Two essential properties of [formula omitted]-Bernstein–Bézier curves

Abstract

The (q,h)-Bernstein–Bézier curves are generalizations of both the h-Bernstein–Bézier curves and the q-Bernstein–Bézier curves. We investigate two essential features of (q,h)-Bernstein bases and (q,h)-Bézier curves: the variation diminishing property and the degree elevation algorithm. We show that the (q,h)-Bernstein bases for a non-empty interval [a,b] satisfy Descartes' law of signs on [a,b] when q>−1, q≠0, and h≤min⁡{(1−q)a,(1−q)b}. We conclude that the corresponding (q,h)-Bézier curves are variation diminishing. We also derive a degree elevation formula for (q,h)-Bernstein bases and (q,h)-Bézier curves over arbitrary intervals [a,b]. We show that these degree elevation formulas depend only on the parameter q and are independent of both the parameter h and the interval [a,b]. We investigate the convergence of the control polygons generated by repeated degree elevation. We show that unlike classical Bézier curves, the control polygons generated by repeated degree elevation for (q,h)-Bézier curves with 0<q<1 do not converge to the original (q,h)-Bézier curve, but rather to a piecewise linear curve with vertices that depend only on q and the monomial coefficients of the q-Bézier curve having the same control points as the original (q,h)-Bézier curve. A similar result holds when q>1. Here the control polygons generated by repeated degree elevation converge to a piecewise linear curve that depends only on q and the monomial coefficients of the 1/q-Bézier curve with the control points of the original (q,h)-Bézier curve in reverse order.