I am currently working on my master thesis and part of my master thesis involves the following problem:
Given a circular shape (doesn't need to be a circle, just anything that closed). Does specifying the following values uniquely identify the circular shape?
$$ s_x = \oint_S x dS \\ s_y = \oint_S y dS \\ A = \oint_S 1 dS \\ A_m = \oint_S (x,y) \times dS \\ I_x = \oint_S \left(x-\frac{s_x}{A}\right)^2 dS \\ I_y = \oint_S \left(y-\frac{s_y}{A}\right)^2 dS $$
I am not sure if the equation for $A_m$ is correct but it should equal the enclosed area.
I am not interested in finding an equation which identifies the shape. My gut feeling tells me that there must be multiple shapes that satisfy this. Ideally infinitely many but I have no idea on how to even remotely prove this. It's even sufficient to simply find two shapes which have the same properties as listed above.
I came up with shapes that have similar properties but they were always transformable into the other by mirroring and rotating.
I am wondering what methods I could employ or is there some trick you guys are seeing?
I appreciate any input, Finn
Different thoughts (too long for a mere comment).
You are conscious that such a finite set (here 6 of them) cannot "capture" the infinite varieties of closed curves (i.e., homeomorphic to a circle). As you say that you would like to have an argument against "finiteness", think to the technique called "Fourier descriptors". See for example here. The idea is simple, consider your curve parametrized by its curvilinear abscissa traditionally called $s$, which is a periodic function (once you have made a complete turn, you begin again...). Make a Fourier expansion of $x(s)$ and y(s) : we need all coefficients $a_k$ and $b_k$ for the reconstruction of the curve... But it is true also that by taking a finite sufficient number among the first ones, you will have a satisfying reconstruction.
I think you should normalize your data, by having your curves all with unit area, in order in particular to be able to compare them. Indeed, enclosed area doesn't give any information about the shape itself.
You could consider a "small world approach" by taking a subset of all possible curves in order to better understand them, and possibly be able to characterize them by your parameters ? Consider for example closed splines or closed NURBS defined for example by a few points (the same ones for all curves).