Graphics Reference
In-Depth Information
c
t
=
cos
t
,
s
t
=
sin
t
,
c
b
=
cos
b
,
s
b
=
sin ,
b
c
a
=
cos
a
,
and
s
a
=
sin
a
,
the matrix for R
3
R
2
R
1
would be
cc
tb
sc
tb
-
s
b
Ê
ˆ
Á
Á
˜
(2.37)
cs s
tba
-
sc
ta
cs s
tba
+
cc
ta
c s
ba
˜
.
Ë
¯
cs c
tba
+
ss
ta
cs c
tba
-
cs
ta
c c
ba
To prove the theorem, all we have to do is set the matrices in (2.36) and (2.37) equal
to each other and solve for a, b, and t. This is not hard using the first row and last
column. If cbπ0, then
(
)
a=
a
tan
2
a
,
a
,
(2.38a)
23
33
(
)
(2.38b)
2
2
b=
a
tan
2
a
,
a
+
a
,
13
11
12
(
)
t=
a
tan
2
a
,
a
,
(2.38c)
12
11
where atan2(y,x) is basically the arctangent tan
-1
(y/x), except that the sign of both x
and y are used to determine into which quadrant the angle falls. See Appendix A for
a precise definition.
The case of an arbitrary frame F is an easy consequence of this case that we leave
as an exercise. The theorem is proved.
Note.
In the proof of Theorem 2.5.1.1, only one choice had to be made in defining
a, b, and t and that was the choice of the positive square root of the sum of squares
of a
11
and a
21
. This amounts to restricting b to lying in the interval [-p/2,p/2]. With
this restriction, the a, b, and t are uniquely determined for R.
Definition.
Given a rotation R, the angles a, b, and t in Theorem 2.5.1.1 are called
the
roll, pitch, and yaw angles of R
, respectively. The tuple [a,b,t,
p
] is called a
roll-
pitch-yaw representation of the rotation R
(with respect to the frame F). [a,b,t] will
denote the roll-pitch-yaw representation in the case where
p
is the origin.
The terminology of “roll,” “pitch,” and “yaw” comes from aviation and navigation.
Roll is the twisting motion about the lengthwise axis of a ship or airplane. Pitch is
the dipping or rising motion of the bow or nose. Yaw is the side-to-side twisting
motion in its horizontal plane about a vertical axis. The note following the theorem
above shows that the roll, pitch, and yaw angles are unique if the pitch angle lies in
the interval [-p/2,p/2].
Instead of rotating about the axes of a fixed coordinate system as is done in
the case of the roll-pitch-yaw representation of a rotation we can do our rotations
about axes in each successive new coordinate system. The choice of axes is up
to us.
2.5.1.2. 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 S
1
about
