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for some functions k
1
(t) and k
2
(t), where v(t) = |p¢(t)|. This follows from the definition
of relatively parallel and a translation of the proof of Theorem 9.16.6 in [AgoM05]
into this context. One then simply pieces together the unique local solutions defined
by their initial conditions.
11.13.5 Theorem.
The following relationships exist between the curvature and
torsion of a curve p(t) and the functions k
1
(t) and k
2
(t) in equations (11.136) for its
parallel transport frames:
2
2
()
=
()
+
()
k
t
kt
kt
1
2
()
()
d
dt
q
kt
kt
Ê
Ë
ˆ
¯
2
1
()
=
()
()
=
t
t
t
,
where
q
t
arctan
.
Proof.
See [Bish75]. The formula for k(t) is clear from the definition of the curva-
ture function. To get the formula for torsion function t(t) assume that we have arc-
length parameterization and write the principal normal N in the form
=
(
)
+
(
)
N
cos
q
n
sin
q
n
.
(11.137)
1
2
Assuming that the frame field (T,n
1
,n
2
) has been oriented correctly, one gets that the
binormal B is defined by
(
)
+
(
)
B
=-
sin
q
n
cos
q
n
,
1
2
since that is a vector orthogonal to the N (and T). Differentiating equation (11.137) gives
(
(
)
+
(
)
)
NT
¢=-
k
k
+ ¢
q
-
sin
q
n
cos
q
n
1
2
=-
TB
+ ¢
q
.
(11.138)
Finally, comparing equation (11.138) with the Serret-Frenet formulas shows us that
t =q¢.
Here are some points to keep in mind when deciding on frames for curves. The
advantage of Frenet frames is that they are defined locally. Their disadvantage is that
one cannot use them if second derivatives of curves are zero. The advantage of par-
allel transport frames is that they are defined for arbitrary regular curves. Their dis-
advantage is that, being basically solutions to differential equations, errors may
accumulate as one moves far from the start point. One can of course switch back and
forth between Frenet and parallel transport frames as appropriate. If one uses curves
and Frenet frames to define tubes, one will observe a great deal of twisting near points
on the curve with small curvature or large torsion. This undesirable property is readily
explained by the generalized Serret-Frenet formulas (Theorem 9.4.8 in [AgoM05])
since the derivatives of the normal and binormal vectors will be large at those places.
Reducing such twisting is one reason for using parallel transport frames. On the other
