Path: utzoo!utgpu!jarvis.csri.toronto.edu!mailrus!cs.utexas.edu!natinst!rpp386!woody
From: woody@rpp386.cactus.org (Woodrow Baker)
Newsgroups: comp.lang.postscript
Subject: Re: Splines Underli
Message-ID: <17837@rpp386.cactus.org>
Date: 1 Feb 90 13:25:30 GMT
References: <7897@shlump.nac.dec.com> <17811@rpp386.cactus.org> <38227@apple.Apple.COM>
Organization: River Parishes Programming, Plano, TX
Lines: 42

In article <38227@apple.Apple.COM>, kchen@Apple.COM (Kok Chen) writes:
> amanda@mermaid.intercon.com (Amanda Walker) writes:
> Actually it may be much easier than that (without invoking C. F. Gauss :-),
> and (A',B',C',D').  Where D = A'.
> 
> Now, construct a point U such that BCU are collinear and BC = CU.
> Similarly, construct a point U' s.t. C'B'U' are collinear and 
> B'C' = C'U'.  
> 
> If U != U', the two Beziers cannot be reduced (without error) to a 
> single Bezier.  You are done, and the guy who gave you the original
> two Beziers lied to you :-).
> 
> However, if U = U', construct point S s.t. ABS are collinear and 
> AB = BS.  Construct S' s.t. S'C'D' are collinear and S'C' = C'D'.
> 
> You have it - ASS'D' are the control points of a cubic that is the 
> concatenation of the original two Beziers.
> 
> Solving for approximate answers within some flatness criteria is 
> left as an exercise for the student.  Or, you can be real brave
> (a fine line separates bravery from stupidity :-) and ignore the
> U = U' check.
> 
> "Tricks" like the above are "played" all the time when you try
> to approximate/represent an n-th order Bezier with the control 
> points of an m-th order Bezier. Much more difficult are problems
> that deal with the offsets of Bezier curves.  You see attempts
> at solving those periodically.
> 
> This is having less and less to do with comp.lang.postscript 
> (INFO-POSTSCRIPT?).

Thanks for the info.  As to the above sentence, I'll have to disagree
with you.  Splines and transformation matrixes are the heart and soul
of Postscript. They are what *REALLY* set ps apart.  A complete understanding
of them leads to not only better postscript code, but bettter applications
that use postscript.

Cheers
Woody