Graphics Reference
In-Depth Information
We isolate by multiplying Eq. (9.10) throughout by
m
m
·
l
+
m
·
m
=
m
·
p
−
d
k
=
=
m
·
p
−
d
k
(9.11)
But from Eq. (9.8), we find that
=
+
+
m
t
21
i
t
22
j
t
23
k
Therefore,
=
t
21
i
+
t
22
j
+
t
23
k
·
p
−
d
k
=
t
21
i
+
t
22
j
+
t
23
k
·
x
P
i
+
y
P
j
+
z
P
k
−
dt
23
and substituting gives
dt
33
=
t
21
i
+
t
22
j
+
t
23
k
·
x
P
i
+
y
P
j
+
z
P
k
−
dt
23
x
P
t
31
+
y
P
t
32
+
z
P
t
33
x
P
t
21
+
y
P
t
22
+
z
P
t
23
=
dt
33
z
P
t
33
−
dt
23
x
P
t
31
+
y
P
t
32
+
The reader may wish to confirm that the values of and reproduce the previously computed
values for different transforms relating
lmn
with
ijk
.
Note that the transform relating
lmn
with
ijk
is derived by concatenating the yaw, roll, and
pitch transforms associated with changing coordinates between two frames of reference. For
instance, the individual transforms are given by
⎡
⎤
⎡
⎤
⎡
⎤
cos roll sinroll 0
−
1
0
0
cos yaw 0
−
sinyaw
⎣
⎦
⎣
⎦
⎣
⎦
sinroll cos roll 0
0
0
cos pitch sinpitch
0
1
0
0
1
0
−
sinpitch cos pitch
sinyaw
cos yaw
and are concatenated as follows:
⎡
⎤
⎡
⎤
l
m
n
i
j
k
⎣
⎦
=
⎣
⎦
rollpitchyaw
⎡
⎤
⎡
⎤
⎡
⎤
l
m
n
t
11
t
12
t
13
t
21
t
22
t
23
t
31
t
32
t
33
i
j
k
⎣
⎦
=
⎣
⎦
·
⎣
⎦
where
t
11
=
cos yaw cos roll
+
sinyaw sinpitch sinroll
t
12
=
cos pitch sinroll
t
13
=−
sinyaw cos roll
+
cos yaw sinpitch sinroll
