Good morning,
I want to prove that all points (A, B, C, D), as shown in the figure Image, compute the same center of gravity of the square.
Is there a way to prove that by using the local coordinates of the points (A, B, C, D) and translating them into the global system using the matrix and its inverse so they have the same center of gravity? The points do not necessarily have the same orientation.
The way to compute the gravity center is $$ \bar c_i [t] = \frac{1}{N}\sum_j \bar r_j [t] $$
and ri[t] is the coordinates of (A for exemple) at a moment t. N is the total number of points. So all the points use their own position coordiantes and the other points to calculate the same gravity center.