Conic Section as Bézier Curve

Any conic section can be represented as a rational Bézier curve of degree two defined by , where are the Bernstein polynomials and the control points. It is always possible to write the expression in a standard form such that . From such a form it is easy to determine the type of the conic section: if , it is a hyperbola; if , it is a parabola; and if , it is an ellipse.

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We show how to obtain the standard form (i.e., to make the first and last weights equal to 1) of a rational Bézier curve of degree . Let be defined by , where are the Bernstein polynomials. We neither change the curve nor its degree by applying a rational linear transformation , yielding .
The curve is thus represented in standard form by the original control points and the new weights by choosing .
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