Subdivision Algorithm for Bézier Curves

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This Demonstration illustrates the convergence of the de Casteljau-based subdivision algorithm for Bézier curves.

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


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Details

The subdivision algorithm follows from the de Casteljau algorithm that calculates a current point , for , of a polynomial Bézier curve , for , where are the control points, by applying the following recurrence formula:

for

for .

For any value of between and , we have

.

The subdivision algorithm associates to the polygon the two polygons and . They constitute the control polygons of the two parts of the curve , respectively for in and :

and

.

When , varies over and varies over .

This Demonstration illustrates repeated application of the above procedure (the control "k" denotes the number of iterations). Obviously, the resulting polygon sequence converges very quickly to the curve .



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