Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

Topic: Cubic Bezier spline as a limit (or enveloped by a limit, or
something...)

Replies: 1   Last Post: Feb 19, 2012 10:50 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Peter Percival

Posts: 1,301
Registered: 10/25/10
Re: Cubic Bezier spline as a limit (or enveloped by a limit, or
something...)

Posted: Feb 19, 2012 10:50 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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.


--
Using Opera's revolutionary email client: http://www.opera.com/mail/



Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.