Graphics Reference
In-Depth Information
Figure 9.2.
Curve length is inde-
pendent of change of
parameters.
G(s) = F(f(s)) = F(t)
G
F
f
a
t
b
c
s
d
vation we want to make now will have some important consequences later. The
formula in Proposition 9.2.1 shows that the length of a parametric curve is deter-
mined by the lengths of its tangent vectors. Tangent vectors lie in ordinary vector
spaces and so the essential ingredient in a definition of the length of a curve is a notion
of the length of vectors in vector spaces. We shall see in Sections 9.8 and 9.17 that
abstractly it is better not to think of length as a generalization of line segment length
but rather to think of it more generally as derived from an inner product on vector
spaces.
From Example 9.2.2 we see that three different parameterizations of the segment
from [0,0] to [1,1] produced the same length. This is of course what we want. We do
not want length to be something that depends on the parameterization of a set. To
what extent is this true? If we think of the parameter as time and the parametric curve
itself as describing a path along which one is walking, then two paths are going to
have different lengths if one of them backtracks and the other does not. But if we are
interested in the length of a set then we do not want to allow any backtracking anyway
and so it is not unreasonable to restrict ourselves to comparing two parametric curves
that are both, at least locally, one-to-one functions.
Let
C
be the underlying set of two C
1
curves F : [a,b] Æ
C
and G : [c,d] Æ
C
. Let
f : [c,d] Æ [a,b] be a function such that G(s) = F(f(s)), and let t =f(s). See Figure 9.2.
The map f can be thought of as a change in coordinates. We have the following chain
of equalities
d
d
Ú
Ú
¢
()
lG
=
G s ds
c
d
(
()
)
¢
()
=¢
Fs s s
ff
c
d
Ú
Ú
(
()
)
¢
()
(
)
=¢
Fs
ff
s s
if
f
¢>
0
c
b
=¢
()
(
)
Ft dt
change of variables
a
b
=
lF
.
This shows what we were trying to show, namely, that the length of a parametric curve
is basically an invariant of the underlying set. The next two facts allow us to define
the length of this set precisely.
