  Links in $S^3$ as well as Legendrian links in the standard tight contact
structure on $S^3$ can be encoded by grid diagrams. These consist of a
collection of points on a toroidal grid, connected by vertical and horizontal
edges. Blackwell, Gay and second author studied triple grid diagrams, a
generalization where the points are connected by vertical, horizontal and
diagonal edges. In certain cases, these determine Lagrangian surfaces in
$\mathbb{CP}^2$. However, it was difficult to construct explicit examples of
triple grid diagrams, either by an approximation method or combinatorial
search. We give an elegant geometric construction that produces the moduli
space of all triple grid diagrams. By conditioning on the abstract graph
underlying the triple grid diagram, as opposed to the grid size, the problem
reduces to linear algebra and can be solved quickly in polynomial time.
