Download Lecture 5: 3D Transformations

CS-184: Computer Graphics
Lecture #5: 3D Transformations and
Rotations
Prof. James O’Brien
University of California, Berkeley
V2009-F-05-1.0
Today
• Transformations
in 3D
• Rotations
Matrices
• Euler angles
• Exponential maps
• Quaternions
•
2
3D Transformations
• Generally, the
extension from 2D to 3D is straightforward
Vectors get longer by one
Matrices get extra column and row
• SVD still works the same way
• Scale, Translation, and Shear all basically the same
•
•
• Rotations
get interesting
3
Translations

1
à = 0
0

1 0
0 1
à = 
0 0
0 0

0 tx
1 ty
0 1

0 tx
0 ty 

1 tz 
0 1
For 2D
For 3D
4
Scales

sx
à =  0
0

sx 0
 0 sy
à = 
0 0
0 0
0
sy
0
0
0
sz
0

0
0
1

0
0

0
1
For 2D
For 3D
(Axis-aligned!)
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Shears

1
à = hyx
0

1 hxy
hyx 1
à = 
hzx hzy
0 0
hxy
1
0

0
0
1

hxz 0
hyz 0

1 0
0 1
For 2D
For 3D
(Axis-aligned!)
6
Shears


1 hxy hxz 0
hyx 1 hyz 0

à = 
hzx hzy 1 0
0 0 0 1
Shears y into x
7
Rotations
• 3D
•
•
Rotations fundamentally more complex than in 2D
2D: amount of rotation
3D: amount and axis of rotation
-vs2D
3D
8
Rotations
• Rotations
•
still orthonormal
Det(R) = 1 != −1
• Preserve
• 3D
lengths and distance to origin
rotations DO NOT COMMUTE!
• Right-hand
• Unique
rule
DO NOT COMMUTE!
matrices
9
Axis-aligned 3D Rotations
• 2D
axis
rotations implicitly rotate about a third out of plane
10
Axis-aligned 3D Rotations
• 2D
axis
rotations implicitly rotate about a third out of plane
!
"
cos(θ) − sin(θ)
R=
sin(θ) cos(θ)


cos(θ) − sin(θ) 0
R =  sin(θ) cos(θ) 0
0
0
1
Note: looks same as R̃
11
Axis-aligned 3D Rotations


1
0
0
R = 0 cos(θ) − sin(θ)
0 sin(θ) cos(θ)


cos(θ) 0 sin(θ)
1
0 
Rŷ =  0
− sin(θ) 0 cos(θ)


cos(θ) − sin(θ) 0
Rẑ =  sin(θ) cos(θ) 0
0
0
1
x̂
ŷ
x̂
ẑ
12
Axis-aligned 3D Rotations


1
0
0
R = 0 cos(θ) − sin(θ)
0 sin(θ) cos(θ)


cos(θ) 0 sin(θ)
1
0 
Rŷ =  0
− sin(θ) 0 cos(θ)


cos(θ) − sin(θ) 0
Rẑ =  sin(θ) cos(θ) 0
0
0
1
x̂
“Z is in your face”
ŷ
x̂
ẑ
12
Axis-aligned 3D Rotations


1
0
0
R = 0 cos(θ) − sin(θ)
0 sin(θ) cos(θ)

 Also right handed “Zup”
cos(θ) 0 sin(θ)
ẑ
1
0 
Rŷ =  0
− sin(θ) 0 cos(θ)
ŷ


cos(θ) − sin(θ) 0
x̂

Rẑ = sin(θ) cos(θ) 0
0
0
1
x̂
13
Axis-aligned 3D Rotations
• Also
known as “direction-cosine” matrices


1
0
0
Rx̂= 0 cos(θ) − sin(θ)
0 sin(θ) cos(θ)


cos(θ) 0 sin(θ)
1
0 
Rŷ=  0
− sin(θ) 0 cos(θ)


cos(θ) − sin(θ) 0
R ẑ=  sin(θ) cos(θ) 0
0
0
1
14
Arbitrary Rotations
• Can
be built from axis-aligned matrices:
• Result
R = Rẑ · Rŷ · Rx̂
due to Euler... hence called
Euler Angles
• Easy to store in vector
• But
NOT a vector.
R = rot(x, y, z)
15
Arbitrary Rotations
R = Rẑ · Rŷ · Rx̂
Rx̂
Rŷ
Rẑ
R
16
Arbitrary Rotations
• Allows
• Euler
tumbling
angles are non-unique
• Gimbal-lock
• Moving
•
-vs- fixed axes
Reverse of each other
17
Exponential Maps
• Direct
representation of arbitrary rotation
• AKA: axis-angle, angular
displacement vector
• Rotate
θ degrees about some axis
• Encode θ by length of vector
r̂
θ = |r|
θ
18
Exponential Maps
• Given
vector r , how to get matrix
R
• Method from text:
1. rotate about x axis to put r into the x-y plane
2. rotate about z axis align r with the x axis
3. rotate θ degrees about x axis
4. undo #2 and then #1
5. composite together
19
Exponential Maps
r
x⊥
x
r
x⊥
x
• Vector
expressing a point has two parts
• x does not change
• x rotates like a 2D point
⊥
20
Exponential Maps
r
x⊥
x
r
x⊥
x
x! = r̂ × x
x!
θ
−x⊥ = r̂ × (r̂ × x)
x⊥
x
x! = x|| + x" sin(θ) + x⊥ cos(θ)
x! sin(θ)
−x⊥ cos(θ)
21
Exponential Maps
• Rodriguez
Formula
x! = r̂(r̂ · x)
+sin(θ)(r̂ × x)
−cos(θ)(r̂ × (r̂ × x))
r
x!
x
x!
r
x
Actually a minor variation ...
22
Exponential Maps
• Rodriguez
Formula
x! = r̂(r̂ · x)
+sin(θ)(r̂ × x)
−cos(θ)(r̂ × (r̂ × x))
r
x!
x
r
x!
Linear in x
x
Actually a minor variation ...
22
Exponential Maps
• Building
the matrix
x! = ((r̂r̂t) + sin(θ)(r̂×) − cos(θ)(r̂×)(r̂×)) x


0 −r̂z r̂y
0 −r̂x
(r̂×) =  r̂z
−r̂y r̂x 0
Antisymmetric matrix
(a×)b = a × b
Easy to verify by expansion
23
Exponential Maps
• Allows
• No
tumbling
gimbal-lock!
• Orientations
• Nearly
unique representation
• Singularities
• Nice
are space within π-radius ball
on shells at 2π
for interpolation
24
Exponential Maps
• Why
exponential?
• Recall
series expansion of
ex
x x2 x3
e = 1+ + + +···
1! 2! 3!
x
25
Exponential Maps
• Why exponential?
• Recall series expansion of
• Euler: what happens if you
ex
put in iθ for x
iθ −θ2 −iθ3 θ4
e = 1+ +
+
+ +···
1!
2!
3!
4!
iθ
!
" !
"
−θ2 θ4
θ −θ3
= 1+
+ +··· +i
+
+···
2!
4!
1!
3!
= cos(θ) + i sin(θ)
26
Exponential Maps
• Why
(r̂×)θ
e
exponential?
(r̂×)θ (r̂×)2θ2 (r̂×)3θ3 (r̂×)4θ4
= I+
+
+
+
+···
1!
2!
3!
4!
3
But notice that: (r̂×) = −(r̂×)
(r̂×)θ
e
(r̂×)θ (r̂×)2θ2 −(r̂×)θ3 −(r̂×)2θ4
= I+
+
+
+
+···
1!
2!
3!
4!
27
Exponential Maps
e(r̂×)θ = I +
(r̂×)θ (r̂×)2θ2 −(r̂×)θ3 −(r̂×)2θ4
+
+
+
+···
1!
2!
3!
4!
!
"
! 2
"
3
4
θ
θ
θ
θ
e(r̂×)θ = (r̂×)
− + · · · + I + (r̂×)2 + − + · · ·
1! 3!
2! 4!
e(r̂×)θ = (r̂×) sin(θ) + I + (r̂×)2(1 − cos(θ))
28
Quaternions
• More
popular than exponential maps
• Natural
• Due
•
•
iθ
extension of e = cos(θ) + i sin(θ)
to Hamilton (1843)
Interesting history
Involves “hermaphroditic monsters”
29
Quaternions
• Uber-Complex
Numbers
q
= (z1, z2, z3, s) = (z, s)
q
= iz1 + jz2 + kz3 + s
i2 = j2 = k2 = −1
i j = k ji = −k
jk = i k j = −i
ki = j ik = − j
30
Quaternions
• Multiplication
q·p
natural consequence of defn.
= (zqs p + z psq + z p × zq , s psq − z p · zq)
• Conjugate
∗
q
= (−z, s)
• Magnitude
||q||2 = z · z + s2 = q · q∗
31
Quaternions
• Vectors
as quaternions
v
• Rotations
= (v, 0)
as quaternions
θ
θ
= (r̂ sin , cos )
2
2
• Rotating a vector
r
!
∗
x = r·x·r
• Composing rotations
r
= r1 · r2
Compare to Exp. Map
32
Quaternions
• No
tumbling
• No
gimbal-lock
• Orientations
• Surface
• Nice
are “double unique”
of a 3-sphere in 4D
for interpolation
||r|| = 1
33
Interpolation
34
Rotation Matrices
• Eigen
system
One real eigenvalue
Real axis is axis of rotation
• Imaginary values are 2D rotation as complex number
•
•
• Logarithmic
formula
(r̂×) = ln(R) =
θ = cos
−1
!
θ
(R − RT)
2 sin θ
Tr(R) − 1
2
"
Similar formulae as for exponential...
35
Rotation Matrices
• Consider:
 rxx

RI = ryx
 rzx

rxy
ryy
rzy
rxz  1 0 0

ryz  0 1 0
rzz  0 0 1
• Columns
are coordinate axes after transformation (true
for general matrices)
• Rows
are original axes in original system (not true for
general matrices)
36
Note:
• Rotation
stuff in the book is a bit weak... luckily you have
these nice slides!
37