# Averaging transformation of a closed plane curve

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 $C=A^{-1}(C_r)$ ?

• It looks to me that there's no nice necessary geometric criterion for the curve $P$ and the "delay" $\Delta t$ under which the CG is stable. (There may be reasonably nice sufficient conditions, but the CG is certainly not stable in general.) – Andrew D. Hwang Jun 10 '16 at 1:30
• @Andrew D. Hwang What about if $\Delta{t}$ would be of such nature that it would give equal length of segments of the curve as in the case of circle and rectangle? At the case of curves it is easy enough to present reasoning for $2n$ symmetrical curves like hexagon, octogon or even more complicated like circle modifies by adding to its contour for example $cos(2k{\beta})$. Any sum of such figures has stabile GG. – Widawensen Jun 10 '16 at 5:59
• This transformation is "linear" and , it seems it preserves CG for any linear combinations of objects which have the same CG in the frame where origin of the coordinates frame is in this CG. – Widawensen Jun 10 '16 at 6:13
• In this frame we can add two curves $C_1$, $C_2$ obtaining $C_{new}$ by operation $x_{new}=x_1+x_2$ and $y_{new}=y_1+y_2$ where coordinates are assigned to the same $t$, which can be for example the same angle with axis $0x$ for vectors of curve points. – Widawensen Jun 10 '16 at 6:28
• Just time for a quick note: If you parametrize a circle at non-constant speed (even if the parametrization is smooth over the whole real line), the "average" doesn't seem to be well-behaved. Particularly, the set of points traced by $P_{r}$ depends not only on the closed plane curve (as a point set), but on its parametrization, and the image of $P_{r}$ can be rather "misshapen". – Andrew D. Hwang Jun 10 '16 at 12:10