Colorado School of Mines
Computer Vision
Professor William Hoff
Dept of Electrical Engineering &Computer Science
Colorado School of Mines
Computer Vision
http://inside.mines.edu/~whoff/
1
3D Rotations
Colorado School of Mines
Computer Vision
Topics
• We have looked at how to represent 3D rotations
using XYZ fixed angles
• Here we will look at:
–
–
–
–
Rotations using small angles
Euler angle representation
Axis-angle representation
Quaternions
Colorado School of Mines
Computer Vision
3
XYZ Fixed Angles
• Start with {B} coincident with {A}. First rotate {B} about xA by
angle qX, then rotate it about yA by qY, then rotate about zA by
qZ.
A
B
RXYZ q X , qY , q Z   RZ (q Z ) RY (qY ) RX (q X )
0 
 cz  sz 0   cy 0 sy   1 0




  sz cz 0   0 1 0   0 cx  sx 
0
0 1    sy 0 cy   0 sx cx 

where
cx  cos(q X ), sy  sin(qY ), etc
 cz cy

A


R
q
,
q
,
q

 sz cy
B XYZ
X
Y
Z
  sy

Each rotation
takes place
relative to the
fixed frame
{A}
cz sy sx  szcx cz sy cx  sz sx 

sz sy sx  czcx sz sy cx  cz sx 

cy sx
cy cx

4
Colorado School of Mines
Computer Vision
Small Angle Approximation
• If rotation angles are small
– Eg., object is rotating slowly in a video sequence
– Or we are looking at the effect of small angle perturbations on the
rotation
• Then rotation matrix simplifies
 cos q Z cos qY

A


R
q
,
q
,
q

 sin q Z z cos qY
B XYZ
X
Y
Z
  sin q
Y

cos q Z sin qY sin q X  sin q Z cos q X
sin q Z z sin qY sin q X  cos q Z cos q X
cos qY sin q X
cos q Z sin qY cos q X  sin q Z sin q X 

sin q Z sin qY cos q X  cos q Z sin q X 

cos qY cos q X

• Let cos q ≈ 1, sin q ≈ q for small q
 1

A


R
q
,
q
,
q

 qZ
B XYZ
X
Y
Z
 q
 Y
qZ
1
qX
qY 

q X 
1 
5
Colorado School of Mines
Computer Vision
Possible Combinations for Rotations
• There are 12 possible combinations for rotations about the fixed axes:
–
–
–
–
–
–
–
–
–
–
–
–
RXRYRZ
RXRZRY
RYRXRZ
RYRZRX
RZRXRY
RZRYRX
RXRYRX
RXRZRX
RYRXRY
RYRZRY
RZRXRZ
RZRYRZ
Colorado School of Mines
We will use this
convention in this course
• So for a given 3D rotation, the values
of the 3 angles depends on the
rotation convention you use
• However, a given 3D rotation always
has a unique rotation matrix
Computer Vision
6
Euler angles – a different rotation convention
• ZYX Euler angles
– Start with {B} coincident with {A}. First rotate {B} about zB by angle qZ,
then rotate it about yB by qY, then rotate about xB by qX.
– Each rotation takes place relative to the moving frame {B}
– There are 12 angle set conventions for Euler angles
Colorado School of Mines
Computer Vision
7
Equivalent Angle-Axis
•
k
A general rotation can be expressed as a
rotation q about an axis k
 k x k x vq  cq

Rk q    k x k y vq  k z sq
 k k vq  k sq
y
 x z
k x k y vq  k z sq
k y k y vq  cq
k y k z vq  k x sq
k x k z vq  k y sq 

k y k z vq  k x sq 
k z k z vq  cq 
where
{B}
cq  cos q , sq  sin q , vq  1  cos q
T
kˆ  k , k , k 
x
y
q
{A}
z
•
The inverse solution (i.e., given a rotation
matrix, find k and q):
•
The product of the unit vector k and angle
q , w = q k = (wx, wy, wz) is a minimal
representation for a 3D rotation
 r11  r22  r33  1 

2


 r32  r23 


1
kˆ 
r

r
 13 31 
2 sin q 

 r21  r12 
q  acos 
Note that (-k,-q)
is also a solution
8
Colorado School of Mines
Computer Vision