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Re: Cubic Bezier spline as a limit (or enveloped by a limit, or
something...)
Posted:
Feb 19, 2012 10:50 AM
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On Tue, 14 Feb 2012 13:34:44 -0000, Peter Percival
<peterxpercival@hotmail.com> wrote:
> Let a cubic Bezier spline "depart from" P_1 in the direction P_2 and
> arrive at P_4 from the direction P_3. Let
>
> P_{12} bisect P_1 P_2
> P_{23} bisect P_2 P_3
> P_{34} bisect P_3 P_4.
>
> Let
>
> P_{1223} bisect P_{12} P_{23}
> P_{2334} bisect P_{23} P_{34}.
>
> Let
>
> P_{112} bisect P_1 P_{12}
> P_{121223} bisect P_{12} P_{1223}
> etc.
>
> Etc.
>
> In the limit, the family of line segments defined by those Ps approaches
> (in some sense or other) the curve
>
> (1 - t)^3 P_1 + 3(1 - t)^2 P_2 + 3(1 - t)t^2 P_3 + t^3 P_4 .
>
> But the limiting process (if that's what it is) is not of the kind that
> I met in my mathematics degree, so what exactly do I mean by the claim
> "In the limit..."? (Supposing that I've defined the sequence of Ps
> correctly ("defined" isn't the right word: I've just hinted at what they
> are).) Also, it's not so much that the segments approach the curve,
> rather the segments approach the tangents to the curve... or something...
>
> I cannot remember where I came across this construction, so I cannot
> return to it to see if the answer's there. I'll be very grateful if
> someone can tell me what I'm talking about.
I have found mention of this at the start of Chapter 3 of Knuth's 'The
METAFONTbook'. He wries 'The recursive midpoint rule for curve-drawing
was discovered in 1959 by Paul de Casteljau...'. But still I have found
no precise account of the limiting process and how it relates to the curve.
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