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% SUMMARY: Guru Don Lancaster gives us three new PostScript methods to subdivide
% a cubic spline into individual chord segments.
%
% These include the CONSTANT ERROR method useful for vector
% conversion, the CONSTANT T PARAMETER method useful for
% various art applications, and Don's new and nonobvious
% CONSTANT CHORD LENGTH method for placing glyphs along an
% arbitrary path.
%
%*****************************************************************
% There are lots of intriguing PostScript opportunities that emerge
% whenever you subdivide a cubic spline into smaller pieces.

% There are three methods that yield different results for different
% purposes:

% (A) The CONSTANT ERROR method uses the flattenpath operator
% to break the spline up into chords of usually uneven
% length. Any and all chords will always be within a
% given error distance to the real curve.
%
% One use is to convert PostScript curves into vectors
% for use on a plotter, Santa Claus machine, engraver,
% embroidery or other non-PostScript device.

% (B) The CONSTANT "t" PARAMETER method uses the equations
% behind cubic splines to break the spline up into chords
% of usually uneven length. The chords will be somewhat
% longer along the "more bent" portions of the curve.
%
% Two uses are for avuncular sleezoids or similar "artsy
% craftsy" stuff.

% (C) The CONSTANT CHORD method uses successive approximations
% of method (B) to break the spline up into chords of
% virtually identical length. This method is non obvious
% and computationally intensive. But the results are
% easily distilled and compiled.
%
% One major use is to nonlinear map border elements or
% other glyphs along a serpentine or otherwise arbitrary
% path. More on this in a followup file.

% Let's look at each method in turn, generating some working utilities
% along the way. We will use my gonzo utilities to quickly and conveniently
% show the results. Most of the actual tools do not necessarily require
% gonzo for their final use.

% The GENERAL PROCEDURE for using PostScript-as-language is as
% follows:

% (1) Carefully READ the entire sourcecode file inside your
%     favorite WP or Editor as an ASCII textfile.
%
% (2) Carefully MODIFY the filenames and variables and such
%     so they apply to you.
%
% (3) SAVE the modified code AS A STANDARD ASCII TEXTFILE       using a unique filename.
%
% (4) RUN the modified code by dragging and dropping it into
%     Acrobat Distiller. GhostScript may alternately be used.
%
% (5) TEST, INTERPRET, and MODIFY your results.
%

%%%%%%%%%%%% (A) INSTALL AND RUN GONZO UTILITIES %%%%%%%%%%%%%%%%%%

% The demo uses my gonzo utilities from
% http://www.tinaja.com/psutils/gonzo20.ps

% (C:\\windows\\desktop\\gonzo\\gonzo.ps) run % run the gonzo utilities

% gonzo begin               % activate the gonzo utilities
% ps.util.1 begin
% % printerror
% nuisance begin

% Build a graph space...

100 100 10 setgrid          % set up a working grid
30 40 showgrid              % and show part of it

%%%%%%%%%%%%%%% (B) CONSTANT ERROR METHOD %%%%%%%%%%%%%%%%%%%%%%

% The CONSTANT ERROR method uses the flattenpath operator to break
% the spline up into chords of usually uneven length. Any and all
% chords will always be within a given error distance to the real
% curve. The chords will typically be different lengths.

gsave 0 5 translate                   % position on graph
/maxpixelerror 5 def                  % set maximum error in pixels
line1                                 % set drawing width
beige 0.25 setgray                    % nice color

gsave 5 0 mt 10 10 20 0 25 10 curveto % define the curve

gsave stroke grestore                 % draw the curve

gsave maxpixelerror setflat           % replace curve with chords
flattenpath

mark                                  % create xy array for use
{}{}{}{} pathforall ]
/xyvalues exch def grestore

% showflatchords plots the endpoints of your chords

/showflatchordends {0 2 xyvalues length 1 sub
 {/posn exch def xyvalues posn get xyvalues posn
 1 add get mt dot} for} def

showflatchordends                       % show chord ends as dots

/cstretch 0.04 def                     

% and Gonzo label the curve

/sstretch 0.2 def
/font1 /Helvetica 0.85 gonzofont font1
black

12 3 ((A) Constant error method) cl grestore

%%%%%%%%%%%%%%% (C) CONSTANT "t" PARAMETER METHOD %%%%%%%%%%%%%%%%%%%

% The CONSTANT "t" PARAMETER method uses the equations behind cubic
% splines to break the spline up into chords of usually uneven length.
% The chords will be somewhat longer along the "more bent" portions
% of the curve.

% Let's start with three Bezier general curve utilities

% grabcoeffs takes the eight spline xy values and changes them into
% some intermediate variables handy to calculate x and y as a function
% of t...

/grabcoeffs {

/y3a exch def          % final y
/x3a exch def          % final x
/y2a exch def          % 2nd influence y
/x2a exch def          % 2nd influence x
/y1a exch def          % 1st influence y
/x1a exch def          % 1st influence x
/y0a exch def          % initial y
/x0a exch def          % initial x

/A x3a x1a x2a sub 3 mul add x0a sub store      % as in A(x)t^3
/B x2a x1a dup add sub x0a add 3 mul store      % as in B(x)t^2
/C x1a x0a sub 3 mul store                      % as in C(x)t
/D y3a y1a y2a sub 3 mul add y0a sub store      % as in D(y)t^3
/E y2a y1a dup add sub y0a add 3 mul store      % as in E(y)t^2
/F y1a y0a sub 3 mul store                      % as in F(y)t

} def

% And this is the "magic" code to find x and y given tt from 0 to 1..

/ytt { dup dup D mul E add mul F add mul y0a add} def /xtt { dup dup A mul B add mul C add mul x0a add} def

gsave 0 10 translate               % set position on graph
line1                              % set the linewidth
beige 0.25 setgray                 % nice color

5 0 10 10 20 0 25 10 grabcoeffs    % grab curveto values

5 0 moveto 10 10 20 0 25 10        % draw the curve curveto stroke

/numsegs 14 def                    % select number of segments

x0a y0a mt                         % plot the dots

0 1 numsegs div 1.0001 {dup xtt
exch ytt lineto currentpoint dot mt} for

% Note that the tt values are 0, 1 numsegs div, 2 numsegs div, ... 1
% An explicit custom array is probably not required

/cstretch 0.04 def                 % and Gonzo label the curve
/sstretch 0.2 def
/font1 /Helvetica 0.85 gonzofont
font1 black

12 3 ((B) Constant "t" parameter method) cl grestore

grestore

%%%%%%%%%%%%%%% (D) CONSTANT CHORD METHOD %%%%%%%%%%%%%%%%%%%

% The CONSTANT CHORD method uses successive approximations of method
% (B) to break the spline up into chords of virtually identical length.
% This method is non obvious and computationally intensive. But the
% results are easily distilled and compiled.

% bezierlength finds the length of a Bezier curve and leaves the length
% on the stack. It works by breaking the curve up into straight line
% approximate segments.

/bezierlength {/oldx x0a store /oldy y0a store /blength 0 store 0 1 numlenpts 1 sub div 1.0001 {dup xtt exch ytt /newy exch store /newx exch store newx oldx sub dup mul newy oldy sub dup mul add sqrt blength add /blength exch store /oldy newy store /oldx newx store} for }def

gsave 0 20 translate              % set position on graph
line1                             % set the linewidth beige
0.25 setgray                      % nice color

/numlenpts 400 def                % # of samples for length

5 0 10 10 20 0 25 10 grabcoeffs   % grab curveto values

bezierlength                      % get 23.9937 in this example

/segments 14 def                  % number of equal length chords
/chord blength segments div def   % length along curve between repeats
/numtrials 100 def                % needed accuracy of chording

% grabprevxy saves the previous x and y values

/grabprevxy { dup dup xtt /prevx exch store ytt /prevy exch store} def

% findchordguess measures the current chord

/findchordguess {dup dup xtt exch ytt prevy sub dup mul exch prevx sub dup mul add sqrt} def

% guessnextttt estimates the chord length by assuming it is equal to
% the change in t. This greatly reduces the number of trials needed...

/guessnexttt {1 copy 1 segments div add findchordguess chord lt} def

% findtvalues creates an array of the t values needed for virtually
% equal length chords

/findtvalues {mark 0 grabprevxy segments {guessnexttt /guessdir exch store {1 segments numtrials mul div guessdir {add}{sub} ifelse findchordguess chord guessdir {gt}{lt} ifelse {exit} if dup dup 0 lt exch 1 gt or {exit} if} loop grabprevxy} repeat 1 gt {1} if ] /ttvalues exch def} def

findtvalues                     % creates an array of tt values

% plotchordvalues puts a dot at each tt location

/plotchordvalues { ttvalues { dup xtt exch ytt mt dot} forall } def

gsave 0 5 translate

line1 x0a y0a moveto x1a y1a x2a y2a x3a y3a curveto stroke plotchordvalues

/cstretch 0.04 def              % and Gonzo label the curve
/sstretch 0.2 def
/font1 /Helvetica 0.85 gonzofont
font1 black

12 3 ((C) Constant chord length method) cl grestore

showpage

Click here for the Downloadable PostScript Sourcecode for this file.
Click here for the Acrobat PDF Graphic Demo Figure for this file.

Remember to first READ the file in an editor, MODIFY it to match your
needs, SAVE it as an ORDINARY ASCII TEXTFILE, and then RUN
it by dragging and dropping into Acrobat Distiller.

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