Graphics Reference
In-Depth Information
and
d
N
G
Ê
Ë
ˆ
¯
()
=
(
)
p
p
1
-
=
pd
1
-
k
,
(9.79b)
d
v
v
2
v
where k
i
are the principal normal curvatures of
S
. Let E
d
, F
d
, and G
d
be the coeffi-
cients of the first fundamental form for
S
d
. Their definition and equations (9.79)
imply that
2
2
(
)
(
)
EEd
=-
1
k
,
F
=
0
,
GGd
=
1
-
k
.
(9.80)
d
1
d
d
2
Next, let N
d
(u,v) be the normal vector to the surface
S
d
at p
d
(u,v) and
(
)
Nuv
Nuv
,
,
d
d
(
)
=
n
d
uv
,
.
(9.81)
(
)
Then again using equations (9.79),
(
)
(
(
)
=
()( )
¥
()( )
=-
(
)
2
)
N uv
,
p
uv
,
p
uv
,
1
kk kk
+
d
+
d Nuv
,
d
d
u
d
1
2
1
2
v
that is,
9.14.1. Theorem.
(
)
(
(
)
=-
2
)
Nuv
,
12
HdKdNuv
+
,
,
(9.82)
d
where K and H are the Gauss and mean curvatures of the surface
S
at p(u,v), respec-
tively.
Since the expression 1 - 2Hd + Kd
2
in equation (9.82) factors into (1 - dk
1
)(1 - dk
2
),
we always need to choose a d so that neither of these factors is zero. In fact, the map
SS
Æ
d
(
)
Æ
(
)
+
(
)
puv
,
puv
,
d
n
uv
,
is then one-to-one in a neighborhood of p(u,v). Furthermore, note that the parallel
surface could be oriented in the opposite direction from the original surface. That
happens when
2
12
-
Hd
+
Kd
<
0
.
Define s to be the sign of that expression, that is,
nn
d
=s .
(9.83)
Next, the definition of the coefficients L
d
, M
d
, and N
d
of the second fundamental
form of
S
d
and the formulas above imply that
