3 D transformation translation, scaling

1. 3D TRANSFORMATION
(Translation & Scaling)
By:
Arvind Kumar
Assistant Professor
(Vidya College of Engineering)

2. Contents
By: Arvind Kumar
3D
Translation
Scaling

3. 3D Translation
By: Arvind Kumar
3D
Translation of Object
zyx tzztyytxx  ',','





































11000
100
010
001
1
'
'
'
z
y
x
t
t
t
z
y
x
z
y
x
x
z
y

4. 3D Scaling
By: Arvind Kumar
3D
Scaling of object
zyx szzsyysxx  ',','





































11000
000
000
000
1
'
'
'
z
y
x
s
s
s
z
y
x
z
y
x
x
z
y

5. 3D Composite Scaling
By: Arvind Kumar
3D
1. Translate the fixed point to the Origin
2. Scale the Object.
3. Translate the Fixed point back to its position.
Scaling with a Fixed Point
Steps:
Inverse Translate
z
y y y y
Original
position
x x x xzzz
Translate Scaling

6. 3D Composite Scaling
By: Arvind Kumar
3D
Scaling with a Fixed Point

































































11000
100
010
001
1000
000
000
000
1000
100
010
001
1
'
'
'
),,(),,(),,(
z
y
x
t
t
t
s
s
s
t
t
t
z
y
x
tttTsssStttT
z
y
x
z
y
x
z
y
x
zyxzyxzyx
     PP  xyzS

7. 3D Composite Scaling
By: Arvind Kumar
3D
Q1. Given a 3D object with coordinate points A(0, 3, 3),
B(3, 3, 6), C(3, 0, 1), D(0, 0, 0). Apply the scaling
parameter 2 towards X axis, 3 towards Y axis and 3
towards Z axis and obtain the new coordinates of the
object with fixed point D(0, 0, 0).
Solution:
• Coordinates of the object = A (0, 3, 3), B(3, 3, 6), C(3,
0, 1), D(0, 0, 0)
• Scaling factor along X axis = 2
• Scaling factor along Y axis = 3
• Scaling factor along Z axis = 3

8. 3D Composite Scaling
By: Arvind Kumar
3D
Here Fixed point is already origin , so no need to
translate the fixed point.
tx=0 ; ty=0; tz=0
On Putting the values in above matrix

































































11000
100
010
001
1000
000
000
000
1000
100
010
001
1
'
'
'
),,(),,(),,(
z
y
x
t
t
t
s
s
s
t
t
t
z
y
x
tttTsssStttT
z
y
x
z
y
x
z
y
x
zyxzyxzyx

9. 3D Composite Scaling
By: Arvind Kumar
3D
































































1111
0163
0033
0330
1000
0100
0010
0001
1000
0300
0030
0002
1000
0100
0010
0001
1
'
'
'
z
y
x

























1111
03189
0099
0660
1
'
'
'
z
y
x
New coordinates are:
A’(0, 9, 9), B’(6, 9, 18),
C’(6, 0, 3), D’(0, 0, 0)