Graphics Reference
In-Depth Information
We again define
q
in terms of
p
,sowelet
q
=
p
where is a scalar that requires defining.
As
q
and
p
have the same projection on
n
, we have
n
·
q
=
n
·
p
and
n
·
d
d
n
·
k
=
p
=
p
n
·
n
·
We now define
n
in terms of
ijk
. Therefore, from Eq. (9.8), we find
=
+
+
n
t
31
i
t
32
j
t
33
k
and
dt
33
=
x
P
t
31
+
y
P
t
32
+
z
P
t
33
and Eq. (9.9) becomes
+
=
−
l
m
p
d
k
(9.10)
We can isolate by multiplying Eq. (9.10) throughout by
l
l
·
l
+
l
·
m
=
l
·
p
−
d
k
=
l
·
p
−
d
l
·
k
But from Eq. (9.8) , we have
l
=
t
11
i
+
t
12
j
+
t
13
k
Therefore,
=
t
11
i
+
t
12
j
+
t
13
k
·
p
−
d t
11
i
+
t
12
j
+
t
13
k
·
k
=
t
11
i
+
t
12
j
+
t
13
k
·
x
P
i
+
y
P
j
+
z
P
k
−
dt
13
=
x
P
t
11
+
y
P
t
12
+
z
P
t
13
−
dt
13
Finally, substituting gives
dt
33
=
x
P
t
11
+
y
P
t
12
+
z
P
t
13
−
dt
13
x
P
t
31
+
y
P
t
32
+
z
P
t
33
dt
33
x
P
t
11
+
y
P
t
12
+
z
P
t
13
=
z
P
t
33
−
dt
13
x
P
t
31
+
y
P
t
32
+
