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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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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