 Tripartite graph encoding of a grid

For $A\subseteq [n{]}^2$, use three disjoint labelled vertex classes $X=[n]$, $Y=[n]$, $Z=[2n]$. Each $(x,y)\in A$ inserts the three edges $xy,x(x+y),y(x+y)$, producing pairwise edge-disjoint triangles. Every triangle in a graph with labels $(x,y,z)$ corresponds to the three points $(x,y),(x,z-x),(z-y,y)$ of $A$. It is canonical when $z=x+y$; otherwise it gives a corner in an integer grid with nonzero displacement $z-x-y$.