suppose I have a 1-periodic vector function $\vec{r}(t)$. In general, that'll be a closed loop in $\mathbb{R}^3$. But if $\vec{r}(t)$ lies in some plane then that's a closed loop in $\mathbb{R}^2\subset\mathbb{R}^3$.
in that case, I can pick orthogonal coordinates $(x,y)$ on this plane. What I'm trying to figure out is the best way to say the following: I can identify $(x,y)\in\mathbb{R}^2$ with $z=x+i y\in\mathbb{C}$.