Provided by: libmath-planepath-perl_122-1_all 
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.
Level Methods
"($n_lo, $n_hi) = $path->level_to_n_range($level)"
Return "(0, 9**$level - 1)".
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, 2014, 2015 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/>.