Closed-form Quadrangulation of N-Sided Patches

We analyze the problem of quadtragulating a $n$-sided patch, with prescribed numbers of edge-subdivisions at its boundary, with a single internal irregular vertex (none, when $n = 4$) breaking the otherwise fully regular lattice. We derive, in analytical closed-form, (1) the necessary and sufficient conditions that a patch must meet to admit this quadrangulation, and (2) a full description of the resulting tessellation.