Algorithms and identities for (p,q)-Bézier curves via (p,q)-Blossom

04/09/2016

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by   Khalid Khan, et al.

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In this paper, a new variant of the blossom, the (p,q)-blossom, is introduced, by altering the diagonal property of the standard blossom. This (p,q)-blossom has been adapted for developing identities and algorithms for (p,q)-Bernstein bases and (p,q)-Bézier curves. We generate several new identities including an explicit formula representing the monomials in terms of the (p,q)-Bernstein basis functions and a (p,q)-variant of Marsden's identity by applying the (p,q)-blossom. We also derive for each (p,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 (p,q)-Bézier curves.