Conic Section as Bézier Curve

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.

Contributed by: Isabelle Cattiaux-Huillard and Gudrun Albrecht (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

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 .



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send