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 A0 (for t = 0) and An (for t= 1) and has its portion joining these points traced in the convex envelope of the control points ; the tangent at A0 is (A0A1) and the one at An is (An-1An).

When the weights (ak) 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 Ãk defined by .
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.