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RATIONAL BEZIER CURVE

Affine parametrization:
(i.e. )
where
are the Bernstein polynomials: .
Rational algebraic curve of degree £ n. |

Given a sequence of weighted points - the control
points - ,
the associated rational Bezier curve is the curve with the aforementioned
parametrization ; the curve passing through *A*_{0}
(for *t* = 0) and *A** _{n} *(for

When the weights (a* _{k}*)
are equal, we get polynomial Bezier curves.

This curve is the conical projection (transformation ) of the 3D polynomial Bezier curve whose control points are the points

Therefore, rational Bezier curves include all rational curves.

The parabola being the only polynomial conic, simple Bezier curves cannot represent a circle exactly, but it is possible with rational Bezier curves.

For example, if you take a control polygon formed by two orthogonal line segments of equal length, then the simple Bezier curve is a parabola; if you double the weight of one of the extremities, you get a circle:

In bold, the rational Bezier curve: a circle.

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© Robert FERRÉOL
2017