Formula omitted]-Blossoming: A new approach to algorithms and identities for [formula omitted]-Bernstein bases and [formula omitted]-Bézier curves

Abstract

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