Lecture Notes for 120 - UCLA Department of Mathematics

1.2. ARCLENGTH AND LINEAR MOTION 14
This is easily shown to be independent of the parameter t as long as the reparametrization
is in the same direction. One also easily checks that a curve on [a, b] is stationary
if and only if its speed vanishes on [a, b]. Weusuallysuppresstheinterval
and instead simply write L (q).
Example 1.2.1. If q (t) =q 0 + v 0 t is a straight line, then its speed is constant
|v 0 | and so the arclength over an interval [a, b] is |v 0 | (b a).
Example 1.2.2. If q (t) =R (cos t, sin t)+c is a circle of radius R centered at
c, thenthespeedisalsoconstantR and so again it becomes easy to calculate the
arclength.
Example 1.2.3. Consider the hyperbola x 2 y 2 =1. It consists of two components
separated by the y-axis. The component with x>0 can be parametrized
using hyperbolic functions q (t) = (cosh t, sinh t). The speed is
ds p
p
dt = sinh 2 t + cosh 2 t = 2sinh 2 t +1= p cosh 2t
While this is both a fairly simple curve and a not terribly difficult expression for
the speed it does not appear in any way easy to find the arclength.
Definition 1.2.4. Acurveiscalledregular if it is never stationary, or in
other words the speed is always positive. A curve is said to be parametrized by
arclength if its speed is alway 1. Suchaparametrizationisalsocalledaunit speed
parametrization.
Lemma 1.2.5. A regular curve q (t) can be reparametrized by arclength.
Proof. If we have a reparametrization q (s) of q (t) with ds
dt
> 0 that has unit
speed, then
dq ds
ds dt = dq
dt = v
so it follows that
ds
dt = dq = |v|
dt
must be the speed of q (t).
This tells us that we should define the reparametrization s = s (t) as the antiderivative
of the speed:
ˆ t1
dq
s (t 1 )=s (t 0 )+ dt
t 0
dt
It then follows that
ds
dt = dq > 0
dt
Thus it is also possible to find the inverse relationship t = t (s) and we can define
the reparametrized curve as q (s) =q (s (t)) = q (t).
This reparametrization depends on specifying an initial value s (t 0 ) at some
specific parameter t 0 . For simplicity one often uses s (0) = 0 if that is at all
reasonable.
⇤
To see that arclength really is related to our usual concept of distance we show: