Provided by: libmath-planepath-perl_122-1_all 
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 by Walter Wunderlich,
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 going
from one corner 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 and
the transposing in that case applies to ever smaller parts. But for the integer version
here the start direction is 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 which is
similar to the "PeanoCurve" and 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
Walter Wunderlich "Uber Peano-Kurven", Elemente der Mathematik, 28(1):1-10, 1973.
<http://sodwana.uni-ak.ac.at/geom/mitarbeiter/wallner/wunderlich/>
<http://sodwana.uni-ak.ac.at/geom/mitarbeiter/wallner/wunderlich/pdf/125.pdf> (scanned
copy, in German)
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/>.