I want to confirm if I am correct. Due to the rotational symmetry of open unit disc $D$, it suffices to study only translations on $D$ out of the four plane isometries? This is because, every other isometric transformation(like reflection, rotation, glide reflection) can just be viewed as the effect of a translation. Am I correct? The same will hold for the punctured disc?
EDIT: I am sorry question is not clear. I am giving my exact problem here. I want to evaluate certain integral say $\int f(x) dx$ on on every congruent copy of punctured disc in the plane. This means I need to evaluate $\int f(g(x)) dx$ on the unit punctured disc where $g$ is the isometry of a plane, right? What my question is that whether it suffices to take $g$ as the translation only because the effect of other three isometries on punctured disk can be viewed as the special cases of translation, due to the 'nice' symmetry of punctured disc? Hope this clarifies my question.