Quantum Bernstein bases and quantum Bézier curves

The purpose of this paper is to investigate the most general quantum Bernstein bases and quantum Bézier curves. Classical Bernstein bases satisfy a two-term formula for their classical derivatives; quantum Bernstein bases satisfy a two-term formula for their quantum derivatives. To study the properties of these general quantum polynomial schemes, a new variant of the blossom, the quantum blossom, is introduced by altering the diagonal property of the classical blossom. The significance of the quantum blossom is that the quantum blossom provides the dual functionals for quantum Bézier curves over arbitrary intervals. Using the quantum blossom, several fundamental identities involving the quantum Bernstein bases are developed, including a quantum variant of the Marsden identity and the partition of unity property. Based on these properties of quantum Bernstein bases, quantum Bézier curves are shown to be affine invariant, and under certain conditions lie in the convex hull of their control points. In addition, for each quantum Bézier curve of degree nn, a collection of n!n!, affine invariant, recursive evaluation algorithms are derived. Using two of these recursive evaluation algorithms, a recursive subdivision procedure for quantum Bézier curves is constructed. This subdivision procedure generates a sequence of control polygons that converges rapidly to the original quantum Bézier curve.