\(q\)-Bernstein polynomials and Bézier curves (Q1861311)

Bézier curve techniques are extended by using a generalization of the Bernstein basis, called the \(q\)-Bernstein basis. A one-parameter family of generalized Bernstein polynomial is defined. It is proved that the approximation to a convex function by its \(q\)-Bernstein polynomials is one sided. It is shown also that the difference of two consecutive \(q\)-Bernstein polynomials has a representation involving second-order divided differences. The connection with the power basis, the Bernstein basis and the \(q\)-Bernstein basis is revealed by deriving their transformation matrices. Parametric curves are constructed using the \(q\)-Bernstein basis; their shape properties are discussed using the total positivity. Moreover, a degree elevation algorithm for \(q\)-Bernstein parametric curves is presented.