%% Copyright 2013 Kevin Ryde
%% This file is part of Math-PlanePath.
%% Math-PlanePath is free software; you can redistribute it and/or modify it
%% under the terms of the GNU General Public License as published by the Free
%% Software Foundation; either version 3, or (at your option) any later
%% version.
%% Math-PlanePath is distributed in the hope that it will be useful, but
%% WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
%% or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
%% for more details.
%% You should have received a copy of the GNU General Public License along
%% with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.

%% Usage: tex dragon-pgf-latex.tex
%%        xdvi dragon-pgf-latex.dvi
%% This a dragon curve drawn with the PGF lindenmayersystems library.
%%     http://sourceforge.net/projects/pgf/
%% The PGF manual includes examles of Koch snowflake, Hilbert curve and
%% Sierpinski arrowhead.  In the ``spy'' library section there's some
%% magnifications of the Koch and of a quadric curve too.
%% In the rule here \symbol{S} is a second drawing symbol.  It draws a
%% line segment the same as F, but the two different symbols let the
%% rules distinguish odd and even position line segments.
%% F and S are always in pairs, F first and S second, F_S_F_S_F_S_F_S.
%% At each even position F expands to a left bend, ie with a "+" turn.
%% At each odd position S expands to a right bend, ie with a "-".
%% This is the "successive approximation" method for generating the
%% curve where each line segment is replaced by a bend to the left or
%% right according as it's at an even or odd position.
%% The sequence of + and - turns resulting in the expansion follows
%% the "bit above lowest 1-bit" rule.  This works essentially because
%% the bit above obeys an expansion rule
%%    if k even
%%      bitabovelowest1bit(2k)   = bitabovelowest1bit(k)
%%      bitabovelowest1bit(2k+1) = 0         # the "+" in F -> F+S
%%    if k odd
%%      bitabovelowest1bit(2k)   = bitabovelowest1bit(k)
%%      bitabovelowest1bit(2k+1) = 1         # the "-" in S -> F-S

\input tikz.tex

\pgfdeclarelindenmayersystem{Dragon curve}{
  \rule{F -> F+S}
  \rule{S -> F-S}
  [lindenmayer system={Dragon curve, step=10pt, axiom=F, order=8}]
  lindenmayer system;