Formulas and algorithms for quantum differentiation of quantum Bernstein bases and quantum Bézier curves based on quantum blossoming

• Published in: Graphical models Vol. 74; no. 6; pp. 326 - 334
• Main Authors: Goldman, Ron, Simeonov, Plamen
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.11.2012
• Subjects: Algorithms; Bezier; Derivatives; Differentiation; Mathematical models; Quantum Bernstein basis; Quantum blossom; Quantum Bézier curve; Quantum derivative; Quantum blossom; Quantum Bézier curve; Quantum Bernstein basis; Quantum derivative
• ISSN: 1524-0703; 1524-0711
• DOI: 10.1016/j.gmod.2012.04.004

Two quantum derivative algorithms for cubic q-Bézier curves P(t). Here the product notation u v w represents the q-blossom value p(u, v, w; q). In addition, δ = (1, 0), t = (t, 1), qka = (qka, 1) and qkb = (qkb, 1), for k = 0, 1, 2. The value at each node is computed by multiplying the value on one level lower directly to the left and to the right by the corresponding linear function along the arrow exiting that node and then adding the two products. [Display omitted] ► We introduce the homogeneous variants of the quantum blossoms. ► We derive relationships between the homogeneous quantum blossoms and the quantum derivatives. ► We present explicit formulas for the quantum derivatives of the quantum Bern stein bases. ► We generate recursive algorithms for the quantum derivatives of quantum Bézier curves. Using the homogeneous version of the quantum blossom, we derive formulas and algorithms for the quantum derivatives of quantum Bernstein bases and quantum Bézier curves.