NAME

       Math::PlanePath::WunderlichMeander -- 3x3 self-similar "R" shape

SYNOPSIS

        use Math::PlanePath::WunderlichMeander;
        my $path = Math::PlanePath::WunderlichMeander->new;
        my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

       This is an integer version of the 3x3 self-similar meander from

           Walter Wunderlich, "Uber Peano-Kurven", Elemente der Mathematik, volume 28, number 1,
           1973, pages 1-10.  <http://sodwana.uni-ak.ac.at/geom/mitarbeiter/wallner/wunderlich/>,
           <http://sodwana.uni-ak.ac.at/geom/mitarbeiter/wallner/wunderlich/pdf/125.pdf>

             8     20--21--22  29--30--31  38--39--40
                    |       |   |       |   |       |
             7     19  24--23  28  33--32  37  42--41
                    |   |       |   |       |   |
             6     18  25--26--27  34--35--36  43--44
                    |                               |
             5     17  14--13  56--55--54--53--52  45
                    |   |   |   |               |   |
             4     16--15  12  57  60--61  50--51  46
                            |   |   |   |   |       |
             3      9--10--11  58--59  62  49--48--47
                    |                   |
             2      8   5-- 4  65--64--63  74--75--76
                    |   |   |   |           |       |
             1      7-- 6   3  66  69--70  73  78--77
                            |   |   |   |   |   |
           Y=0->    0-- 1-- 2  67--68  71--72  79--80-...

                   X=0  1   2   3   4   5   6   7   8

       The base pattern is the N=0 to N=8 section.  It works as a traversal of a 3x3 square
       starting in one corner and going along one side.  The base figure goes upwards and it's
       then used rotated by 180 degrees and/or transposed to go in other directions,

           +----------------+----------------+---------------+
           | ^              |              * | ^             |
           | |              |  rotate 180  | | |   base      |
           | |     8        |       5      | | |     4       |
           | |   base       |              | | |             |
           | *              |              v | *             |
           +----------------+----------------+---------------+
           | <------------* | <------------* | ^             |
           |                |                | |             |
           |       7        |       6        | |     3       |
           |   rotate 180   |   rotate 180   | |   base      |
           |  + transpose   |  + transpose   | *             |
           +----------------+----------------+---------------+
           |                |                | ^             |
           |                |                | |             |
           |       0        |       1        | |     2       |
           |   transpose    |   transpose    | |   base      |
           | *----------->  | *------------> | *             |
           +----------------+----------------+---------------+

       The base 0 to 8 goes upwards, so the across sub-parts are an X,Y transpose.  The transpose
       in the 0 part means the higher levels go alternately up or across.  So N=0 to N=8 goes up,
       then the next level N=0,9,18,.,72 goes right, then N=81,162,..,648 up again, etc.

       Wunderlich's conception is successive lower levels of detail as a space-filling curve.
       The transposing in that case applies to ever smaller parts.  But for the integer version
       here, the start direction is held fixed and the successively higher levels alternate.  The
       first move N=0 to N=1 is rightwards per the "Schema" shown in Wunderlich's paper (and like
       various other "PlanePath" curves).

FUNCTIONS

       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

       "$path = Math::PlanePath::WunderlichMeander->new ()"
           Create and return a new path object.

       "($x,$y) = $path->n_to_xy ($n)"
           Return the X,Y coordinates of point number $n on the path.  Points begin at 0 and if
           "$n < 0" then the return is an empty list.

       "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
           The returned range is exact, meaning $n_lo and $n_hi are the smallest and biggest in
           the rectangle.

   Level Methods
       "($n_lo, $n_hi) = $path->level_to_n_range($level)"
           Return "(0, 9**$level - 1)".

SEE ALSO

       Math::PlanePath, Math::PlanePath::PeanoCurve

HOME PAGE

       <http://user42.tuxfamily.org/math-planepath/index.html>

LICENSE

       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.