Graphics Reference
In-Depth Information
Assume that g (0) =
p
Œ
S
. First of all, since g≤ = k
N
, the normal curvature k
n,g
(
p
) is
now also defined by the formula
()
=≤
()
∑
()
k
n
p
g
t
n p
.
,
g
Next, to simplify the notation, let
n
(s) be shorthand for
n
(g (s)). Then
n
(s) • g¢(s) = 0
and differentiating both sides of this identity gives
¢
()
∑
()
=-
()
∑≤
()
n
s
g
s
n
s
g
s
.
(9.44)
It follows that
(
¢
()
)
=-
()
¢
(( )
∑
()
=- ¢
()
∑
()
=
Q
g
0
D
np
g
0
g
0
II
n
00
g
()
∑≤
()
(
(
)
)
n
n
0
g
0
by Equation
9 44
.
()
∑
() ()
=
000
k
N
()
=
k
p
,
n
,
g
and we have shown
9.9.6. Theorem.
The value of the second fundamental form at a vector
v
in the
tangent space to a surface at some point
p
is the normal curvature of any regular
curve in the surface through
p
with tangent vector
v
there.
Theorem 9.9.6 proves Meusnier's theorem, which we restate now in the following
way:
9.9.7. Theorem.
(Meusnier) All regular curves lying in a surface
S
passing through
a point
p
and having the same tangent vector at
p
have the same normal curvatures
there.
Note that by definition
(
¢
()
)
=
(
¢
()
)
∑
()
Q
II
g
0
S
g
0
g
0
.
p
Therefore, the normal curvature of a curve can be derived from the shape operator.
In fact, by normalizing vectors, we get information about the curvature of the surface.
Definition.
Let
u
be a unit vector in the tangent space to
S
at
p
. Then the quantity
()
=
()
∑
k
n
u
S
u
u
p
is called the
normal curvature
of
S
in the direction
u
.
Note:
There will never be any confusion when using the term “normal curvature” at
a point
p
of a surface
S
. Although we have two variants, one, the normal curvature
k
n,g
of a curve g, and the other, the normal curvature k
n
(
u
) in a direction defined by a
