Affine Transformations Mô hình hoá - Modelling Ví dụ Phép biến đổi ...

Khoa CNTT - DDHBK Hà nội
hunglt@it-hut.edu.vn
8682595
Identity as a Coordinate Transform
Translation
x
u
(1,1)
x
u’
(1,1)
x
u
(1,1)
⎡1
0 tx⎤
Q =
⎢
0 1
⎥
⎢
ty
⎥
P
⎢⎣
0 0 1 ⎥⎦
x
u’
(1+tx,1+ty)
⎡1
Q =
⎢
⎢
0
⎢⎣
0
0
1
0
v
0⎤
0
⎥
⎥
P
1⎥⎦
y
v’ y
(c) SE/FIT/HUT 2002 31
⎡1
⎢
⎢
0
⎢⎣
0
0 tx⎤⎡0⎤
⎡tx⎤
1 ty
⎥⎢
⎥
=
⎢ ⎥
⎥⎢
0
⎥ ⎢
ty
⎥
0 1 ⎥⎦
⎢⎣
1⎥⎦
⎢⎣
1 ⎥⎦
v y
⎡tx⎤
⎢ ⎥
⎢ ⎥
⎢⎣
1 ⎥⎦
⎡1
⎢
⎢
0
⎢⎣
0
0 tx⎤⎡1⎤
⎡ 1+
tx⎤
1 ty
⎥⎢
⎥
=
⎢ ⎥
⎥⎢
0
⎥ ⎢
ty
⎥
0 1 ⎥⎦
⎢⎣
1⎥⎦
⎢⎣
1 ⎥⎦
⎡1
⎢
⎢
0
⎢⎣
0
v’
y
0 tx⎤⎡0⎤
⎡ tx ⎤
1 ty
⎥⎢
⎥
=
⎢ ⎥
⎥⎢
1
⎥ ⎢
1+
ty
⎥
0 1 ⎥⎦
⎢⎣
1⎥⎦
⎢⎣
1 ⎥⎦
origin
O = ty
v (1, 0, 0) u (0, 1, 0)
(c) SE/FIT/HUT 2002 32
Rotation
x
u
(1,1)
u’
x
⎡0⎤
O =
⎢ ⎥
⎢
0
⎥
⎢⎣
1⎥⎦
Scaling
x
u
(1,1)
x
u
(sx*1,sy*1)
v’
v
⎡cosθ
− sinθ
Q =
⎢
⎢
sinθ
cosθ
⎢⎣
0 0
y
0⎤
0
⎥
⎥
P
1⎥⎦
y
⎡cosθ
⎤ ⎡− sinθ
⎤
v =
⎢ ⎥
⎢
sinθ
⎥
u =
⎢ ⎥
⎢
cosθ
⎥
⎢⎣
1 ⎥⎦
⎢⎣
1 ⎥⎦
(c) SE/FIT/HUT 2002 33
⎡sx
0
Q =
⎢
⎢
0 sy
⎢⎣
0 0
v
0⎤
0
⎥
⎥
P
1⎥⎦
y
y v
⎡0⎤
⎡sx⎤
⎡ 0 ⎤
=
⎢ ⎥
⎢ ⎥
O
⎢
0
⎥
v =
⎢ ⎥ u =
⎢
sy
⎢
0
⎥
⎥
⎢⎣
1⎥
⎢ ⎥
⎦ ⎢ ⎥ ⎣ 1
⎣ 1 ⎦
⎦
(c) SE/FIT/HUT 2002 34
Composite Transformations
x
u
(1,1)
v y
x
O =
u’
v
’
y
v =
⎡x1(1
− cosθ
) + y1
sinθ
⎤
⎢
⎥
⎢
y1(1
− cosθ
) − y1
sinθ
⎥
⎢⎣
1 ⎥⎦
⎡cosθ
+ x1(1
−cosθ
) + y1
sinθ
⎤
⎢
⎥
⎢
sinθ
+ y1
(1 −cosθ
) − y1
sinθ
⎥
⎢⎣
1 ⎥⎦
Modeling Transformations
• To make full use of the computational optimisation made
possible by composite transforms, we only want to apply the
transformations to points at the very end
• i.e. the transformation operation (multiplying point p by
transform matrix is the very last thing we do in the modelling
phase)
⎛cosθ
− sinθ
⎜
M = ⎜ sinθ
cosθ
⎜
⎝ 0 0
x1
(1 − cosθ
) + y1
sinθ
⎞
⎟
y1(1
− cosθ
) − x1
sinθ
⎟
1 ⎟
⎠
u =
⎡−sinθ
+ x1
(1 −cosθ
) + y1
sinθ
⎤
⎢
⎥
⎢
cosθ
+ y1
(1 −cosθ
) − y1
sinθ
⎥
⎢⎣
1
⎥⎦
Specify points
in local coords
Specify
Transformations
(composite if necessary)
Send to
Pipeline
(c) SE/FIT/HUT 2002 35
(c) SE/FIT/HUT 2002 36
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