If you have a square two-dimensional grid, that are lots of ways it could be triangulated -- for an individual square one could add an edge from upper-right to lower-left, or from lower-right to upper-left. For an ensemble of squares, a choice of extra-edge direction for each square, so lots of choices.
I can think of a couple of reasons to triangulate a grid, to perform a linear interpolation on simplicies, and I read that algorithms for finding sinks and sources generally assume a triangular mesh. For these (and possibly for other) applications, are there better choices of square-grid triangulations? Pointers to the literature on this would be much appreciated.