This is my first question on the math stack exchange. I am currently a second-year undergraduate student taking an elementary course on Curves and Surfaces. My question is as follows.
In the parametrization $γ : (−2π, 2π) \to\Bbb R^2$ given by $γ(t) = (\cos t − \sin t, \cos t + \sin t)$, why is the domain of $t$ given by the open interval $(−2π, 2π)$ and not $[−2π, 2π]$. Isn't $t$ clearly defined at $−2π$ and $2π?$ In other words why is an open interval taken to describe a parametric curve?
Any help would be highly appreciated.
There are various possibilities for the domains of parameterized curves; both open intervals and closed intervals are common choices for the domain.
For the specific curve $\gamma$ mentioned in your question, it is indeed true that the same formula for $\gamma(t)$ can be used with the domain $(-2\pi,+2\pi)$ and with the domain $[-2\pi,+2\pi]$, and in fact you could use that formula with the domain $(-\infty,+\infty)$ or with any subinterval thereof.
Different applications might require different choices of the domain. Lacking any knowledge of what application you (or your textbook or lecturer) have in mind for $\gamma$, it is not possible to answer your question of why one might choose $(-2\pi,+2\pi)$ over $[-2\pi,+2\pi]$.