In approximation theory and computer aided geometric design an important problem is to find the approximation of points with the use of polynomials and parametric polynomial curves. This can be achieved with Bernstein aproximation polynomials, which were introduced in the proof of Stone-Weierstrass theorem, and with Bézier curves, which are the basic objects in curve modelling. The work focuses on their generalization which can be achieved with the introduction of parameter $q$.
Bernstein basis polynomials and their generalization to $q$-Bernstein basis polynomials, with which we can define $q$-Bernstein approximation polynomials and $q$-Bézier curves, are introduced. The elementary characteristics for the standard example, when $q=1$, are derived, as well as the elementary characteristics of a more general example.
The generalization of de Casteljau algorithm, which is a stable algorithm for the calculation of points on the Bézier curve is also presented, as well as the generalisation of the curve degree elevation procedure and calculation of derivatives of the $q$-Bézier curves.
Theoretic examples are illustrated with a variety of numerical examples.
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