qq-Blossoming: A new approach to algorithms and identities for qq-Bernstein bases and qq-Bézier curves

We introduce a new variant of the blossom, the qq-blossom, by altering the diagonal property of the standard blossom. This qq-blossom is specifically adapted to developing identities and algorithms forqq-Bernstein bases and qq-Bézier curves over arbitrary intervals. By applying the qq-blossom, we generate several new identities including an explicit formula representing the monomials in terms of the qq-Bernstein basis functions and a qq-variant of Marsden’s identity. We also derive for each qq-Bézier curve of degree nn, a collection of n!n! new, affine invariant, recursive evaluation algorithms. Using two of these new recursive evaluation algorithms, we construct a recursive subdivision algorithm for qq-Bézier curves.