I have a cognitive science problem, but the solution appears to be related to physics, so I thought I'd ask here.

I have many (1000) data points, each with an individual x and y position, and all of the same mass. I made these points translation invariant by subtracting the mean x and mean y position (i.e. the 'center of mass') from the respective coordinates of each point, so no matter where the points are, they will be translated so that the center of mass is (0,0). Now I need to do the same to make them rotation invariant in Euclidean space.

Because translation invariance involved 'center of mass,' the instructor has suggested the rotation version will involve moment of inertia and rotating to the parallel axis representation. I have no experience with this. So, given the information about the points above, can someone tell me how I can find the moment of inertia of the points and rotate to the parallel axis representation?

To summarize, the problem is how to operate on the points/coordinates such that they are rotationally invariant.

If you guys are interested in the source of the problem, see this text. The problem is #4 of section 10.2 (pages 107-110 in the pdf).

http://www.ejwagenmakers.com/BayesCourse/BayesBookWeb.pdf