Arc Length
Philippe B. Laval
KSU
Today
Philippe B. Laval (KSU)
Arc Length
Today
1 / 12
Introduction
In this section, we discuss the notion of curve in greater detail and
introduce the very important notion of arc length and learn how to
reparametrize a curve with respect to arc length.
Philippe B. Laval (KSU)
Arc Length
Today
2 / 12
Parametrizations of a Curve
We have described the curve C with position vector
!
r (t) = hf (t) ; g (t) ; h (t)i for t 2 [a; b] as the set of points (x; y ; z)
where
8
< x = f (t)
y = g (t)
:
z = h (t)
as t varies in the interval [a; b]. In fact, a curve is a little more than just a
set of points. It is a succession of points traversed in a certain order. As t
varies from a to b, the points obtained will trace the curve in a certain
order. We say that a parametrized curve is an oriented curve.
Often, the parameter represents time. The equations of the curve tell us
where we are on the curve as a function of time. Other parameters can be
used. For example, we could have equations which tell us where we are on
the curve depending on how far we have traveled on the curve.
Philippe B. Laval (KSU)
Arc Length
Today
3 / 12
Parametrizations of a Curve
Given a curve C with position vector !
r (t), t 2 [a; b], if t can be
expressed in terms of another parameter say u with the function t = f (u)
then we can write !
r (t) = !
r (f (u)). We simply replaced t by f (u) since
the two are equal. The result is a function of u. In other words, we have
changed the parameter of the curve. In order to describe the same portion
of the curve, we will also need to change the interval of values of the
parameter. Such an operation is called a reparametrization of the curve
C . The new parametrization describes the same curve, simply written in a
di¤erent form.
If f is an increasing function, then C will be traversed in the same
direction. If f is a decreasing function then C will be traversed in opposite
direction.
Philippe B. Laval (KSU)
Arc Length
Today
4 / 12
Parametrizations of a Curve
Example
Consider the curve !
r (t) = hcos t; sin t; ti, t 2 [0; 2 ]. Reparametrize it
with the parameter u given by t = f (u) = u. Is the curve traversed in
the same order?
Example
Consider the curve !
r (t) = hcos t; sin t; ti, t 2 [0; 2 ]. Reparametrize it
with the parameter v given by t = g (v ) = 2 u. Is the curve traversed in
the same order?
Example
Consider the curve !
r (t) = hcos t; sin t; 0i, t 2 [0; 2 ]. Reparametrize it
with the parameter u given by t = h (u) = 2
u. Is the curve traversed
in the same order?
Philippe B. Laval (KSU)
Arc Length
Today
5 / 12
Arc Length
Of all the possible parameters for a curve, one parameter plays a very
important role. It is arc length. We …rst give the de…nition of the length of
the portion of a curve !
r (t) between t = a and t = b, valid for 2-D curves
as well as 3-D curves. Though it is given here as a de…nition, the formula
can actually be proven.
De…nition (Arc Length)
Let C be a smooth curve with position vector !
r (t) for t 2 [a; b]. The
length of the portion of the curve between t = a and t = b is
L=
Z
a
b
!
r 0 (t) dt
It can also be proven that L as computed above does not depend on the
parametrization used.
Philippe B. Laval (KSU)
Arc Length
Today
6 / 12
Arc Length
Example
Compute the circumference of a circle of radius a, a > 0.
Example
Compute the length of the arc of the circular helix !
r (t) = hcos t; sin t; ti
from the point (1; 0; 0) to the point (1; 0; 2 ).
Philippe B. Laval (KSU)
Arc Length
Today
7 / 12
Arc Length
We saw above that curves could be parametrized di¤erent ways. For
motion along a curve, time is often the parameter of choice. However, to
describe the geometric properties of a curve time is not the most
appropriate parameter. The parameter of choice is arc length. We explain
how this is done and why.
De…nition (Arc length Function)
Let C be a smooth curve with position vector !
r (t) for t 2 [a; b] and
suppose C is traversed only once as t increases from a to b. We de…ne the
arc length function from t = a, denoted s or s (t) to be
Z t
!
s (t) =
r 0 (u) du
a
Philippe B. Laval (KSU)
Arc Length
Today
8 / 12
Arc Length
Geometrically, s (t) is the length of the arc of C between !
r (a) and !
r (t).
So, it is always a positive quantity which increases as t increases. From
the de…nition of t and using the fundamental theorem of calculus, we have
the following theorem.
Theorem
If s is the arc length function de…ned above then:
ds
0
= !
r (t)
dt
We now show how a curve can be parametrized with respect to arc length.
Goal: Given a smooth curve C with position vector !
r (t), t 2 I for some
interval I , we want to reparametrize C with respect to arc length that is
we need to …nd a relationship between t and s. To do this, we compute
s (t) using the above formula for s (t). We illustrate this procedure with a
few examples.
Philippe B. Laval (KSU)
Arc Length
Today
9 / 12
Arc Length
Example
Reparametrize the helix !
r (t) = hcos t; sin t; ti with respect to arc length
measured from the point (1; 0; 0).
Example
Reparametrize a circle of radius a centered at the origin given by
!
r (t) = ha cos t; a sin ti, t 2 [0; 2 ] with respect to arc length.
We invite the reader to check that in both examples above, we have
!0
R (s) = 1. In fact, it can be proven that this is always true. We state
this result as a theorem we will not prove.
Philippe B. Laval (KSU)
Arc Length
Today
10 / 12
Arc Length
Theorem
Let C be a smooth curve with position vector !
r (s) where s is the arc
length parameter. Then
!
r 0 (s) = 1
Furthermore, if t is any parameter such that !
r 0 (t) = 1, then t must be
the arc length parameter.
Corollary
Let C be a smooth curve with position vector !
r (s) where s is the arc
length parameter. Then
!
T (s) = !
r 0 (s)
Philippe B. Laval (KSU)
Arc Length
Today
11 / 12
Exercises
See the problems at the end of my notes on vector functions: arc length.
Philippe B. Laval (KSU)
Arc Length
Today
12 / 12