Before I begin, I must confess: I myself don't know what am I looking for over here. Please bear with me.
I am looking forward to knowing if I can find a symmetry out of some seemingly random points on a plane? I want to map or transform them to some regular polygon so that I can study the properties of the configuration.
Actually, this question has borne out of my ongoing research which belongs to the domain of Physics. The propeties that I'd talked about, pertains to the physical properties. I believe my question belongs to the domain of Graph Theory or is it Topology?
Any experts? Thanks.
Given a set of any 4 points $\{(x_i,y_i)\}_{i=1}^4$ you can define $$ f(x,y) = \begin{cases} (-1,0), & (x,y) = (x_1,y_1)\\ (1,0), & (x,y) = (x_2,y_2)\\ (0,-1), & (x,y) = (x_3,y_3)\\ (0,1), & (x,y) = (x_4,y_4)\\ (x,y) & \text{otherwise} \end{cases} $$ which satisfies the condition you wrote in the question.