Graphics Reference
In-Depth Information
Note that the direction vector
u
that is chosen for the axis matters because
(
p
,-
u
,q) = (
p
,
u
,-q).
Definition.
The rotation defined by the triple (
p
,
u
,q) is called the
rotation about the
directed line
defined by the point
p
and direction vector
u
through the angle
q.
We can represent a rotation more compactly by incorporating the angle of the
rotation in the length of the direction vector for the axis.
A compact axis-angle representation of a rotation:
Here a pair (
p
,
v
) represents
the rotation whose axis-angle representation is (
p
,
v
/|
v
|,|
v
|).
The next two characterizations of rotations are in terms of rotating about coor-
dinate axes. Fortunately, when it comes to rotations we only need to know the equa-
tions of the rotations about the coordinate axes by heart. It is therefore worthwhile
to summarize those before moving on since the equations for all other rotations can
be derived from them.
The equations and matrices for the rotations about the coordinate axes:
rotation about x-axis
rotation about y-axis
rotation about z-axis
xx
y
¢=
¢=
xx
¢=
cos
q
+
z
sin
q
xx
¢=
cos
q
-
y
sin
q
y
cos
q
-
z
sin
q
y
¢=
¢=-
y
yx
¢=
sin
q
+
z
cos
q
(2.34)
z
¢=
y
sin
q
+
z
cos
q
z
x
sin
q
+
z
cos
q
z
¢=
z
cos
q
s
in
q
0
0
cos
0
01 0
0
q
-
s
in
q
10
0
Ê
ˆ
Ê
ˆ
Ê
ˆ
Á
Á
˜
˜
Á
Á
˜
˜
Á
Á
˜
˜
(2.35)
-
s
in
q
cos
q
0
0
cos
q
s
in
q
Ë
¯
Ë
¯
Ë
¯
0
0
1
s
in
q
cos
q
-
s
in
q
cos
q
Note that the minus sign in the equation and matrix of a rotation about the y-axis is
different from the other rotations. The reason is that we are expressing things in world
coordinates and when looking down the y-axis, the x-axis is pointing to the left which is
the wrong direction because angles are oriented according to the basis (-
e
1
,
e
3
).
2.5.1.1. Theorem.
Consider a coordinate system specified by a frame F =
(
u
1
,
u
2
,
u
3
,
p
). If R is a rotation about
p
, then R is the composite of a rotation R
1
about
the x-axis of F through an angle a, a rotation R
2
about the y-axis of F through an
angle b, and finally a rotation R
3
about the z-axis of F through an angle t.
Proof.
Assume first that we are rotating about the origin and F is the standard frame.
With this hypothesis, R is a linear transformation and has matrix
aaa
aaa
aaa
Ê
ˆ
11
12
13
Á
Á
˜
˜
.
(2.36)
21
22
23
Ë
¯
31
32
33
The rotations R
1
, R
2
, and R
3
would also be linear transformations and we know their
matrices from (2.35). Therefore, using the abbreviations
