Beaches, bays and headlands

We consider local and global combinatorial constraints on embeddings of a simple closed curve in a rectangular array of unit squares in which the curve lies entirely within the rectangular array and passes through each unit square once. Geographical terminology is convenient to describe the embeddings. We study shortest segments of the embedded curve with no clockwise quarter turns (bays) and a prescribed number of anti clockwise quarter turns (headlands). We also study embeddings which either maximise or minimise the number of quarter turns. Enumerative questions related to these embeddings have recently been studied by Kwong and Rogers, as Hamiltonian cycles in grid graphs.