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Cubic Bezier spline as a limit (or enveloped by a limit, or something...)
Posted:
Feb 14, 2012 8:34 AM


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.
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