Path as Parameterization of Contour
Please fix formatting and $\LaTeX$ errors and inconsistencies.
It may also need to be brought up to our standard house style.
If you have done this or know it has been done, please remove {{Tidy}} from the code.
Generalise to Real Cartesian Space
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
Theorem
Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.
Let $\gamma: \left[{a \,.\,.\, b}\right] \to \C$ be a path.
Let there exist $n \in \N$ and a subdivision $\left\{{a_0, a_1, \ldots, a_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$ such that:
- $\gamma {\restriction_{ \left[{a_{k - 1} \,.\,.\, a_k}\right] } }$ is a smooth path for all $k \in \left\{ {1, \ldots, n}\right\}$
where $\gamma {\restriction_{\left[{a_{k - 1} \,.\,.\, a_k}\right]} }$ denotes the restriction of $\gamma$ to $\left[{a_{k - 1} \,.\,.\, a_k}\right]$.
Then there exists a contour $C$ with parameterization $\gamma$ and these properties:
- $(1): \quad$ If $\gamma$ is a closed path, then $C$ is a closed contour.
- $(2): \quad$ If $\gamma$ is a Jordan arc, then $C$ is a simple contour.
- $(3): \quad$ If $\gamma$ is a Jordan curve, then $C$ is a simple closed contour.
Proof
Put $\gamma_k = \gamma {\restriction_{ \left[{a_{k-1} \,.\,.\, a_k}\right] } } : \left[{a_{k-1} \,.\,.\, a_k}\right] \to \C$.
By definition, it follows that there exists a directed smooth curve $C_k$ with parameterization $\gamma_k$.
For all $k \in \left\{ {1, \ldots, n-1}\right\}$, we have $\gamma_k \left({a_k}\right) = \gamma_{k + 1} \left({a_k}\right)$.
Define the contour $C$ as the concatenation $C_1 \cup C_2 \cup \ldots \cup C_n$.
Then, it follows by definition that $\gamma$ is a possible parameterization of $C$.
$\Box$
Suppose that $\gamma$ is a closed path.
Then $\gamma \left({a}\right) = \gamma_1 \left({a_0}\right) = \gamma_n \left({a_n}\right)$, so $C_1$ has start point $\gamma \left({a}\right)$, and $C_n$ has end point $\gamma \left({a}\right)$.
By definition, it follows that $C$ is a closed contour.
$\Box$
Suppose that $\gamma$ is a Jordan arc.
Let $k_1, k_2 \in \left\{ {1, \ldots, n}\right\}$, and $t_1 \in \left[{a_{k_1 - 1} \,.\,.\, a_{k_1} }\right), t_2 \in \left[{a_{k_2 - 1} \,.\,.\, a_{k_2} }\right)$.
Then $\gamma \left({t_1}\right) \ne \gamma \left({t_2}\right)$ by the definition of Jordan arc, so $\gamma_{k_1} \left({t_1}\right) \ne \gamma_{k_2} \left({t_2}\right)$.
Let instead $k \in \left\{ {1, \ldots, n}\right\}$ and $t \in \left[{a_{k-1} \,.\,.\, a_k}\right)$ with $t \ne a_1$.
Then $\gamma \left({t}\right) \ne \gamma \left({a_n}\right)$ by the definition of Jordan arc, so $\gamma_k \left({t}\right) \ne \gamma_n \left({a_n}\right)$.
By definition, it follows that $C$ is a simple contour.
$\Box$
Suppose that $\gamma$ is a Jordan curve.
As a Jordan curve by definition is both a Jordan arc and a closed path, it follows from what is shown above that $C$ is a simple closed contour.
$\blacksquare$