The image $M=\gamma(S)$ is a Mobius Strip, what are the missing points of the parameterization of the interior and what is another parameterization?

Let $$S=\{(r,\theta) | -\frac12 and $$S'=\{(r,\theta) | -\frac12 so $$S'$$ is the interior of $$S$$. Define $$\gamma : S \rightarrow \mathbb{R}^3$$ by: $$\gamma(r,\theta) = \begin{bmatrix}2cos\theta+rcos(\frac\theta2)\\2sin\theta+rsin(\frac\theta2)\\rsin(\frac\theta2)\end{bmatrix}$$ Then the image $$M=\gamma(S)$$ is a Mobius Strip. We may assume that $$\gamma$$ is injective on $$S$$ and $$D(\gamma)(\overrightarrow c)$$ is injective for all $$\overrightarrow c \in S$$ without having to prove this. But then this makes $$\gamma$$ an invalid parametrization because $$S$$ is not open, so we restrict the domain to $$S'$$. Which points of $$M$$ are missing from the image $$\gamma(S')$$? Also, how would I find another open set $$T\subseteq\mathbb{R}^2$$ and another parameterization $$\phi:T\rightarrow M$$ such that $$\phi(T)$$ is the "most of" M (in the same sense that $$\gamma(S')$$ is most of M), but the set of points missing from $$\phi(T)$$ is different from the set of points missing from $$\gamma(S')$$ and describe the missing points of M. (We don't need to prove that $$\phi$$ and $$D(\phi)$$ are injective)