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2d/3D transformations in computer graphics(Computer graphics Tutorials)
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12. Scaling and Rotation
Translation
How to uniformly treat transformations?
=
′
′
y
x
dc
ba
y
x
Observation
=
′
′
1
10
01
y
x
f
e
y
x
13. 1
( , , ) ( , , 1) ( / , / ) for 0
( , , ) ( / , / ,1) , 0
T T T
T T
P AP
a b c x
P d e f y
g h i
P x y w x w y w x w y w w
x y w x w y w Rα α α
′ =
′ =
′ ′ ′ ′ ′ ′ ′≡ ⇔ ≡ ≠
′ ′ ′ ′⇔ ∀ ∈ ≠
3. Homogeneous Coordinate System
w
x
y
( , , )T
x y wα ′ ′
1=w
( , ,1)T
x w y w′ ′
(0,0,1)T
14. ( , ) ( , ,1) ( , , )T T T
x y x y wx wy w= ⇔
0≠w
normalized(standard)
No unique homogeneous representation of a Cartesian point !!!
15. What if ? ( , ,0)T
x y0=w
lim lim( , ,1)
lim( , ,1 )
( , ,0)
T
t
t t
T
t
T
p at bt
a b t
a b
→∞ →∞
→∞
=
=
=
( , ,1)T
tp at bt=
w
• Ideal points !!!
points at infinity
well, ...
( , ,1)T
a b(0,0,1)T
0 0
0 0
1 1 1 0 1
1
t t
a a at
p s t b p t b bt
s t
÷ ÷ ÷ ÷ ÷
= + ⇒ = + = ÷ ÷ ÷ ÷ ÷
÷ ÷ ÷ ÷ ÷
+ =
16. Projective Space
Points in the Euclidean space + Points at infinity
Positions Vectors
Positions and vectors are treated homogeneously!!!
( , , ) , 0T
x y w w ≠ ( , , ) , 0T
x y w w =
17. 0 0
0 0
lim lim( , ,1)
lim( , ,1 )
( , ,0)
T
t
t t
T
t
p at x bt y
a x t b y t t
a b
→∞ →∞
→∞
′ = + +
= + +
=
0 0( , ,1)T
tp at x bt y′ = + +
0 0( , ,1)T
x y
( , ,1)T
tp at bt=
(0,0,1)T
( , ,1)T
a b
lim lim( , ,1) lim( , ,1 ) ( , ,0)T T T
t
t t t
p at bt a b t a b
→∞ →∞ →∞
= = =
18. In a Euclidean space two lines intersect each other
If they are not parallel
Non-homogeneous treatment !!!∴
However, in a projective space,
two lines do always intersects.
∴ Homogeneous treatment !!!
24. ( , )T
F Fx y
(a) Original position of
object and fixed point
(b) Translate object so that
fixed point (xF, yF) is at origin
(c) Scale object with
respect to origin
(d) Translate object so that
fixed point is returned to position (xF, yF)T
( , )T
F Fx y
][ rT
][ cS 1
][ −
rT
25. ( , )T
R Rx y
(a) Original position of
object and pivot point
(b) Translation of object so that
the pivot point (xR, yR) is at origin
(c) Rotation about origin (d) Translation of object so that
the pivot point is returned to
position (xR, yR)T
( , )T
R Rx y
][ rT
][ tR 1
][ −
rT