/w, z/w) in this coordinate system.
12. Object coordinates are (00,10,11,01) Perform uniform expansion of this
object. Take scaling factor from your side.Draw final Figure also.
12
13. Q1. Write a 2X2 transformation matrix for each
of the following scaling transformation.
(1)The entire picture is 3 times as large.
(2)The entire picture is 1/3 as large.
(3)The X direction is 4 times as large and the y
direction unchanged.
(4) The x direction reduced to ¾ the original
and y direction increased by 7/5 times.
13
14. Translate the square ABCD whose coordinates are A(0,0), B(3,0),c(3,3),
D(0,3).
Translate this square 1.5 unit in x direction and 0.5 unit in y direction.
After translation performing scaling with scaling factor sx=1.5 and
sy=0.5
14
15. Find the transformation matrix that transforms
the square ABCD whose center is at (2,2) is
reduced to half of its size, with center still
remaining at (2,2). The coordinate of square
ABCD are A(0,0), B(0,4), C(4,4),D(4,0). Find the
coordinates of new square.
15
16. Q: How can we represent translation as a
3x3 matrix?
y
x
tyy
txx
+=
+=
'
'
=
1
010
001
tytx
ranslationT
18. Prove that two scaling transformation is
commutative
Commutative property is
S1.s2=s2.s1
Prove that two successive translation are additive.
Prove that two successive scaling is multiplicative
18
19. Composite 2D Translation (Two successive
Translation)
),(
),(),(
2121
2211
yyxx
yxyx
tttt
ttttT
++=
⋅=
T
TT
+
+
=
⋅
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Two Successive Translation is Additive
20. Composite 2D Scaling (two successive
Scaling)
),(
),(),(
2121
2211
yyxx
yxyx
ssss
ssssT
S
SS
=
⋅=
⋅
⋅
=
⋅
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
Two Successive Scaling is Multiplicative
25. Rotate a point (10,0) in anticlockwise direction
Angle is 90 degree.
After rotating this point rotate this point in
clockwise direction.
25
−
=
′
′
θθ
θθ
cossin
sincos
y
x
y
x
−
=
′
′
θθ
θθ
cossin
sincos
y
x
y
x
Rotation (Clockwise)Rotation (Anti-Clockwise)
26. Example
Find the transformed point, P’, caused by
rotating P= (5, 1) about the origin through an
angle of 90°.
⋅+⋅
⋅−⋅
=
•
−
θθ
θθ
θθ
θθ
cossin
sincos
cossin
sincos
yx
yx
y
x
⋅+⋅
⋅−⋅
=
90cos190sin5
90sin190cos5
⋅+⋅
⋅−⋅
=
0115
1105
−
=
5
1
28. What happens when you apply a rotation
transformation to an object that is not at the
origin?
Solution:
Translate the center of rotation to the origin
Rotate the object
Translate back to the original location
29. In matrix form, it can be shown as
Here
TR is Translation Matrix (Translation towards Origin)
RƟ is Rotation Matrix (Rotation Matrix can be clockwise and
anticlockwise Rotation)
TR
-1
is Translation matrix (Away from origin)
29
[ ] [ ][ ][ ]
1
R RT T R Tθ
−
=
31. Consider the square A(1,0) B(0,0) C(0,1) D(1,1).
Rotate the square ABCD by 90 degree
clockwise about A(1,0). Apply rotation about
arbitrary point to solve this question.
31
32. Translation.
P′=T + P
Scale
P′=S ⋅ P
Rotation
P′=R ⋅ P
We would like all transformations to be
multiplications
32
34. 34
Shearing Transformation
The shearing transformation when applied to the object it results
distortion of shape.
Types of Shearing Transformation
X- shear: In X-shear y coordinate remain unchanged, but x is
changed.
Y- shear: In Y-shear x coordinate remain unchanged, but y is
changed
' 1
' 0 1
x x Shy
y y
=
X- Shear Y- Shear
' 1 0
' 1
x x
y y shx
=
35. Shear following object 2 unit in x direction and
2 unit in y direction.
Object coordinates are (00,10,11,01).
35
36. Basic 2D transformations as 3x3 matrices
ΘΘ
Θ−Θ
=
1100
0cossin
0sincos
1
'
'
y
x
y
x
=
1100
10
01
1
'
'
y
x
t
t
y
x
y
x
=
1100
01
01
1
'
'
y
x
sh
sh
y
x
x
y
Translate
Rotate Shear
=
1100
00
00
1
'
'
y
x
s
s
y
x
y
x
Scale
37. Transformations can be combined by
matrix multiplication
ΘΘ
Θ−Θ
=
w
y
x
sy
sx
ty
tx
w
y
x
100
00
00
100
0cossin
0sincos
100
10
01
'
'
'
p’ = T(tx,ty) R(Θ) S(sx,sy) p
38. Rotation with respect to a pivot point (x,y)
* ( , ) ( ) ( , )
1 0 cos sin 0 1 0
* 0 1 sin cos 0 0 1
0 0 1 0 0 1 0 0 1
Object T x y R T x y
x x
Object y y
θ
θ θ
θ θ
− − × ×
− −
÷ ÷ ÷
= − × × ÷ ÷ ÷
÷ ÷ ÷
39. Steps for Fix point Scaling
Translate point to origin (Fig (b))
Perform Scaling Fig(c) Expansion or
compression
Inverse Translation Fig(d)
39
40. Scaling with respect to a fixed point (x,y)
* ( , ) ( , ) ( , )
1 0 0 0 0 1 0 0
* 0 1 0 0 0 0 1 0
1 0 0 1 1
x y
x
y
Object T x y S s s T x y
s
Object s
x y x y
− − × ×
÷ ÷ ÷
= × × ÷ ÷ ÷
÷ ÷ ÷− −
41. Q1. Magnify the triangle with vertices A(0,0),
B(1,1), C(5,2) to twice its size while keeping
c(5,2) fixed.
41
