Complex Bézier curves and the geometry of polygons

In this paper, we associate to every planar polygon a complex polynomial, in which the blossom of the polynomial function captures the process in which linear transformations applied to the polygon lead to regular structures. In particular, we prove, in a purely algebraic way several well-known theorems on polygons such as the Napoleon–Barlotti Theorem, the Petr–Douglas–Neumann Theorem, and the Fundamental Decomposition Theorem of polygons to regular polygons.