Math Forum Discussions

Peter Percival

Posts: 20
Registered: 10/25/10

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

Plain Text

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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