Graphics Reference
In-Depth Information
9.4.4. Theorem.
t(s) = 0 if and only if the curve F(s) is a planar curve.
Proof.
See [Lips69].
9.4.5. Proposition.
If the curve F(s) is an arc-length parameterization, then
¢
()
≤
()
¢≤
()
Fs
Fs
Fs
Ê
ˆ
1
1
Á
Á
˜
˜
()
=
(
¢
()
¥≤
()
)
∑¢≤
()
=
t s
Fs
F s
F s
det
.
(9.16)
2
2
()
()
k
s
k
s
Ë
¯
Proof.
This is a straightforward computation using the various definitions and
vector identities, namely,
t =- ¢∑
=-
BN
TN N
(
)
¢∑
[
]
∑=- ¥¢
(
)
∑
¥
= -
TN T NN
¥
¢+
¢ ¥
TN N
¢
T
¢
T
¢
k
TT
≤-
¢ ¢
k
T
TT
≤
¢
¯
∑
Ê
ˆ
˜
=-
Ê
Ë
Ê
Ë
ˆ
¯
¢
ˆ
¯
∑
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
Ê
Ë
ˆ
¯
∑
Ê
Ë
ˆ
¯
=-
T
¥
=-
T
¥
T
¥
Á
k
k
2
k
k
k
k
1
2
1
(
)
∑¢=-
(
)
∑¢≤
=-
TT T
¥≤
F F F
¢¥≤
.
2
k
k
Next, we want to determine the well-known equations that relate the vector fields T¢,
N¢, and B¢ to the vector fields T, N, and B along the curve. We already know that T¢=kN
and B¢=-tN. Let
N TbNcB
¢=
+
+
.
Since N • N = 1, it follows that N¢ •N = 0, that is, N¢ is orthogonal to N. This means
that b = 0. But N • T = 0 implies that
aNT NT
=¢ ∑ =-∑¢ =-∑
N N
k
=-
k,
and N • B = 0 implies that
(
)
=
cNB NB
=¢ ∑ =-∑¢ =-∑-
N
t
N
t.
Collecting these facts gives us the well-known theorem below due to Serret (1851) and
Frenet (1847):
9.4.6. Theorem.
(The Serret-Frenet Formulas) The following equations hold for
arc-length parameterizations:
T N
NT
¢=
¢=-
k
k
+
t
B
B
¢=
-
t
N
