Assume that you have a circle with radius $r_0$, then you keep adding cosine modes as below:
if you plot this as below by matlab:
r0=1; a1=0.2; a2=0.2; a3=0.2; a4=0.2; th=0:0.01:2*pi; r=r0+a1cos(th)+a2cos(2*th)+a3cos(3*th)+a4cos(4*th); x=r*cos(th); y=r*sin(th); plot(x,y);
You will see that you can get different shapes by changing the number of modes (i.e. here n=4) or coefficients (i.e. $a_i$) and even omitting some modes (i.e. $a_k=0$) you will get different shapes.
My question is that how can you decompose for instance an oval or a square in cosine modes, and find their coefficients (somehow in the way of fourier decomposition), is it ever possible?
But I think making any shape would be possible by my method mentioned above, but I am interested to know how I can come from an arbitrary shape to its corresponding cosine modes.