Winding number of $\mathrm{Re}(c)\,p+\mathrm{Im}(c)\,q$ for an orientation-preserving frame $(p,q)$

Let $c,p,q:\mathbb{R}\to\mathbb{C}$ be continuous $1$-periodic loops such that $\mathrm{Im}\big(\overline{p(t)}\,q(t)\big)>0$ for every $t$, i.e. for each $t$ the real-linear map $x+iy\mapsto x\,p(t)+y\,q(t)$ is an orientation-preserving isomorphism of $\mathbb{C}$. If $c$ has winding number $k$ and $p$ has winding number $m$, then the loop

has winding number $k+m$.

This computes the winding of a vector field written in a moving, possibly non-orthonormal, positively oriented frame: for instance $\pi\partial_s$ of a pseudoholomorphic curve composed with a reparametrisation of the domain, read in a global trivialisation of $\xi$.

Formalization Note Loops are ℝ → ℂ; "winding number $k$" is HasLoopWind f k. The hypotheses on $c$ and $p$ include continuity, periodicity and non-vanishing; for $q$ only continuity and periodicity are assumed.