Provided by: libmath-planepath-perl_113-1_all bug

NAME

       Math::PlanePath::KochelCurve -- 3x3 self-similar R and F

SYNOPSIS

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

DESCRIPTION

       This is an integer version of the Kochel curve by Herman Haverkort.  It fills the first
       quadrant in a 3x3 self-similar pattern made from two base shapes.

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

                   X=0  1   2   3   4   5   6   7   8   9  10  11  12  13  14

       The base shapes are an "R" and an "F".  The R goes along an edge, the F goes diagonally
       across.

                 R pattern                      F pattern   ^
           +------+-----+-----+           +------+-----+----|+
           |2   | |3\   |4    |           |2   | |3\   |8   ||
           |  R | |  F  |   R |           |  R | |  F  |  R ||
           |    | |   \ |-----|           |    | |   \ |    ||
           +------+-----+-----+           +------+-----+-----+
           |1  /  |6    |5  / |           |1  /  |4  / |7  / |
           |  F   | Rrev|  F  |           |  F   |  F  |  F  |
           | /    |-----| /   |           | /    | /   | /   |
           +------+-----+-----+           +------+-----+-----+
           |0|    |7\   |8    |           |0|    |5\   ||6   |
           | |Rrev|  F  |  R  |           | |Rrev|  F  ||Rrev|
           | o    |   \ |------>          | o    |   \ ||    |
           +------+-----+-----+           +------+-----+-----+

       "Rrev" means the R pattern followed in reverse, which is

           +------+-----+-----+
           |8<----|7\   |6    |    Rrev pattern
           |   R  |  F  | Rrev|
           |      |   \ |-----|    turned -90 degrees
           +------+-----+-----+    so as to start at
           |1  /  ||2   |5  / |    bottom left
           |  F   || R  |  F  |
           | /    ||    | /   |
           +------+-----+-----+
           |0|    |3\   ||4   |
           | |Rrev|  F  ||Rrev|
           | o    |   \ ||    |
           +------+-----+-----+

       The F pattern is symmetric, the same forward or reverse, including its sub-parts taken in
       reverse, so there's no separate "Frev" pattern.

       The initial N=0 to N=8 is the Rrev turned -90, then N=9 to N=17 is the F shape.  The next
       higher level N=0,N=9,N=18 to N=72 is the Rrev too, as is any N=9^k to N=8*9^k.

   Fractal
       The curve is conceived by Haverkort for filling a unit square by descending into ever-
       smaller replacements, like other space-filling curves.  For that the top-level can be any
       of the patterns.  To descend any of the shapes can be used for the start, but for the
       outward expanding version here the starting pattern must occur at the start of its next
       higher level, which means Rrev is the only choice as it's the only start in any of the
       three patterns.

       But all the patterns can be found in the path at any desired size.  For example the "1"
       part of Rrev is an F, which means F to a desired level can be found at

           NFstart = 1 * 9^level
           NFlast = NFstart + 9^level - 1
                  = 2 * 9^level - 1
           XFstart = 3^level
           YFstart = 0

       level=3 for N=729 to N=1457 is a 27x27 which is the level-three F shown in Haverkort's
       paper, starting at XFstart=27,YFstart=0.

FUNCTIONS

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

       "$path = Math::PlanePath::KochelCurve->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.

SEE ALSO

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

       Herman Haverkort, "Recursive Tilings and Space-Filling Curves with Little Fragmentation",
       Journal of Computational Geometry, 2(1), 92-127, 2011.

           <http://jocg.org/index.php/jocg/article/view/68>
           <http://jocg.org/index.php/jocg/article/download/68/20>
           <http://arxiv.org/abs/1002.1843>

           <http://alexandria.tue.nl/openaccess/Metis239505.pdf> (slides)
           <http://www.win.tue.nl/~hermanh/stack/h-rtslf-eurocg2010-talk.pdf> (short form)

HOME PAGE

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

LICENSE

       Copyright 2011, 2012, 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/>.