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Revision difference : rectifiable curve |
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Let $\alpha:[a,b] \rightarrow \mathbb{R}^k$ be a curve in $\mathbb{R}^{k}$ and $P = \{ a_{0}, ..., a_{n} \}$ is a partition of the interval $[a, b]$, then the points in the set
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If $\alpha$ is a curve in $\mathbb{R}^{k}$ and $P = \{ a_{0}, ..., a_{n} \}$ is a partition of the interval $[a, b]$, then the points in the set
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| $$\{ \alpha(a_{0}), \alpha(a_{1}), ..., \alpha(a_{n}) \}$$ |
$$\{ \alpha(a_{0}), \alpha(a_{1}), ..., \alpha(a_{n}) \}$$ |
| are called the \textbf{vertices of the inscribed polygon} $\Pi (P)$ determined by $P$. |
are called the \textbf{vertices of the inscribed polygon} $\Pi (P)$ determined by $P$. |
| A curve is \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. |
A curve is \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. |
| 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. |
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. |
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