q-Bernstein polynomials and Bézier curves

H Oruç, George Mcartney Phillips

Research output: Contribution to journalArticlepeer-review

120 Citations (Scopus)


We define q-Bernstein polynomials, which generalize the classical Bernstein polynomials, and show that the difference of two consecutive q-Bernstein polynomials of a function f can be expressed in terms of second-order divided differences of f. It is also shown that the approximation to a convex function by its q-Bernstein polynomials is one sided.

A parametric curve is represented using a generalized Bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve. We study the nature of degree elevation and degree reduction for this basis and show that degree elevation is variation diminishing, as for the classical Bernstein basis. (C) 2002 Elsevier Science B.V. All rights reserved.

Original languageEnglish
Pages (from-to)1-12
Number of pages12
JournalJournal of Computational and Applied Mathematics
Publication statusPublished - 1 Feb 2003


  • generalized Bernstein polynomial
  • shape preserving
  • total positivity
  • degree elevation


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