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'rectifiable'
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| Title of object: |
rectifiable |
| Canonical Name: |
Rectifiable |
| Type: |
Definition |
| Created on: |
2002-07-30 10:17:35 |
| Modified on: |
2006-03-31 02:37:46 |
| Classification: |
msc:51N05 |
Revision comment (for changes between this and next version):
| Changes for correction #7516 ('a short intuitive explanation'). |
Preamble:
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Content:
Let $f:[a,b] \rightarrow \mathbb{R}^k$ be a simple curve in $\mathbb{R}^{k}$ and let $P = (s_{0}, ..., s_{n})$ with $a \le s_0 < s_1 < \cdots s_n \le b$ be a partition of the interval $[a, b]$; then the points in the set
\[\{ f(s_{0}), f(s_{1}), ..., f(s_{n}) \}\]
are called the \textbf{vertices of the inscribed polygon} $\Pi (P)$ determined by $P$.
A curve is said to be \textbf{rectifiable} if there exists a positive number $M$ such that the length of the inscribed polygon $\Pi (P)$ is less than $M$ for all possible partitions $P$ of $[a, b]$, where $[a, b]$ is the interval the curve is defined on. The length of the inscribed polygon is defined as $\sum_{t=1}^{n} | \alpha(a_{t}) - \alpha(a_{t-1}) |$.
If $\alpha$ is rectifiable then the \textbf{length} of $\alpha$ is defined as the least upper bound of the lengths of inscribed polygons taken over all possible partitions.
Note: If one does not demand the curve to simple (i.e. that $f$ be one-to-one), then one may still proceed as was done here, taking the supremum of the lengths of inscribed in the curve. However, what one will obtain is not necessarily the length of the curve, but the distance travelled along the curve. For instance, if a porion of the curve is traced more than once, then the length of that portion of the curve will be counted more than once.
Note: Although this definition mentions a parameterization of the (simple) curve, what is being defined actually is independant of the choice of parameterization. In fact, all the information that is used is the ordering of points along the curve, which is invariant under reparameterization. Also, because the length of a line segment does not depend on how one might choose to orient the line segment, what is being defined here is invariant under reversing the parameterization of the curve. Hence, this is a geometrical property of the curve. |
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