This document discusses 3D transformations in computer graphics. It covers 3D translation, rotation, and scaling. Translation moves an object by adding offsets to the x, y, and z coordinates. Rotation in 3D can occur around any axis and is represented using 4x4 matrices. Scaling enlarges or shrinks an object along the x, y, and z axes by multiplying coordinates by scale factors. More complex transformations can be achieved by combining multiple simple transformations.

1. Computer Graphics
3D Transformations
Total Slides 17

2. From 2D to 3D
• Translation is simple as in 2D
• Use of Homogeneous coordinate in 3D
– In 3D transformation always use Matrices: 4x4
• All transformation in 3D is simple but only
Rotation transformation is complex in 3D
transformation.

3. 3D Translation
Translate using (tx, ty,tz):
x’=x+ tx, y’=y+ ty , z’=z+ tz
or
x
y
P
P+T
T
z










=










=










=
+=
z
y
x
t
t
t
z
y
x
z
y
x
TPP
TPP'
en,
'
'
'
'
met,

4. 3D Translation 2
In 4D homogeneous coordinates:
x
y
P
P+T
T
z
11000
100
010
001
1
'
'
'
or,




























=














=
z
y
x
t
t
t
z
y
x
z
y
x
MPP'

5. 3D Rotation 1
z
P
x
y
P’
α
1000
0100
00cossin
00sincos
)(
with,)(
Or
'
cossin'
sincos'
:as-aroundangleoverRotate













 −
=
=
=
+=
−=
αα
αα
α
α
αα
αα
α
z
z
zz
yxy
yxx
z
R
PRP'

6. 2D Rotation about the origin.
y
x
r
r
P’(x’,y’)
P(x,y)
θ
φ
y
φ
φ
sin.
cos.
ry
rx
=
=
x
θφθφφθ
θφθφφθ
cos.sin.sin.cos.)sin(.
sin.sin.cos.cos.)cos(.
rrry
rrrx
+=+=′
−=+=′

7. 3D Rotation 2
z
x
y
Rotation around axis:
- Counterclockwise, viewed from rotation axis
z
x
y z
x
y

8. 3D Rotation 3
z
x
y
z
x
y z
x
y
zy
yx
xz
→
→
→
xz
zy
yx
→
→
→
Rotation around axes:
Cyclic permutation coordinate axes
xzyx →→→

9. 3D Rotation
zy
yx
xz
→
→
→
1000
0100
00cossin
00sincos
)(
with,)(
Or
'
cossin'
sincos'
:as-aroundangleoverRotate













 −
=
=
=
+=
−=
αα
αα
α
α
αα
αα
α
z
z
zz
yxy
yxx
z
R
PRP'
1000
0cossin0
0sincos0
0001
)(
with,)(
Or
'
cossin'
sincos'
:as-aroundangleoverRotate














−
=
=
=
+=
−=
αα
αα
α
α
αα
αα
α
x
x
xx
zyz
zyy
x
R
PRP'

10. 11000
000
000
000
1
'
'
'
or,




























=














=
z
y
x
s
s
s
z
y
x
z
y
x
SPP'
3D scaling
Scale with factors sx, sy,sz:
x’= sx x, y’= sy y, z’= sz z
or

11. 3D scaling

12. 3D shearing

13. 3D rotation Example

14. 3D rotation Example

15. Combine 3D Transformation

16. 3D Combine Transformations

17. Thank You!!