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The Bernstein polynomial Bi, n (x) has some useful properties:
- normalization between 0 and 1
- single unique local maximum at x=i/n
- positive
   
The polynomial occurred as result of the work of Sergei Natanovich
Bernstein, an Ukrainian mathematician (1880-1968). Bernstein
introduced the curve in 1911, using it for a constructive proof for the Stone-Weierstrass
approximation theorem.
Paul de Faget de Casteljau used the polynomials for the approximation
of curves, for the French car company Citroën.
Working for Citroëns competitor Renault, Pierre Bézier (1910-1999) came
to the same curves, which he mentioned Bézier curves.
His initial idea was to construct a basic curve as the intersection of two
elliptic cylinders, later he moved to polynomials.
It was AR. Forrest who realized that Faget de Casteljau and Bézier
did the same job
Therefore we call the Bézier curve also the Bernstein-Bézier
curve.
It can be expressed as Bernstein polynomials Bi,n, given a set of n+1 control points
Pi (Pix, Piy):
The curve has the following properties:
- there is no line that has more intersections with a Bézier curve than
with the curve composed by the line segments through the points.
- the curve can be translated and rotated by performing these operations on
the control points.
- a numerical instability for large numbers of control points.
- moving a single control point changes the global shape of the curve.
This can be avoided by smoothly patching together low-order Bézier curves.
Another generalization of the curve is the B-spline.
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