Suppose that I have a function $SI_p(\theta, \Gamma)$, $\Gamma : [0, 2 \pi) \rightarrow \mathbb{R}^2$ is an oriented closed and continuous curve in $\mathbb{R}^2$, where $SI_p(\theta, \Gamma)$ gives the total amount of intersections with a ray at $p$ oriented by $\theta \in [0, 2 \pi)$. $p \in \mathbb{R}^2$. We can assume $p = (0,0)$ without loss of generality because we can just translate $\Gamma$. The sign of an intersection can be determined whether the ray intersects from the left or the right of the curve. The picture below shows what I mean by "Signed intersection" (albeit on an open-curve, but the concept is the same)
Now also defined $I_p(\theta, \Gamma)$ where $I_p$ is the total amount of intersections (ignores the sign).
If we start at $\theta = 0$ and increase it, for a given $p$ and $\Gamma$, can we show that the value of $SI_p$ can only change when the value of $I_p$ changes? Further, I think $SI_p$ is constant for closed curve (up to intersections where the curve is tangent to the ray) and only changes at the $\theta$ pointing towards the endpoints for open-curves. But I'm not sure how to show this.