Definition:Smooth Path/Simple/Complex Plane

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Let $\gamma : \left[{a \,.\,.\, b}\right] \to \C$ be a smooth path in $\C$.

$\gamma$ is a simple smooth path if and only if:

$(1): \quad \gamma$ is injective on the half-open interval $\left[{a \,.\,.\, b}\right)$
$(2): \quad \forall t \in \left({a \,.\,.\, b}\right): \gamma \left({t}\right) \ne \gamma \left({b}\right)$

That is, if $t_1, t_2 \in \left({a \,.\,.\, b}\right)$ with $t_1 \ne t_2$, then $\gamma \left({a}\right) \ne \gamma \left({t_1}\right) \ne \gamma \left({t_2}\right) \ne \gamma \left({b}\right)$.

Also see

Compare with the topological definition of an arc which requires that $\gamma$ is injective on the closed interval $\left[{a \,.\,.\, b}\right]$.

A simple smooth path $\gamma$ is injective on $\left[{ a \,.\,.\, b}\right]$ if and only if $\gamma$ is not closed.