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Raising the Degree for Bézier Curves

A Bézier curve in the plane is given by parametric equations of the form , where are points in the plane called control points and is the Bernstein polynomial of degree . This parametrization can be changed (without changing the curve) via a recursive procedure outlined in the Details section that generates a new set of control points larger by 1 at each iteration. The figure shows the Bézier curve, the original control polygon (obtained by connecting the control points with line segments), and the control polygon determined by the control points after some iterations of the recursive procedure. This artificial process is useful in CAGD to increase the flexibility for the designer. It may also be used in numerous contexts where several Bézier curves need to be represented with the same degree, for example to join two curves by a Bézier surface, or to compare two curves, etc.

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DETAILS

A Bézier curve is defined by
,
where the are the Bernstein polynomials. Applying the following recursive formula for the control points raises the degree of the Bernstein basis:
,
, for ,
.
The curve can then be parametrized by
.
Indeed:
.
Although the curve has a new control polygon, its parametrization remains the same. For further details, see:
G. Farin, Curves and Surfaces for Computer Aided Geometric Design, Boston: Academic Press, 1992.
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