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(The following is believed correct. Please report any errors or differing experiences.)

%! % THREE CUBIC SPLINE GENERATION ALGORITHMS % ======================================== % by Don Lancaster

% SUMMARY: Tutorial and demo looks at three different ways to generate
% cubic splines. Which is to use the PostScript -curveto- operator,
% to brute force plot the equations in t space, or to use an
% elegant "differencing method" that only needs repeated additions.

% Copyright c. 1998 by Don Lancaster and Synergetics, Box 809,
% Thatcher AZ, 85552 (520) 428-4073. synergetics@tinaja.com
% All commercial rights and all electronic media rights are
% *fully* reserved. Linking welcome. Reposting is expressly forbidden.

% Consulting services available.
% Further support on www.tinaja.com/info01.html

% 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

%%%%%%%%%%%% (B) USE THE POSTSCRIPT -CURVETO- OPERATOR %%%%%%%%%%%%%%

% first establish a rubbergrid...

60 175 8 setgrid 65 50 showgrid

% pick your control points. This would probably be a matrix later on...

/x0  0 def  /y0  0 def
/x1 10 def  /y1 45 def
/x2 50 def  /y2 40 def
/x3 60 def  /y3 10 def

% Plot the spline as a standard PostScript curveto...

x0 y0 moveto x1 y1 x2 y2 x3 y3 curveto line1 stroke

% Show the control points

x0 y0 mt dot x1 y1 mt dot x2 y2 mt dot x3 y3 mt dot

%%%%%%%%%%%%%(C) BRUTE FORCE T-PARAMETER PLOTTING %%%%%%%%%%%%%%%

% Find t-power coefficients a-d given control points. See HACK62.PDF for
% more details. This moves you from graph space to math space..

/ax x3 x2 3 mul sub x1 3 mul add x0 sub def
/ay y3 y2 3 mul sub y1 3 mul add y0 sub def

/bx x2 3 mul x1 6 mul sub x0 3 mul add def
/by y2 3 mul y1 6 mul sub y0 3 mul add def

/cx x1 3 mul x0 3 mul sub def
/cy y1 3 mul y0 3 mul sub def

/dx x0 def
/dy y0 def

% check for identical results by actually plotting the t equations
% one unit above original...

gsave 0 1 translate % move new curve up by one for visibility.

0 0 moveto 0 0.01 1 {/tt exch store tt ax mul bx add tt mul cx add tt
mul dx add tt ay mul by add tt mul cy add tt mul dy add lineto} for

stroke grestore

%%%%%%%%%%% (D) ELEGANT ADDITION-ONLY DIFFERENCING METHOD %%%%%%%%

% How the method works: Given a point on the spline curve, the next
% point on the curve can be found by simple addition of running terms.

% The method is device and language independent, letting you generate
% a cubic spline **without** using the PostScript -curveto- operator!

/numincs 100 def % number of steps to generate the spline

% next, precalculate delta-t increment, square, and cube

/dt1 1 numincs div store       % delta t
/dt2 dt1 dup mul store         % delta t squared
/dt3 dt1 dt2 mul store         % delta t cubed

% initialize some stuff...

/xx dx store
/yy dy store
/ux 0 store
/uy 0 store
/vx 0 store
/vy 0 store

/mx1 ax dt3 mul store
/my1 ay dt3 mul store

/lx1 bx dt2 mul store
/ly1 by dt2 mul store

/kx mx1 lx1 add cx dt1 mul add store
/ky my1 ly1 add cy dt1 mul add store

/mx mx1 6 mul store
/my my1 6 mul store

/lx mx lx1 2 mul add store
/ly my ly1 2 mul add store

% now generate the fake curveto...

0 2 translate % tempory offset

xx yy moveto

numincs { /xx xx ux add kx add store /yy yy uy add ky add store
/ux ux vx add lx add store /uy uy vy add ly add store
/vx vx mx add store /vy vy my add store
xx yy lineto } repeat

line2 stroke

% Note that successive points along the curve require only *ten* repeated
% additions. There are no multiplies or other time-intensive operators.
% Note that this is **not** an approximation, but an exact solution!

% Try changing -numincs- to 3 to prove this to yourself.

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.

Consulting services available.
Further support on www.tinaja.com/info01.html  

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