Composition of parametrizations, using the paired algebras of forms and sites

An elegant mathematical setting for Bézier curves and surfaces, proposed by Lyle Ramshaw, consists of two copies of a polynomial algebra and a pairing between them. One copy is used for the component functions of the parametrization. The other, in dimension one, represents the domain points. Elements of higher degree in the second algebra are called sites and are useful for explaining blossoming and other constructions performed on Bézier curves and surfaces. This paper extends these definitions to mixed polynomials in two sets of variables, to give an elegant description of the composition of two Bézier parametrizations. In case the first transformation is linear, the construction is related to a classical group representation.