On G1 stitched bi-cubic Bézier patches with arbitrary topology

Lower bounds, mandating a minimal number and degree of polynomial pieces, represent a major achievement in the theory of geometrically smooth (G) constructions. On one hand, they establish a floor when searching for optimal constructions, on the other they can be used to flag complex constructions for potential flaws. In particular, quadrilateral meshes of arbitrary topology can not in general be converted to G-connected Bézier patches of bi-degree 3 with one piece per quad or use just linear reparameterizations. This note illustrates how lower bounds indicate otherwise difficult-to-find flaws in a complex new surface construction.