arclength as filtered limit
The length (http://planetmath.org/Rectifiable) of a rectifiable curve may be phrased as a filtered limit. To do this, we will define a filter of partitions of an interval . Let be the set of all ordered tuplets of distinct elements of whose entries are increasing:
We shall refer to elements of as partitions of the interval . We shall say that is a refinement of a partition if . Let be the set of all subsets of such that, if a certain partition belongs to then so do all refinements of that partition.
Let us see that is a filter basis. Suppose that and are elements of . If a partition belongs to both and then every one of its refinements will also belong to both and , hence .
Next, note that, if a partition of is a refinement of a partition of then, by the triangle inequality, the length of is greater than the length of . By definition, for every , we can pick a partition such that the length of differs from the length of the curve by at most . Since the length of for any partition refining lies between the length of and the length of the curve, we see that the length of will also differ by at most , so the length of the curve is the limit of the length of polygonal lines according to the filter generated by .
|Title||arclength as filtered limit|
|Date of creation||2013-03-22 15:49:34|
|Last modified on||2013-03-22 15:49:34|
|Last modified by||rspuzio (6075)|