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From: hemphill@cit-vax.Caltech.Edu (Scott Hemphill)
Newsgroups: comp.lang.postscript
Subject: Re: Splines Underli
Message-ID: <13711@cit-vax.Caltech.Edu>
Date: 1 Feb 90 11:41:23 GMT
References: <7897@shlump.nac.dec.com> <17811@rpp386.cactus.org> <1990Jan31.213001.11641@intercon.com> <38227@apple.Apple.COM>
Reply-To: hemphill@cit-vax.UUCP (Scott Hemphill)
Organization: California Institute of Technology
Lines: 39

woody@rpp386.cactus.org (Woodrow Baker) writes:
> Now, a followup question.  Given the resulting array of points, 
> created by the subdivision of the spline, is is possible to reverse the
> proceedure, and work back to the original set of corrdinates.  If so,
> what are the pitfalls, and where might there be code or references?
> It looks possible, given that every thing was done with shifts.

amanda@mermaid.intercon.com (Amanda Walker) writes:
>It should be possible, since four points will uniquely define a cubic
>curve.  It's been a while since I took Analytic Geometry :-), but you

But *six* points are necessary to uniquely determine a *parametric* cubic curve.
The following PostScript fragment shows 4 points with two different Bezier
curves passing through them.

%!
/in {72 mul} bind def
4.25 in 5.5 in translate
/dot {currentpoint 0.05 in 0 360 arc fill} bind def
-3 in -1 in moveto dot
-1 in 1 in moveto dot
1 in 1 in moveto dot
3 in -1 in moveto dot
-3 in -1 in moveto
-1 in 2 in 1 in 2 in 3 in -1 in curveto stroke
-3 in -1 in moveto
7 in 9 div 23 in 9 div -7 in 9 div 23 in 9 div 3 in -1 in curveto stroke
showpage

kchen@Apple.COM (Kok Chen) writes:
>Actually it may be much easier than that (without invoking C. F. Gauss :-),
>[interesting algorithm deleted]

>This is having less and less to do with comp.lang.postscript 

These sound like the questions one asks to figure out how to reverse-engineer
PostScript fonts.
-- 
Scott Hemphill	hemphill@csvax.caltech.edu	...!ames!elroy!cit-vax!hemphill