q-Bézier Curves

Visualization of Bézier curve and its generalization in -calculus. The Bézier curve, introduced by Pierre Bézier, is widely used in computer graphics for modelling smooth curves and also in the design of planes, sport cars, etc. This concept is generalized in -calculus to obtain a whole family of curves parametrized by in the unit interval. The original Bézier curve is then obtained in the limit as goes to 1.


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For a given set of control points, the Bézier curve is defined by the relation P(t)=pi Bi,n(t) where Bi,n(t) are Bernstein polynomials. Control points form the so-called control polygon. The -calculus generalization of a Bézier curve, called the -Bézier curve, is defined similarly by , where the are -Bernstein polynomials. This concept was first introduced by G. M. Phillips.
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