Definition:Smooth Path/Simple/Complex Plane

Definition
A smooth path $\gamma : \left[{ a \,.\,.\, b }\right] \to \C$ is a simple smooth path iff $\gamma$ is injective on the half-open interval $\left[{a\,.\,.\,b}\right)$, and for all $t \in \left({ a \,.\,.\, b }\right)$, we have $\gamma \left({t}\right) \ne \gamma \left({b}\right)$.

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