Bézier Curve by de Casteljau's Algorithm

As r changes from 1 to 3 a sequence of linear interpolations shows how to construct a point on the cubic Bézier curve when there are four control points. The parameter t controls the proportion of the distance along an interpolating line. As t varies between 0 and 1 the entire curve is generated.

THINGS TO TRY

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DETAILS

For n+1 control points, (Subscript[p, 0]), (Subscript[p, 1]),...,(Subscript[p, n]), the Bézier curve can be constructed by the recurrence relation (Subscript[B, i]^r)(t)=(1-t) (Subscript[B, i]^(r-1))(t)+t (Subscript[B, i+1]^(r-1))(t) where (Subscript[B, i]^1)=(1-t) (Subscript[p, i])+t (Subscript[p, i+1]) is the linear interpolation between control points (Subscript[p, i]) and (Subscript[p, i+1]). The recursion level r goes from 1 to n and i runs from 0 to n-r.

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Contributed by: Bruce Atwood

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