Graphics Reference
In-Depth Information
¢
()
¥≤
()
¢
()
¥≤
()
Ft
F t
()
=
(2)
Bt
Ft
F t
(3) N(t) = B(t) ¥ T(t)
¢
()
¥≤
()
¢
()
3
Ft F t
Ft
()
=
(4)
k t
¢
()
≤
()
¢≤
()
Ft
Ft
Ft
Ê
ˆ
(
¢
()
¥≤
()
)
∑¢≤
()
¢
()
¥≤
()
Ft
F t
F t
1
()
=
Á
Á
˜
˜
t t
=
det
.
(5)
2
2
¢
()
¥≤
()
Ft
F t
Ft
F t
Ë
¯
Proof.
This is easy if one copies what we did for arc-length parameterizations above.
See [Gray98] or [Spiv70b].
An immediate corollary of Theorems 9.4.4 and 9.4.9 is
9.4.10. Theorem.
The regular curve F(t) is a planar curve if and only if
¢
()
≤
()
¢≤
()
Ft
Ê
ˆ
(
¢
()
¥≤
()
)
∑¢≤
()
=
Á
Á
˜
˜
Ft
F t
F t
det
F t
Ft
=
0
.
Ë
¯
Finally, we note that the Serret-Frenet formulas generalize to curves in
R
n
.
9.4.11. Theorem.
Let F(s) be a curve in
R
n
, n ≥ 3, parameterized by arc-length. If
the vectors F¢(s), F≤(s),...,F
(n)
(s) are linearly independent, then at each point F(s) on
the curve there is an orthonormal basis of vectors u
1
(s), u
2
(s),...,u
n
(s) with the func-
tion u
i
(s) satisfying the differential equations
¢
=
¢
=-
u
k
u
1
1
2
u
k
u
+
k
u
2
1
1
2
3
¢
=-
u
k
u
+
k
u
3
2
2
3
4
.
.
.
.
.
.
¢
u
=-
k
u
+
k
u
n
-
1
n
-
2
n
-
2
n
-
1
n
¢
=-
u
k
u
n
n
--
11
n
Proof.
See [Spiv75].
Definition.
The functions u
1
,u
2
,...,u
n
in Theorem 9.4.11 are called a
Frenet basis
and the functions k
i
are called
generalized curvatures
for the curve F(s).
If n = 3, then k
1
is the ordinary curvature of a curve in 3-space and k
2
is the
torsion.
