Graphics Reference
In-Depth Information
Ta b l e 9 . 3
Vertex coordinates
for the cube in Fig.
9.5
(d)
vertex
0
1
2
3
4
5
6
7
x
0
1
0
1
0
1
0
1
y
00
−
1
−
100
−
1
−
1
z
0
0
0
0
1
1
1
1
coordinates shown in Table
9.1
by the matrix (
9.6
). We show the matrix multiplying
an array of coordinates as before:
⎡
⎤
⎡
⎤
001
0
00001111
00110011
01010101
⎣
⎦
⎣
⎦
10
100
−
⎡
⎤
01010101
00
⎣
⎦
=
1
00001111
−
1
−
100
−
1
−
which agree with the coordinates in Table
9.3
, and we can safely conclude that, in
general, 3D rotation transforms do not commute. Inspection of Fig.
9.5
(d) shows
that the unit cube has been rotated 180° about a vector
T
.
Now let's explore the role eigenvectors and eigenvalues play in 3D rotations.
[
]
101
9.3.1 3D Eigenvectors
In Chap. 4 we examined the characteristic equation used to identify any eigenvectors
associated with a matrix. The eigenvector
v
satisfies the relationship
Av
=
λ
v
where
λ
is a scaling factor.
In the context of a 3D rotation matrix, an eigenvector is a vector scaled by
λ
but not rotated, which implies that it is the axis of rotation. To illustrate this, let's
identify the eigenvector for the composite rotation (
9.3
) above:
⎡
⎤
001
010
⎣
⎦
.
R
90
°
,z
R
90
°
,y
R
90
°
,x
=
−
100
Figure
9.4
(a)-(d) shows the effect of this composite rotation, which is nothing
more than a rotation of 90° about the
y
-axis. Therefore, we should be able to extract
this information from the above matrix.
We begin by writing the characteristic equation for the matrix:
0
−
λ
0
1
01
−
λ
0
=
0
.
(9.7)
−
1
0
0
−
λ
