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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


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 curvedrawing 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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