3d geometric transformations -...
TRANSCRIPT
Sun-Jeong Kim
3D Geometric Transformations
10th Week, 2008
Computer Graphics2
Transformation in 3D
Transformation matrix
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
××
××⇒
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
1131
1333
000 S
LIFC
KHEB
JGDA
3×3 : Scaling, Reflection, Shearing, Rotation3×3 : Scaling, Reflection, Shearing, Rotation
3×1 : Translation3×1 : Translation
1×3 : Homogeneous representation1×3 : Homogeneous representation
1×1 : Uniform global Scaling1×1 : Uniform global Scaling
Computer Graphics3
3D Translation
Translation of a point
zyx tzztyytxx +=+=+= ',','
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
11000
100
010
001
1
'
'
'
z
y
x
t
t
t
z
y
x
z
y
x
xz
y
Computer Graphics4
3D Scaling
Scaling relative to the origin
zyx szzsyysxx ⋅=⋅=⋅= ',','
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
11000
000
000
000
1
'
'
'
z
y
x
s
s
s
z
y
x
z
y
x
xz
y
Computer Graphics5
Relative Scaling
Scaling with a selected fixed position
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=−−−⋅⋅
11000
100
010
001
1000
000
000
000
1000
100
010
001
1
'
'
'
),,(),,(),,(z
y
x
z
y
x
s
s
s
z
y
x
z
y
x
zyxTsssSzyxTf
f
f
z
y
x
f
f
f
fffzyxfff
x x x xzzzz
y y y y
Original position Translate Scaling Inverse Translate
Computer Graphics6
3D Rotation
Coordinate-axes rotationsx-axis rotation
y-axis rotation
z-axis rotation
General 3D rotationsRotation about an axis that is parallel to one of the coordinate axes
Rotation about an arbitrary axis
Computer Graphics7
Coordinate-Axes Rotations
z-axis rotation x-axis rotation y-axis rotation
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ −
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
11000
0100
00cossin
00sincos
1
'
'
'
z
y
x
z
y
x
θθθθ
z
y
x
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
11000
0cossin0
0sincos0
0001
1
'
'
'
z
y
x
z
y
x
θθθθ
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
11000
0cos0sin
0010
0sin0cos
1
'
'
'
z
y
x
z
y
x
θθ
θθ
z
y
xz
y
x
Computer Graphics8
Order of Rotations
Order of rotation affects final positionx-axis z-axis
z-axis x-axis
Computer Graphics9
General 3D Rotations (1)
Rotation about an axis that is parallel to one of the coordinate axes
Translate : the rotation axis coincides with the parallel coordinate axis
Rotation about that axis
Translate : the rotation axis is moved back to its original position
Computer Graphics10
General 3D Rotations (2)
Rotation about an arbitrary axis
Basic Idea
1. Translate (x1, y1, z1) to the origin
2. Rotate (x’2, y’2, z’2) on to the z axis
3. Rotate the object around the z-axis
4. Rotate the axis to the original
orientation
5. Translate the rotation axis to the
original position
Basic Idea
1. Translate (x1, y1, z1) to the origin
2. Rotate (x’2, y’2, z’2) on to the z axis
3. Rotate the object around the z-axis
4. Rotate the axis to the original
orientation
5. Translate the rotation axis to the
original position
(x2,y2,z2)
(x1,y1,z1)
x
z
y
R-1R-1
T-1T-1
RR
TT
( ) ( ) ( ) ( ) ( ) ( )TRRRRRTR αβθβαθ xyzyx111 −−−=
RR
Computer Graphics11
Arbitrary Axis Rotation (1)
Step 1. Translation
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−
=
1000
100
010
001
1
1
1
z
y
x
T
(x2,y2,z2)
(x1,y1,z1)
x
z
y
Computer Graphics12
Arbitrary Axis Rotation (2)
Step 2. Rotate about x axis by α
( )⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
=
1000
0//0
0//0
0001
1000
0cossin0
0sincos0
0001
dcdb
dbdcx αα
αααR
(a,b,c)(0,b,c)
Projected Point α α
Rotated Point
d
c
cb
c
d
b
cb
b
=+
=α
=+
=α
22
22
cos
sin
x
y
z
Computer Graphics13
Arbitrary Axis Rotation (3)
Step 3. Rotate about y axis by β
(a,b,c)
(a,0,d)β
l
d
22
222222
cos,sin
cbd
dacbal
l
d
l
a
+=
+=++=
== ββ
( )⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ −
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ −
=
1000
0/0/
0010
0/0/
1000
0cos0sin
0010
0sin0cos
ldla
lald
y ββ
ββ
βRx
y
Projected Point
zRotated
Point
Computer Graphics14
Arbitrary Axis Rotation (4)
Step 4. Rotate about z axis by the desired angle θ
θ
l
( )⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ −
=
1000
0100
00cossin
00sincos
θθθθ
θzR
y
x
z
Computer Graphics15
Arbitrary Axis Rotation (5)
Step 5. Apply the reverse transformation to place the axis back in its initial position
x
y
l
l
z
( ) ( )
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−
=−−−
1000
0cos0sin
0010
0sin0cos
1000
0cossin0
0sincos0
0001
1000
100
010
001
1
1
1
111
ββ
ββ
αααα
βαz
y
x
yx RRT
( ) ( ) ( ) ( ) ( ) ( )TRRRRRTR αβθβαθ xyzyx111 −−−=
Computer Graphics16
Example (1)
Find the new coordinates of a unit cube 90º-rotated about an axis defined by its endpoints A(2,1,0) and B(3,3,1)
A Unit Cube
Computer Graphics17
Example (2)
Step1. Translate point A to the origin
A’(0,0,0)x
z
y
B’(1,2,1)
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−−
=
1000
0100
1010
2001
T
Computer Graphics18
x
z
y
l
( )
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−=
1000
05
5
5
520
05
52
5
50
0001
αxR
6121
5
5
5
1cos
5
52
5
2
12
2sin
222
22
=++=
==α
==+
=α
lB’(1,2,1)
α
Projected point (0,2,1)
B”(1,0,√5)
Example (3)
Step 2. Rotate axis A’B’ about the x axis by and angle α, until it lies on the xz plane.
Computer Graphics19
x
z
y
l
( )
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡−
=
1000
06
300
6
60010
06
60
6
30
βyR
6
30
6
5cos
6
6
6
1sin
==
==
β
β
β
B”(1,0, √ 5)(0,0,√6)
Example (4)
Step 3. Rotate axis A’B’’ about the y axis by and angle φ, until it coincides with the z axis.
Computer Graphics20
Example (5)
Step 4. Rotate the cube 90° about the z axis
Finally, the concatenated rotation matrix about the arbitrary axis AB becomes,
( ) ( ) ( ) ( ) ( ) ( )TRRRRRTR αββαθ xyzyx °= −−− 90111
( )⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ −
=°
1000
0100
0001
0010
90zR
Computer Graphics21
( )
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−−
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ −
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
1000
560.0167.0741.0650.0
151.1075.0667.0742.0
742.1983.0075.0166.0
1000
0100
1010
2001
1000
05
5
5
520
05
52
5
50
0001
1000
06
300
6
60010
06
60
6
30
1000
0100
0001
0010
1000
06
300
6
60010
06
60
6
30
1000
05
5
5
520
05
52
5
50
0001
1000
0100
1010
2001
θR
Example (6)
Computer Graphics22
Example (7)
Multiplying R(θ) by the point matrix of the original cube
[ ]
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−−−−−
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−
=′
11111111
076.0091.0560.0726.0817.0650.0301.1467.1
483.0409.0151.1225.1184.0258.0484.0558.0
891.2909.1742.1725.2816.2834.1667.1650.2
11111111
10011001
00001111
11001100
1000
560.0167.0741.0650.0
151.1075.0667.0742.0
742.1983.0075.0166.0
P
[ ] ( )[ ]PRP θ=′
Computer Graphics23
Rotations with Quaternions (1)
QuaternionScalar part s + vector part v = (a, b, c)
Real part + complex part (imaginary numbers i, j, k)
Rotation about any axisSet up a unit quaternion (u: unit vector)
Represent any point position P in quaternion notation (p = (x, y, z))
2sin ,
2cos
θθuv ==s
( )pP ,0=
( ) ckbjaissq +++== v ,
Computer Graphics24
Rotations with Quaternions (2)
Carry out with the quaternion operation (q-1=(s, –v))
Produce the new quaternion
Obtain the rotation matrix by quaternion multiplication
Include the translations:
1−=′ qqPP
( )pP ′=′ ,0
( ) ( ) ( )pvvpvvpvpp ××+×+⋅+=′ ss 22
( ) ( ) ( ) ( ) ( ) ( )
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−+−−−−++−−−
=
= −−
22
22
22
11
2212222
2222122
2222221
basabcsbac
sabccascab
sbacscabcb
xyzyxR αβθβαθ RRRRRM
( ) ( )TMTR θθ R1−=
Computer Graphics25
Example
Rotation about z axisSet the unit quaternion:
Substitute a=b=0, c=sin(θ/2) into the matrix:
( )2
sin1 ,0 ,0 ,2
cosθθ
== vs
( )
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−−
=
100
0cossin
0sincos
100
02
sin212
sin2
cos2
02
sin2
cos22
sin21
2
2
θθθθ
θθθ
θθθ
θRM
θθθ
θθ
sin2
sin2
cos2
cos2
sin21 2
=
=−
Computer Graphics26
Other Transformations
Reflection relative to the xy plane
z-axis shearxz
y
x
z
y
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
′′′
11000
0100
0010
0001
1
z
y
x
z
y
x
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
11000
0100
010
001
1
'
'
'
z
y
x
b
a
z
y
x
Computer Graphics27
Coordinate Transformations
Transformation of an object description from one coordinate system to another
Transformation matrixBring the two coordinates systems into alignment
[1] Translation
x
y
z (0,0,0)
y’
z’
x’u'y
u'z
u'x(x0,y0,z0)x
y
z (0,0,0)
u'y
u'z
u'x
( )000 , , zyx −−−T
Computer Graphics28
Coordinate Transformations
[2] Rotation & Scaling
x
y
z (0,0,0)
y’
z’
x’u'y
u'z
u'x x
y
z (0,0,0)
u'y
u'z
u'x x
y
z (0,0,0)
u'y
u'z
u'x
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
′′′′′′′′′
=
1000
0
0
0
321
321
321
zzz
yyy
xxx
uuu
uuu
uuu
R