This question is about an observation regarding winding numbers.
Let $\gamma_1$ and $\gamma_2$ be two closed curves defined on the same interval $[a,b]$ onto $\mathbb{R}^2\setminus \{(0,0)\}$, each continuously differentiable. Then the winding number of the two curves (around $(0,0)$) are the same when $(1-s)\gamma_1(t)+s\gamma_2(t)\neq(0,0)$, $\forall t\in[a,b]$ and $\forall s\in[0,1]$.
The definition we used for winding numbers is $I(\gamma)=(1/2\pi)\oint_{\gamma}\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy$.
My idea is to construct a new curve $\Gamma_s(t)=(1-s)\gamma_1(t)+s\gamma_2(t)$, which is closed since both $\gamma_1$ and $\gamma_2$ are closed. If we could say something about the winding number of this curve, which intuitionally is a function of $s$ and would have something to do with the winding numbers of the two curves in its construction. Moreover, if we could prove that this number is a continuous function regarding $s$, then we could determine that it would be constant because the codomain of the winding number function is always the integers.
Here is the question: Can we write an explicit formula for the winding number of the new curve, or is there an alternative method to prove the continuity of the function?
Thanks for all your help in advance!