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

• Published in: Journal of approximation theory Vol. 164; no. 1; pp. 77 - 104
• Main Authors: Simeonov, Plamen, Zafiris, Vasilis, Goldman, Ron
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 2012
• Subjects: [formula omitted]-Bernstein basis; [formula omitted]-blossom; [formula omitted]-Bézier curve; Marsden’s identity; Subdivision; q -blossom; q -Bézier curve; Marsden’s identity; Subdivision; q -Bernstein basis
• ISSN: 0021-9045; 1096-0430
• DOI: 10.1016/j.jat.2011.09.006

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.