Length and Arc length Parametrization of Curves
Arc length
Consider a differentiable vector-valued function for . It parametrizes a curve in . How can we calculate the arc length of this curve?
First, we divide the interval into "subdivisions" as follows:
and for .
Furthermore, we assume that all are small enough such that
Let and for . We approximate the arc length of the curve by taking the sum of the length of the line segment joining the points on the curve corresponding to the subdivision, as shown in the applet below.
For each line segment, the length is , which can be approximated by . Therefore, we have
(The last equality is the definition of the definite integral using Riemann sum.)
As for parametric curves in , we have the similar formula for its arc length. Suppose the curve is parametrized by for . Then the arc length of the parametric curve is
Exercise: A circular helix in is parametrized by for . Find its arc length.
Reparametrization
Let be a smooth strictly increasing function i.e. is infinitely differentiable and . Given with , which parametrizes a curve in . Then we let and define the vector-valued function of as follows:
with
(Note: is the inverse function of .)
Actually, parametrizes the same curve in . This is called a reparametrization of the parametric curve.
Examples:
Suppose with .
For , with .
For , with .
We would expect the arc length of a parametric curve remains unchanged after a reparametrization. The following theorem confirms this:
Theorem: The arc length of the curve parametrized by from to equals the arc length of the curve parametrized by , where , from to i.e.
Proof:
(since )
(using substitution , we have ).
In the applet below, the right panel shows the reparametrization of the curve when . You can see that the parametric curves for both parametrizations are exactly the same.
Reparametrization by arc length
Given a vector-valued function with , which parametrizes a curve in . We want to find a reparametrization such that the arc length from to for any . Therefore, we have
Take the helix with as an example.
Therefore, , which implies that . The desired reparametrization is as follows:
,
This is called the reparametrization by arc length.
Suppose we want to find the point on the helix such that the arc length from to is . We just need to set i.e.
is the required point on the helix.
Remark: The reparametrization of a curve in is defined in a similar way.
Exercise: Suppose for . Find its reparametrization by arc length.