Closed-form quadrangulation of n-sided patches

We analyze the problem of tessellating an n-sided patch with an ideal, maximally-regular internal quadrangulation, featuring a single irregular vertex (or none, for n=4). We derive, in closed-form, the necessary and sufficient conditions that the boundary of the patch must meet for it to admit an internal quadrangulation of this kind, and, if so, provide a full description of the resulting tessellation(s), as functions of the number of edges subdividing the sides of the patch. The problem has been addressed in previous literature and from multiple angles: our new derivation, which is self-contained and more succinct, is also more complete. In particular, we show that multiple such tessellations can exist, for n=8 (and larger multiples of 4), and enumerate them. This contradicts the commonly held notion that irregular vertices can never be moved in isolation, in a quadrangulated mesh.