Assume I have array of $A_{3\times120}$
Each row of matrix A corresponds to a shape which is generated by its three row elements as below:
$r=1+a_1\cos(\theta)+a_2\cos(2\theta)+a_3\cos(3\theta)$ where $\theta$ varies from 0 to $2\pi$
Where
$x=r\cos(\theta)$; $y=r\sin(\theta)$
Assuming all shapes will have same area, what kind of combinations of $a_i$ do you suggest that I take to see how distributed my shapes are?
Maybe one would say sort them based on $1\times a_1^3+ 2\times a_2^3 +3\times a_3^3$
What way do you suggest?
Sort the shapes based on what to have a stencil implying how different is one shape from the other?
Several ideas:
(more "graphical") calculate the values of these curves on a grid, eg. let $x$ be from $-k$ to $k$ with some step $h=\frac{k}{n}$. Then calculate $y$ values for this set of $x$ coordinates and check in which interval it belongs. To compare two shapes: count matching "pixels".
You can use some measure of difference (metric) between functions (your shapes are in fact functions), eg. $\int_0^{2\pi}|f-g|$ or $\sup\limits_{(0,2\pi)}|f-g|$.
Apart from comparing values in some points (no.1.), you can also compare derivatives in a way described in 1. or in 2.