Graphics Reference
In-Depth Information
The Serret-Frenet formulas are the key to proving the fundamental theorem for
space curves:
9.4.7. Theorem.
Let k, t : [0,L] Æ
R
be continuous functions with k(s) > 0. Then
there is a
unique
(up to a rigid motion) curve F : [0,L] Æ
R
3
parameterized by arc-
length whose curvature and torsion functions are the functions k and t, respectively.
Proof.
The existence part of this theorem involves solving differential equations
associated to the Serret-Frenet formulas. The uniqueness part is easier. See [Lips69]
or [Spiv70b].
Since one is rarely given arc-length parameterizations, it is convenient to have
formulas for the curvature and torsion of space curves with respect to other
parameterizations.
Let F(t) be an arbitrary regular curve and assume that F(t) = G(a(t)), where G(s)
is the arc-length parameterization and s =a(t). Just like in the planar case, we shall
assume that the basic geometric properties defined by G(s), such as curvature and
torsion, can be assumed to be associated to
points
on the curve and hence to F(t). If
k
G
(s) and t
G
(s) and the curvature and torsion functions associated to G(s) and if
(T
G
(s),N
G
(s),B
G
(s)) is the Frenet frame for G(s), then define the corresponding func-
tions for F(t) by
()
=
(
()
)
kk
t
t
a
a
a
a
a
t
,
,
,
G
()
=
(
()
)
t
t
t
G
()
=
(
()
)
Tt
T
t
G
()
=
(
()
)
N t
N
t
,
.
and
G
()
=
(
()
)
Bt
B
t
(9.17)
G
9.4.8. Theorem.
(The generalized Serret-Frenet Formulas) Given a regular curve
F(t), then the functions defined by Equations (9.17) satisfy
T v N
NvT
¢=
¢=-
k
k
+
vB
t
B
¢=
-
v N
t
where v(t) = |F¢(t)| is the speed function of F(t).
Proof.
This follows from the chain rule applied to the functions in equations (9.17)
and Theorem 9.4.6.
9.4.9. Theorem.
If F(t) is an arbitrary regular curve with F≤(t) π 0 (equivalently,
nonzero curvature), then the functions defined by Equations (9.17) satisfy the
following identities:
Ft
Ft
¢
()
¢
()
()
=
(1)
Tt
