computer graphics

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!!