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Let F : [0,L] Æ R n
9.2.6. Proposition.
be the arc-length parameterization of some
path. Then for all s Π[0,L]
(1) |F¢(s)| = 1, and
(2) F¢(s)•F≤(s) = 0.
Proof. Property (1) follows from the Fundamental Theorem of Calculus by differ-
entiating the defining equation for arc-length parameterization. Next, we can restate
(1) in the form
¢ () () = 1.
Fs Fs
Differentiating both sides of this equation gives (2).
What Proposition 9.2.6 says is that if we are dealing with an arc-length parame-
terization of a path, then the length of the tangent vector of that parameterization (or
its speed) is 1 and the tangent vector is orthogonal to its second derivative. As we go
along, we shall see that by using arc-length parameterization we can usually give much
cleaner definitions of geometric concepts associated to paths (sets that we are typi-
cally after and not some accidental parameterization of them) and proofs will be
easier. The practical downside is that paths are unfortunately not usually presented
via such parameterizations, and so the question arises as to how one might find the
arc-length parameterization of a path. We address this question next.
Suppose that F : [a,b] Æ R n is a curve. The curve F may not be the arc-length para-
meterization for C = F([a,b]). To find the arc-length parameterization G(s) for C let us
look for a change in coordinate function such that G(s) = F(f(s)). See Figure 9.2 again.
Our earlier discussion then shows that
s
t
f -
1
() ==
Ú
Ú
ts
G
¢=
F
¢
(9.1)
0
a
This means that we know f -1 and can therefore solve for the function f itself.
9.2.7. Theorem. Every regular parameterization of a compact curve can be repa-
rameterized into an arc-length parameterization for the curve with an orientation-
preserving change of coordinates. The resulting unique arc-length parameterization
is called the induced arc-length parameterization .
Proof.
The theorem follows easily from the observations preceding it.
Equation (9.1) leads to two other useful observations, namely,
ds
dt
()
Ft
(9.2)
and the following shorthand relation between differential operators
d
ds
1
d
dt
=
.
(9.3)
¢ ()
F t
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