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CS580 Computer Graphics
22D TransformationsD Transformations
Sung Yong ShinDept. of Computer Science
KAIST
Daroko blog
Daroko blog
Outline
1. Preliminary2. Basic 2D Transformations3. Homogeneous Coordinate system4. Composite Transformations5. Reflection and Shearing
per-vertex operations &primitive assembly rasterization frame buffer
texture assembly
display lists
evaluators
pixel operations
per-fragmentoperations
geometricdata
pixeldata
OpenGL Rendering Pipeline
Window-to-viewportmapping
2. Basic Transformations
translation
scaling
rotation
(absolute) transformation
relative transformation
yx
SS
yx
ySyxSx
y
x
y
x
00
x
y
( , )TF Fx y : fixed point
1p
2p
3p
y
x
( , )TF Fx y
Scaling
( , )Tx y
( ', ')Tx y
(a)
(b)
Non-uniform scaling
sincossinsincoscos
)cos(
yxrr
rx
sincos xyy
Similarly,
( , )Tx y
( , )Tx y
r
r
sincosryrx
Rotation
sincossincosxyyyxx
yx
yx
cossinsincos
( , )TR Rx y
( , )Tx y
( , )Tx y
r
r
( , )Tx y
( , )Tx y
r
r
( , )TR Rx y
: Pivot point
11001
yx
TT
yx
TyyTxx
y
x
y
x
50 100 150
150
100
50
50 100 150
150
100
50
(a)
(b)
Translation
Scaling and Rotation
Translation
How to uniformly treat transformations?
yx
dcba
yx
Observation
11001
yx
fe
yx
1
( , , ) ( , , 1) ( / , / ) for 0
( , , ) ( / , / ,1) , 0
T T T
T T
P APa b c x
P d e f yg h i
P x y w x w y w x w y w w
x y w x w y w R
3. Homogeneous Coordinate System
w
x
y
( , , )Tx y w
1w( , ,1)Tx w y w
(0,0,1)T
( , ) ( , ,1) ( , , )T T Tx y x y wx wy w
0w
normalized(standard)
No unique homogeneous representation of a Cartesian point !!!
What if ? ( , ,0)Tx y0w
lim lim( , ,1)
lim( , ,1 )
( , ,0)
Ttt t
T
t
T
p at bt
a b t
a b
( , ,1)Ttp at bt
w
• Ideal points !!!points at infinity
well, ...
( , ,1)Ta b(0,0,1)T
0 00 01 1 1 0 1
1
t t
a a atp s t b p t b bt
s t
Projective Space
Points in the Euclidean space + Points at infinityPositions Vectors
Positions and vectors are treated homogeneously!!!
( , , ) , 0Tx y w w ( , , ) , 0Tx y w w
0 0
0 0
lim lim( , ,1)
lim( , ,1 )
( , ,0)
Ttt t
T
t
p at x bt y
a x t b y t t
a b
0 0( , ,1)Ttp at x bt y 0 0( , ,1)Tx y
( , ,1)Ttp at bt(0,0,1)T ( , ,1)Ta b
lim lim( , ,1) lim( , ,1 ) ( , ,0)T T Ttt t tp at bt a b t a b
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 !!!
A
B
No orientation !!!BA
However,
Why?
For detail treatment of this subject, see M. Penna and R. Patterson,
Projective Geometry and its Application to Computer Graphics, 1986.
rotationscalingtranslationperspective transform
P 'P
T ' where
' [ ', ', '] ,
[ , , ] .
T
T
P TP
P x y w
P x y w
2D Transformation Matrices
ihgfedcba
T
X 'X
Now, uniform treatment !!! Rotation
Scaling
Translation
11000cossin0sincos
'''
yx
wyx
11000000
'''
yx
SS
wyx
y
x
11001001
'''
yx
TT
wyx
y
x
]][]...[[][
][]][]...[['
12
12
TTTT
PTPTTTP
n
n
composite transformation matrix
Homework
4. Composite Transformations
( , )TF 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
( , )TF Fx y
][ rT
][ cS 1][ rT
( , )TR 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
( , )TR Rx y
][ rT
][ tR 1][ rT
x
y
Sy
Sx
[Rt]-1[Sc][Rt]
Why if Sx = Sy ?
x
y
(0, 0)T (1, 0)T
(1, 1)T
(0, 1)T
x
y
(0, 0)T
(3/2, 1/2)T
(2, 2)T
(1/2, 3/2)T
45°Sx = 1Sy = 2
FinalPosition
FinalPosition
(a) [Rt][Tr] (b) [Tr][Rt]
Reflection
1
2 3
1’
2’ 3’
Originalposition
Reflectedposition
100010001
1
2
3
1’
2’
3’
Originalposition
Reflectedposition
100010001
x
x
y y
5. Reflection and Shearing
12
3
1’
2’ 3’
Originalposition
Reflectedposition
100001010
1
2
3
1’2’
3’
Originalposition
Reflectedposition
100010001
x
x
y yy = x
45°
[Rt] [Rfx]
45°
[Rt]-1
Reflection about y = x
100001010
1 2
3
1’
2’3’
Originalposition
Reflectedposition
x
yy = -x
(a) (b) (c)
(0, 0) (0, 0) (0, 0)(1, 0)
(1, 1)(0, 1)
(1, 0)
(3, 1)(2, 1)(0, 1)
(1, 2)
(1, 3)
x
y
x
y
x
y
10001001 xSH
10001001
ySH
Shearing
(a) (b)
Raster method for transformation
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