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

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: 1,325
Registered: 10/25/10
Cubic Bezier spline as a limit (or enveloped by a limit, or
something...)

Posted: Feb 14, 2012 8:34 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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.

--
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.