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Jinxiang Chai
CSCE441: Computer GraphicsCoordinate & Composite
Transformations
1
Outline
2D/3D Coordinate transformation
2D/3D Composite transformation
Required readings: HB 7-8, 9-6
2
Image space
Coordinate Transform: 3D Geometry Pipeline
3
Normalized project
space
View spaceWorld spaceObject space
Aspect ratio &
resolution
Focal length
Rotate and translate the
camera
Coordinate Transformation: 3D Modeling/Design
Coordinate transformation from one reference frame to another
4
Coordinate Transformation: Animation/Robotics
How to model 2D movement of animated characters or robots?
5
Click here
Coordinate Transformation
Coordinate transformation from one reference frame to another
6
Coordinate Transformation
Coordinate transformation from one reference frame to another
7
Local reference frame
Coordinate Transformation
Coordinate transformation from one reference frame to another
8
Local reference frame
Global reference frame
Coordinate Transformation
Coordinate transformation from one reference frame to another
?
9
Local reference frame
Global reference frame
Review – Vector Operations
Dot Product
v w
10
Review – Vector Operations
Dot Product: measuring similarity between two vectors
v w
)cos(|||| wvwv
11
Review – Vector Operations
Dot Product: measuring similarity between two vectors
v w
)cos(|||| wvvw
12
Review – Vector Operations
Dot Product: measuring similarity between two vectors
v w
1vvUnit vector:
13
Review – Vector Operations
Dot Product: measuring similarity between two vectors
v w
)cos(|||| wvwv
14
Review – Vector Operations
Dot Product: measuring similarity between two vectors
v w
0wv
15
Review – Vector Operations
Cross Product: measuring the area determined by two vectors
16
v w
Review – Vector Operations
Cross Product: measuring the area determined by two vectors
sin 2*v w v w u area u
17
v w
u
2D Coordinates
2D Cartesian coordinate system:
i
j
o
18
2D Coordinate Transformation
2D Cartesian coordinate system:
i
j
o
1 ii
1 jj
0 ji
19
2D Coordinate Transformation
2D Cartesian coordinate system: any 2D vector can be represented as
i
j
o
1 ii
1 jj
0 ji
20
jyix
2D Coordinate Transformation
2D Cartesian coordinate system:
P: (x,y)
i
j
o x
y
1 ii
1 jj
0 ji
21
jyixop
2D Coordinate Transformation
2D Cartesian coordinate system:
P: (x,y)
i
j
o x
y
22
jyixop
xi
op
op yj
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
23
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
24
Given the coordinates (x’,y’) in i’j’
- how to compute the coordinates (x,y) in ij?
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
25
Given the coordinates (x’,y’) in i’j’
- how to compute the coordinates (x,y) in ij?
0x
0y
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
),( 00 yx
26
Given the coordinates (x’,y’) in i’j’
- how to compute the coordinates (x,y) in ij?
1100
cossin
sincos
10
0
y
x
y
x
y
x
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
jyixop
ji ij
),( 00 yx
27
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
jyixop
jyixpo
ji ij
),( 00 yx
28
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
jyixop
jyixpo
jyixoo
00
ji ij
),( 00 yx
29
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
poooop
ji ij
),( 00 yx
30
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
poooop
jyixop
jyixpo
jyixoo
00
ji ij
),( 00 yx
31
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
poooop
ji ij
jyixjyixjyix
00
),( 00 yx
32
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
poooop
ji ij
jyixjyixjyix
00
jyixjyyixx
)()( 00
),( 00 yx
33
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
poooop
ji ij
jyixjyixjyix
00
jyixjyyixx
)()( 00
y
xji
yy
xxji
0
0),( 00 yx
34
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
y
xji
yy
xxji
0
0
),( 00 yx
35
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
y
xji
yy
xxji
0
0
y
xjiji
yy
xx 1
0
0
),( 00 yx
36
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
y
xji
yy
xxji
0
0
y
xjiji
yy
xx 1
0
0
y
xji
j
i
yy
xxT
T
0
0
),( 00 yx
37
10
01
jjij
jiii
j
iji
T
T
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
y
xji
j
i
yy
xxT
T
0
0
110010
0
y
x
yjjij
xjiii
y
xTT
TT
),( 00 yx
38
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
y
xji
j
i
yy
xxT
T
0
0
),( 00 yx
39
110010
0
y
x
yjjij
xjiii
y
xTT
TT
2D Coordinate Transformation
pj
i
i
j
o
o
ji ij
y
xji
j
i
yy
xxT
T
0
0
110010
0
y
x
yjjij
xjiii
y
xTT
TT
),( 00 yx
40
What does this column vector mean?
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
y
xji
j
i
yy
xxT
T
0
0
110010
0
y
x
yjjij
xjiii
y
xTT
TT
),( 00 yx
41
What does this column vector mean? Vector i’ in the new reference system
ii T
ij T
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
y
xji
j
i
yy
xxT
T
0
0
110010
0
y
x
yjjij
xjiii
y
xTT
TT
),( 00 yx
42
What does this column vector mean?
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
y
xji
j
i
yy
xxT
T
0
0
110010
0
y
x
yjjij
xjiii
y
xTT
TT
),( 00 yx
43
What does this column vector mean? Vector j’ in the new reference system
ji T
jj T
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
y
xji
j
i
yy
xxT
T
0
0
110010
0
y
x
yjjij
xjiii
y
xTT
TT
),( 00 yx
44
What does this column vector mean?
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
y
xji
j
i
yy
xxT
T
0
0
110010
0
y
x
yjjij
xjiii
y
xTT
TT
),( 00 yx
45
What does this column vector mean? The old origin in the new reference system
2D Coordinate Transformation
2D translation
pj
i
i
j
o
o
110010
0
y
x
yjjij
xjiii
y
xTT
TT
),( 00 yx
46
2D Coordinate Transformation
2D translation
pj
i
i
j
o
o
110010
0
y
x
yjjij
xjiii
y
xTT
TT
),( 00 yx
?
? ?
?
47
2D Coordinate Transformation
2D translation
pj
i
i
j
o
o
110010
0
y
x
yjjij
xjiii
y
xTT
TT
1
0
0
1
),( 00 yx
48
2D Coordinate Transformation
2D translation&rotation
pj
i
i
j
o
o
110010
0
y
x
yjjij
xjiii
y
xTT
TT
),( 00 yx
?
49
2D Coordinate Transformation
2D translation&rotation
pj
i
i
j
o
o
1100
cos
10
0
y
x
yjjij
xji
y
xTT
T
),( 00 yx
?
50
2D Coordinate Transformation
2D translation&rotation
pj
i
i
j
o
o
1100
sin
cos
10
0
y
x
yjj
xji
y
xT
T
),( 00 yx
51
2D Coordinate Transformation
2D translation&rotation
pj
i
i
j
o
o
1100
sin
cos
10
0
y
x
yjj
xji
y
xT
T
),( 00 yx
?
52
2D Coordinate Transformation
2D translation&rotation
pj
i
i
j
o
o
1100
sin
sincos
10
0
y
x
yjj
x
y
xT
),( 00 yx
53
2D Coordinate Transformation
2D translation&rotation
pj
i
i
j
o
o
1100
sin
sincos
10
0
y
x
yjj
x
y
xT
),( 00 yx
?
54
2D Coordinate Transformation
2D translation&rotation
pj
i
i
j
o
o
1100
cossin
sincos
10
0
y
x
y
x
y
x
),( 00 yx
55
2D Coordinate Transformation
An alternative way to look at the problem
i
j
o
P=[x,y]j
i
o ),( 00 yx
56
- set up a transformation that superimposes the x’y’ axes onto the xy axis
1
y
x
2D Coordinate Transformation
An alternative way to look at the problem
i
j
o
P=[x,y]j
i
o ),( 00 yx
57
- set up a transformation that superimposes the x’y’ axes onto the xy axis
1100
10
01
0
0
y
x
y
x
2D Coordinate Transformation
An alternative way to look at the problem
i
j
o
P=[x,y]
j
io ),( 00 yx
58
- set up a transformation that superimposes the x’y’ axes onto the xy axis
1100
10
01
100
0cossin
0sincos
0
0
y
x
y
x
2D Coordinate Transformation
An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’)
i
j
o
p
j
io ),( 00 yx
59
1100
10
01
100
0cossin
0sincos
10
0
y
x
y
x
y
x
2D Coordinate Transformation
An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’) How to transform the point from (x’,y’) to
(x,y)?
i
j
o
p
j
io ),( 00 yx
60
1100
10
01
100
0cossin
0sincos
10
0
y
x
y
x
y
x
2D Coordinate Transformation
An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’) How to transform the point from (x’,y’) to
(x,y)? Invert the matrix!
i
j
o
p
j
io ),( 00 yx
61
1100
0cossin
0sincos
100
10
01
1
11
0
0
y
x
y
x
y
x
2D Coordinate Transformation
An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’) How to transform the point from (x’,y’) to
(x,y)? Invert the matrix!
i
j
o
p
j
io ),( 00 yx
62
1100
0cossin
0sincos
100
10
01
1
11
0
0
y
x
y
x
y
x
2D Coordinate Transformation
An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’) How to transform the point from (x’,y’) to
(x,y)? Invert the matrix!
i
j
o
p
j
io ),( 00 yx
63
1100
0cossin
0sincos
100
10
01
1
11
0
0
y
x
y
x
y
x
100
10
01
0
0
y
x
2D Coordinate Transformation
An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’) How to transform the point from (x’,y’) to
(x,y)? Invert the matrix!
i
j
o
p
j
io ),( 00 yx
64
1100
0cossin
0sincos
100
10
01
1
11
0
0
y
x
y
x
y
x
100
10
01
0
0
y
x
2D Coordinate Transformation
An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’) How to transform the point from (x’,y’) to
(x,y)? Invert the matrix!
i
j
o
p
j
io ),( 00 yx
65
1100
0cossin
0sincos
100
10
01
1
11
0
0
y
x
y
x
y
x
100
10
01
0
0
y
x
100
0cossin
0sincos
2D Coordinate Transformation
Same results!
pj
i
i
j
o
o
1100
cossin
sincos
10
0
y
x
y
x
y
x
),( 00 yx
66
2D Coordinate Transformation
2D translation&rotation
pj
i
i
j
o o
1???
???
???
1
y
x
y
x
30
67
2D Coordinate Transformation
2D translation&rotation
pj
i
i
j
o o
1100
030cos30sin
030sin30cos
1
y
x
y
x
30
68
2D Coordinate Transformation
2D translation&rotation
p
j
i
i
j
oo
1???
???
???
1
y
x
y
x
45
69
2D Coordinate Transformation
2D translation&rotation
p
j
i
i
j
oo
1100
045cos45sin
045sin45cos
1
y
x
y
x
45
70
2D Coordinate Transformation
2D translation&rotation
p
j
i
i
j
o
)2,4( o
1???
???
???
1
y
x
y
x
45
71
2D Coordinate Transformation
2D translation&rotation
p
j
i
i
j
o
)2,4( o
1100
245cos45sin
445sin45cos
1
y
x
y
x
45
72
3D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
),,( 000 zyx
k
'k
73
2D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
110010
0
y
x
yjjij
xjiii
y
xTT
TT
),( 00 yx
poooop
74
3D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
1
'
'
'
1000
'
'
'
10
0
0
z
y
x
zkkjkik
ykjjjij
xkijiii
z
y
x
TTT
TTT
TTT
),,( 000 zyx
k
'k
poooop
75
3D Coordinate Transformation
Transform object description from to
pj
i
i
j
o
o
ji ij
1
'
'
'
1000
'
'
'
10
0
0
z
y
x
zkkjkik
ykjjjij
xkijiii
z
y
x
TTT
TTT
TTT
),,( 000 zyx
k
'k
poooop
76
3D Coordinate Transformation
Transform object description from to
i
i
j
o
ji ij
1
'
'
'
1000
'
'
'
10
0
0
z
y
x
zkkjkik
ykjjjij
xkijiii
z
y
x
TTT
TTT
TTT
k
77
p'i
'j
3D Coordinate Transformation
Transform object description from toji ij
78
'i
11000
0cos0sin
0010
0sin0cos
1
z
y
x
z
y
x
x
y
z
Composite 2D Transformation
How to model 2D movement of characters or robots?
79
Click here
Composite 2D Transformation
A 2D lamp character
1
2
3
),,( 0yx
3c
2c
1c
0c
A
80
0
Composite 2D Transformation
A 2D lamp character – skeleton size3l
A2l
1l
0l
81
Composite 2D Transformation
How can we draw the character given the pose ?
1
2
3
),,( 0yx
3c
2c
1c
0c
A
82
),,,,,( 3210 yx
Composite 2D Transformation
How can we draw the character given the pose ?
1
2
3
),,( 0yx
3c
2c
1c
0c
A
83
),,,,,( 3210 yx
- This requires computing the global coordinates for any point on the character.
- But we only have local coordinates of points.
- So how can we map the local coordinates to the global coordinates?
),,( 0yx
3c2c
0c
A1c
Articulated Character
Local reference frames with a default pose (0,0,0,0,0,0)
Composite 2D Transformation
What’s the pose?
)1,5(
85
?)?,?,?,,1,5(
300
Composite 2D Transformation
What’s the pose?
)1,5(
86
)90,0,90,30,1,5(
300
Composite 2D Transformation
A 2D lamp character
1
2
3
3c
2c
1c
A
Given , , how to compute the global position of a point (e.g., A) based on its local coordinates?
),,,,,( 3210 yx
?
),,( 0yx0c
87
Composite 2D Transformation
What’s local coordinate ?
1
2
3
3c
2c
1c
A
?
A
),,( 0yx0c
88
Composite 2D Transformation
What’s local coordinate ?
1
2
3
3c
2c
1c
A
?
A
1
03
0
l
p
),,( 0yx0c
03221100 )()0,()()0,()()0,()(),( pRlTRlTRlTRyxTp 89
Composite 2D Transformation
What’s the current coordinate A ?
1
2
3
3c
2c
1c
A
?
),,( 0yx0c
1
00
l
p
03221100 )()0,()()0,()()0,()(),( pRlTRlTRlTRyxTp
100
010
01
)0,(2
2
l
lT
90
100
0cossin
0sincos
)( 33
33
3
R
Composite 2D Transformation
What’s the current coordinate A ?
1
2
3
3c
2c
1c
A
?
),,( 0yx0c
1
00
l
p
03221100 )()0,()()0,()()0,()(),( pRlTRlTRlTRyxTp
100
010
01
)0,(2
2
l
lT
91
100
0cossin
0sincos
)( 33
33
3
R
Composite 2D Transformation
What’s the current coordinate A ?
1
2
3
3c
2c
1c
A
?
),,( 0yx0c
1
00
l
p
03221100 )()0,()()0,()()0,()(),( pRlTRlTRlTRyxTp
100
010
01
)0,(1
1
l
lT
92
100
0cossin
0sincos
)( 22
22
2
R
Composite 2D Transformation
What’s the current coordinate A ?
1
2
3
3c
2c
1c
A
?
),,( 0yx0c
1
00
l
p
03221100 )()0,()()0,()(),0()(),( pRlTRlTRlTRyxTp
100
10
001
),0( 00 llT
93
100
0cossin
0sincos
)( 11
11
1
R
Composite 2D Transformation
What’s the current coordinate A ?
1
2
3
3c
2c
1c
A
?
03221100 )()0,()()0,()(),0()(),( pRlTRlTRlTRyxTp
),,( 0yx0c
1
00
l
p
100
cossin
sincos
)(),( 00
00
0 y
x
RyxT
94
Composite 2D Transformation
What’s the current coordinate A ?
1
2
3
3c
2c
1c
A
03221100 )()0,()()0,()(),0()(),( pRlTRlTRlTRyxTp
),,( 0yx0c
95
How to Animate the Character?
A 2D lamp character
1
2
3
),,( 0yx
3c
2c
1c
0c
A
96
How to Animate the Character?
Keyframe animation
- Manually pose the character by choosing appropriate values for
- Linearly interpolate the inbetween poses.
- Works for any types of articulated characters!
97
),,,,,( 3210 yx
Composite 3D Transformation
Similarly, we can easily extend composite transformation from 2D to 3D
1
2
3
),,( 0yx
3c
2c
1c
0c
A
98
Composite 3D Transformation
xRRhTRhTRRRhTRRRzyxTp )()()0,,0()()0,,0()()()()0,,0()()()(),,( 332211110000000
99
