FrenetSerret Apparatus
Data
A space curve given by the position vector
r
(t).
Objective
Find the functions for arc length, s(t), and curvature, κ(t). Find the unit vector
functions
T
(t),
N
(t) and
B
(t).
Method
Differentiate
r
with respect to t,
d
r
dt
, which is the velocity vector. Its length is the
speed, so the arc length satisfies
ds
dt
=
∣
d
r
dt
∣
and
s
t
=
∫
t
0
t
∣
d
r
d
∣
d
, using τ as the variable of
integration. This is a strictly increasing function of t and so there is an inverse function t=t(s).
Reparameterize by arc length and we have the result from the definitions.
T
s
=
d
r
ds
,
s
N
s
=
d
T
ds
from which we get
s
=
∣
d
T
ds
∣
and
N
s
=
1
d
T
ds
, then
B
=
T
×
N
completing the work.
Difficulty
The integral
∫
t
0
t
∣
d
r
d
∣
d
can only be expressed in a useful form for a few very
special cases.
Fix
Differentiate with respect to t rather than s and use the chain rule. Doing this generates a
number of formulas which we solve opportunistically for the required objects.
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 Spring '06
 YUKICH
 Arc Length, Derivative, Vectors, Differential geometry of curves

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