Closed-form quadrangulation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e223" altimg="si21.svg"><mml:mi>n</mml:mi></mml:math>-sided patches

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