Graphics Reference
In-Depth Information
9.2.3. Lemma.
Let F and G be C
1
curves that parameterize the same set
C
. If G(s)
= F(f(s)) is a regular reparameterization of
C
, then F and G have the same length.
Proof.
We leave it to the reader to flesh out the discussion above to a complete proof
that handles the case f¢ < 0. (We only know that f¢ π 0.)
9.2.4. Lemma.
Any compact curve
C
admits a proper regular parameterization
F : [a,b] Æ
C
.
Sketch of Proof.
Since
C
is compact, there are a finite number of local
parameterizations that cover
C
. The map F is gotten by piecing together these local
parameterizations.
We need compactness in Lemma 9.2.4 because F is supposed to have a compact
domain.
Definition.
Let
C
be a compact curve in
R
n
. Define the
length
of
C
, denoted by
length(
C
), to be the length of a proper regular parameterization of
C
.
Compactness is needed here because, for example, the x-axis is a curve and it
clearly does not have finite length.
9.2.5. Theorem.
The length of a compact curve
C
is well defined.
Sketch of Proof.
There are two steps involved in the proof. First, one needs to know
that proper regular parameterizations F exist. This is Lemma 9.2.4. Next, one has to
show that the length does not depend on the choice of F. This is done by showing that
any other such map is a regular reparameterization of F and applying Lemma 9.2.3.
We have seen that sets do not have unique parameterizations. However, there is
one parameterization for curves that is particularly nice.
If a differentiable curve F : [a,b] Æ
R
n
(a π b), satisfies the property that
Definition.
t
Ú
Ft
¢=
a
for all t Π[a,b], then F is called the
arc-length parameterization
of the set
C
= F([a,b]).
(Arc-length parameterization is left
undefined
for single-point sets because it is unin-
teresting and is never used.)
Intuitively, arc-length parameterization has the property that at time t we are a dis-
tance t along the curve from the start point. This property also defines arc-length para-
meterizations of a path uniquely (given a starting point). Furthermore, a simple
consequence of the definition is that the lower bound a must be 0 since the integral from
a to a is 0. Also, since b is just the length L of the curve, one usually writes the domain of
the curve as [0,L]. Another convention is that the parameter s is used rather than t.
Note.
In the future,
using s
as the parameter of a parametric curve
will mean
that
we are dealing with an
arc-length
parameterization.
