Page 206 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 18: Theory of Space Curves

18.1 Arc Length Notes

A parametrized curve in Euclidean three-space e is given by a vector function
3
x(t) = (x (t), x (t), x (t))
3
1
2
that assigns a vector to every value of a parameter t in a domain interval [a, b]. The coordinate
functions of the curve are the functions x (t). In order to apply the methods of calculus, we
i
suppose the functions x (t) to have as many continuous derivatives as needed in the following
i
treatment.
For a curve x(t), we define the first derivative x’(t) to be the limit of the secant vector from x(t) to
x(t+h) divided by h as h approaches 0, assuming that this limit exists. Thus,


x'(t) lim   x(t h) x(t)   .

h 0  h 
The first derivative vector x’(t) is tangent to the curve at x(t). If we think of the parameter t as
representing time and we think of x(t) as representing the position of a moving particle at time
t, then x’(t) represents the velocity of the particle at time t. It is straightforward to show that the
coordinates of the first derivative vector are the derivatives of the coordinate functions, i.e.
x’(t) = (x ’(t), x ’(t), x ’(t)).
1
3
2
For most of the curves we will be concerning ourselves with, we will make the “genericity
assumption” that x’(t) is non-zero for all t. Lengths of polygons inscribed in x as the lengths of
the sides of these polygons tend to zero. By the fundamental theorem of calculus, this limit can
be expressed as the integral of the speed s’(t) = |x’(t)| between the parameters of the end-points
of the curve, a and b. That is,

b b 3
'
2

s(b) s(a)   |x'(t)|dt    x (t) dt.
i
a a i 1

For an arbitrary value t  (a, b), we may define the distance function
t
s(t) – s(a) = |x'(t)|dt,

a
which gives us the distance from a to t along the curve.
Notice that this definition of arc length is independent of the parametrization of the curve. If we
define a function v(t) from the interval [a, b] to itself such that v(a) = a, v(b) = b and v’(t) > 0, then
we may use the change of variables formula to express the arc length in terms of the new
parameter v:

b b
v
 |x'(t)|dt   (a) a (b)  b x'(v(t)) v'(t)dt   x'(v) dv.

a v a
We can also write this expression in the form of differentials:
ds = |x’(t)| dt = |x’(v)| dv.

This differential formalism becomes very significant, especially when we use it to study surfaces
and higher dimensional objects, so we will reinterpret results that use integration or

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