Graphics Reference
In-Depth Information
and
F
N
N
P
v
=
P
v
+
F
v
N
+
N
v
To simplify these partial derivatives, it is assumed that the relative magnitude of F is negligible
and can be ignored. Thus,
N
P
u
=
P
u
+
F
u
N
and
N
P
v
=
P
v
+
F
v
N
which permits Eq. (10.7) to be written as
P
u
+
N
P
v
+
N
F
u
F
v
N
=
×
(10.8)
N
N
Basically, Eq. (10.8) has the form of four vectors:
a
+
b
×
c
+
d
which, when expanded, equals
a
+
b
×
c
+
d
=
a
×
c
+
b
×
c
+
a
×
d
+
b
×
d
Expanding Eq. (10.8) along similar lines, we obtain
N
P
u
×
P
u
×
P
v
+
F
u
P
v
F
v
N
F
u
F
v
N
N
N
=
×
+
+
×
N
N
N
N
But
N
N
×
=
0
N
N
and
P
u
×
P
v
=
N
Therefore,
P
u
×
F
u
N
P
v
F
v
N
N
=
N
+
×
+
N
N
or
N
P
u
×
N
F
u
P
v
F
v
×
+
N
=
N
+
N
