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Computer Graphics
3D Transformations
Total Slides 17
From 2D to 3D
• Translation is simple as in 2D
• Use of Homogeneous coordinate in 3D
– In 3D transformation always use Mat...
3D Translation
Translate using (tx, ty,tz):
x’=x+ tx, y’=y+ ty , z’=z+ tz
or
x
y
P
P+T
T
z










=




...
3D Translation 2
In 4D homogeneous coordinates:
x
y
P
P+T
T
z
11000
100
010
001
1
'
'
'
or,















...
3D Rotation 1
z
P
x
y
P’
α
1000
0100
00cossin
00sincos
)(
with,)(
Or
'
cossin'
sincos'
:as-aroundangleoverRotate




...
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(...
3D Rotation 2
z
x
y
Rotation around axis:
- Counterclockwise, viewed from rotation axis
z
x
y z
x
y
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
xzy...
3D Rotation
zy
yx
xz
→
→
→
1000
0100
00cossin
00sincos
)(
with,)(
Or
'
cossin'
sincos'
:as-aroundangleoverRotate




...
11000
000
000
000
1
'
'
'
or,




























=














=
z
y
...
3D scaling
3D shearing
3D rotation Example
3D rotation Example
Combine 3D Transformation
3D Combine Transformations
Thank You!!
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3 d transformations

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3 d transformations

  1. 1. Computer Graphics 3D Transformations Total Slides 17
  2. 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. 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. 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. 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. 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. 7. 3D Rotation 2 z x y Rotation around axis: - Counterclockwise, viewed from rotation axis z x y z x y
  8. 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. 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. 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. 11. 3D scaling
  12. 12. 3D shearing
  13. 13. 3D rotation Example
  14. 14. 3D rotation Example
  15. 15. Combine 3D Transformation
  16. 16. 3D Combine Transformations
  17. 17. Thank You!!

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