User`s guide

Rotation Matrix to VRML Rotation
9-9
Rotation Matrix
A representation of a three-dimensional spherical rotation as a 3-by-3 real, orthogonal
matrix R: R
T
R = RR
T
= I, where I is the 3-by-3 identity and R
T
is the transpose of R.
R
R R R
R R R
R R R
R R R
R R R
xx xy xz
yx yy yz
=
Ê
Ë
Á
Á
Á
ˆ
¯
˜
˜
˜
=
11 12 13
21 22 23
31 32 33
RR R R
zx zy zz
Ê
Ë
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
In general, R requires three independent angles to specify the rotation fully. There are
many ways to represent the three independent angles. Here are two:
• You can form three independent rotation matrices R
1
, R
2
, R
3
, each representing a
single independent rotation. Then compose the full rotation matrix R with respect to
fixed coordinate axes as a product of these three: R = R
3
*R
2
*R
1
. The three angles are
Euler angles.
• You can represent R in terms of an axis-angle rotation n = (n
x
,n
y
,n
z
) and θ with n*n
= 1. The three independent angles are θ and the two needed to orient n. Form the
antisymmetric matrix:
ˆ
J
n n
n n
n n
z y
z x
y x
=
-
-
-
Ê
Ë
Á
Á
Á
ˆ
¯
˜
˜
˜
0
0
0
Then Rodrigues' formula simplifies R:
R J I J J= = + + -exp(

)

sin

( cos )q q q
2
1