We now discuss curves and their lengths in more detail. Let be an arbitrary natural number which is fixed throughout this post. Note that in general one needs to distinguish between a parameterised curve, which is a continuous mapping and the curve which is the image of given by Here we shall discuss parameterised curves. Hence, for instance, the parameterised curve with is a circle traversed twice and has therefore length whereas its image is just a circle which has length
Rectifiable parameterised curves
Let now be a parameterised curve (note that this implies that is continuous). We call a set a partition of the interval (of length ) if The set of all partitions of length shall be denoted by and the set of all partitions by
where denotes the Euclidean norm.
Definition (Rectifiable parameterised curve)
A parameterised curve is called rectifiable if there is a constant such that for all partitions of we have
If is rectifiable, then we define the length of by
Example and Exercise
Assume that the parameterised curve is continuously differentiable, that is, each of its component functions is continuously differentiable. Then is rectifiable and
To show this observe that
We now introduce the variation of a function.
Let be a function. Then the variation of is given by
Let be continuously differentiable. Show that
If then we say that has bounded variation.
On the other hand, there are continuous functions which have unbounded variation.
Example and Exercise
For instance, the function
is continuous, but has unbounded variation. To show that it has unbounded variation consider the partitions The details are left as exercise.
Rectifiable parameterised curves and bounded variation
We have now the following result.
The parameterised curve given by is rectifiable if and only if each of the component functions have bounded variation, i.e. for
For the proof of this result the following inequality is useful: for any real numbers we have
This inequality can be shown by taking the square on each side. (It is also a special case of the important Jensen’s inequality.) The details of the proof the theorem are left as an exercise.
Parameterised curves which are not rectifiable
Find a parameterised curve in , that is, a continuous function , which is not rectifiable (i.e. has infinite length). (Hint: Use the function of unbounded variation defined above to define )
Consider the Koch curve. This curve is simple and closed and encloses a finite area, but the length of the curve is infinite. This shows that there are regions of finite area whose boundary has infinite length.