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Arc length

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Let $f$ be a parametric curve

$$f : {\bf R} \rightarrow {\bf R}^n $$

and $b > a$. Define the arc length of the curve from $f(a)$ to $f(b)$.

We "approximate" the curve with segments. Pick $s$ points

$$A_1, A_2, ..., A_s$$

on the curve with

$A_i = f( a_i), a_i < a_{i+1}$ (note that this is a partition of the interval).

The length of segment $[A_i, A_{i+1}]$ is

$$||A_i - A_{i+1}||.$$

Then the total length is

$$L({A_i}) \leq \displaystyle\sum_{i=1}^s ||A_i - A_{i+1}||.$$


Suppose the "true" length is equal to $l$. Then, by the Triangle Inequality, we have

$$L \leq l.$$

Definition. Define the arc length as the least upper bound of the set

$$L({ A_i }). $$

Consider the length of a curve $f: [a, b] \rightarrow {\bf R}^n$. By theorem, the length always exists. If the length is finite,

$$L( f; [a, b] ) \neq \infty, $$

then $f$ is called rectifiable.

Theorem. Suppose

$$f: [a, b] \rightarrow {\bf R}^n$$

is rectifiable and

$${\gamma}: [c, d] \rightarrow [a, b]$$

is monotone and continuous. Then

$$g = f {\gamma}: [c, d] \rightarrow {\bf R}^n$$

is rectifiable, and

$$L( f; [a, b] ) = L ( g; [c, d] ).$$


This theorem is about an re-parametrization of the curve.

$L( f; [a, b] )$ is approximated by the sum discussed above:

$$L({ A_i }) = \displaystyle\sum_{i=0}^{n-1} || f(a_{i+1}) - f(a_i) ||.$$

The idea is to turn this into an integral

$$\displaystyle\int_a^b h(x) dx.$$

As usual, this is done by recognizing the Riemann sum of some function, $h$:

$$\displaystyle\sum_{i=0}^{n-1} h(a_i) {\Delta}x, {\rm \hspace{3pt} where \hspace{3pt}} {\Delta}x = a_{i+1} - a_i.$$

To get to this point we re-write the sum

$$L({ A_i }) = \displaystyle\sum_{i=0}^{n-1} || f(a_{i+1}) - f(a_i) || \frac{{\Delta}x}{{\Delta}x}$$

to make

$$|| f(a_{i+1}) - f(a_i) || (\frac{1}{\Delta x}) = h(a_i).$$

Further

$$L({ A_i }) = \displaystyle\sum_{i=0}^{n-1} \frac{|| f(a_{i+1}) - f(a_i) ||}{a_{i+1} - a_i} {\Delta}x,$$

so that

$$\frac{|| f(a_{i+1}) - f(a_i) ||}{a_{i+1} - a_i} \rightarrow || f'(x) ||.$$

The limit is the following Riemann integral:

$$L( f; [a, b] ) = \displaystyle\int_a^b || f'(x) || dx.$$

Example. The length of a circle of radius $r$ is equal to $2{\pi}r$. How do we know? Describe the circle as

$$f(t) = r (\cos t, \sin t), t \in [0, 2{\pi}].$$

Now

$$f'(t) = r ( - \sin t, \cos t)$$

and

$$|| f'(t) || = r ( \sin^2 t + \cos^2 t)^{\frac{1}{2}} = r.$$

Then

$$L = \displaystyle\int_0^{2 \pi} r dt = rt |_0^{2 \pi} = r (2{\pi} - 0) = 2{\pi}r.$$