Averaging transformation of a closed plane curve

Question

Let's suppose we have a closed plane curve of some shape whose points are described by the single parametric equation $P(x(t), y(t))$ where $t$ is some increasing parameter (example circle) or by a set of parametric equations for segments of the curve (example rectangle).

Now we are transforming curve by the following operation performed for all points of the closed curve:

$P_r(x_r(t), y_r(t))$ where $x_r(t)=\dfrac{x(t-\Delta {t})+x(t+\Delta {t})}{2}, y_r(t)= \dfrac{y(t-\Delta {t})+y(t+\Delta {t})}{2}$

In this step the bigger value of $\Delta {t}$ we assume the more rounded transformed curve we obtain so index $r$ is used for the new point which is simply a midpoint of segment $P(x(t-\Delta {t})),y(t-\Delta {t})P(x(t+\Delta {t}),y(t+\Delta {t})).$

This transformation of closed curve $C$ I would name "averaging" and denote as $A$ so we have $C_r=A(C)$. Maybe it has some other name, someone knows?

Parametric function $P(x(t), y(t))$ should be of such nature that it would give infinitely many "loops" for the curve (as in the case of circle - period $2{\pi}$) but we "average" the curve only for a single loop, of course.

Main questions:

1. What conditions should be imposed on parameter $t$ and $\Delta {t}$ to be sure that after transformation center of the gravity (CG) of the new closed curve will be exactly the same as that of the old one?

2. Should any conditions be assumed for the original closed curve to have stabile CG? (for rectangles and circles transformation acts properly)

3. Or maybe CG is stabile under described transformation for any closed curve .. but if so .... how to prove it ?

4. And what gives composition of n averaging operations i.e. $C_{r_n}=A^n(C)$ with $n{\to{\infty}}$. It always converges to point and it converges to CG?
(for circle it is true - it can be proved - it is easy to obtain equation for transformed circle)

5. Assuming ${\Delta}t=$ const and is known in what circumstances transformation $A^{-1}$ reverse to the original one exists (for a circle it is only one such transformation)? How to construct $C=A^{-1}(C_r)$ ?