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.\"
.IX Title "Math::PlanePath::WunderlichMeander 3pm"
.TH Math::PlanePath::WunderlichMeander 3pm "2014-08-26" "perl v5.20.1" "User Contributed Perl Documentation"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
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.nh
.SH "NAME"
Math::PlanePath::WunderlichMeander \-\- 3x3 self\-similar "R" shape
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 3
\& use Math::PlanePath::WunderlichMeander;
\& my $path = Math::PlanePath::WunderlichMeander\->new;
\& my ($x, $y) = $path\->n_to_xy (123);
.Ve
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
This is an integer version of the 3x3 self-similar
meander by Walter Wunderlich,
.IX Xref "Wunderlich, Walter"
.PP
.Vb 10
\&      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
.Ve
.PP
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,
.PP
.Vb 10
\&    +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-+\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-+\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-+
\&    | ^              |              * | ^             |
\&    | |              |  rotate 180  | | |   base      |
\&    | |     8        |       5      | | |     4       |
\&    | |   base       |              | | |             |
\&    | *              |              v | *             |
\&    +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-+\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-+\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-+
\&    | <\-\-\-\-\-\-\-\-\-\-\-\-* | <\-\-\-\-\-\-\-\-\-\-\-\-* | ^             |
\&    |                |                | |             |
\&    |       7        |       6        | |     3       |
\&    |   rotate 180   |   rotate 180   | |   base      |
\&    |  + transpose   |  + transpose   | *             |
\&    +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-+\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-+\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-+
\&    |                |                | ^             |
\&    |                |                | |             |
\&    |       0        |       1        | |     2       |
\&    |   transpose    |   transpose    | |   base      |
\&    | *\-\-\-\-\-\-\-\-\-\-\->  | *\-\-\-\-\-\-\-\-\-\-\-\-> | *             |
\&    +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-+\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-+\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-+
.Ve
.PP
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.
.PP
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 \*(L"Schema\*(R" shown in Wunderlich's paper (and which is
similar to the \f(CW\*(C`PeanoCurve\*(C'\fR and various other \f(CW\*(C`PlanePath\*(C'\fR curves).
.SH "FUNCTIONS"
.IX Header "FUNCTIONS"
See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes.
.ie n .IP """$path = Math::PlanePath::WunderlichMeander\->new ()""" 4
.el .IP "\f(CW$path = Math::PlanePath::WunderlichMeander\->new ()\fR" 4
.IX Item "$path = Math::PlanePath::WunderlichMeander->new ()"
Create and return a new path object.
.ie n .IP """($x,$y) = $path\->n_to_xy ($n)""" 4
.el .IP "\f(CW($x,$y) = $path\->n_to_xy ($n)\fR" 4
.IX Item "($x,$y) = $path->n_to_xy ($n)"
Return the X,Y coordinates of point number \f(CW$n\fR on the path.  Points begin
at 0 and if \f(CW\*(C`$n < 0\*(C'\fR then the return is an empty list.
.ie n .IP """($n_lo, $n_hi) = $path\->rect_to_n_range ($x1,$y1, $x2,$y2)""" 4
.el .IP "\f(CW($n_lo, $n_hi) = $path\->rect_to_n_range ($x1,$y1, $x2,$y2)\fR" 4
.IX Item "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
The returned range is exact, meaning \f(CW$n_lo\fR and \f(CW$n_hi\fR are the smallest
and biggest in the rectangle.
.SS "Level Methods"
.IX Subsection "Level Methods"
.ie n .IP """($n_lo, $n_hi) = $path\->level_to_n_range($level)""" 4
.el .IP "\f(CW($n_lo, $n_hi) = $path\->level_to_n_range($level)\fR" 4
.IX Item "($n_lo, $n_hi) = $path->level_to_n_range($level)"
Return \f(CW\*(C`(0, 9**$level \- 1)\*(C'\fR.
.SH "SEE ALSO"
.IX Header "SEE ALSO"
Math::PlanePath,
Math::PlanePath::PeanoCurve
.PP
Walter Wunderlich \*(L"Uber Peano-Kurven\*(R", Elemente der Mathematik, 28(1):1\-10,
1973.
.Sp
.RS 4
<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)
.RE
.SH "HOME PAGE"
.IX Header "HOME PAGE"
<http://user42.tuxfamily.org/math\-planepath/index.html>
.SH "LICENSE"
.IX Header "LICENSE"
Copyright 2011, 2012, 2013, 2014 Kevin Ryde
.PP
This file is part of Math-PlanePath.
.PP
Math-PlanePath is free software; you can redistribute it and/or modify it
under the terms of the \s-1GNU\s0 General Public License as published by the Free
Software Foundation; either version 3, or (at your option) any later
version.
.PP
Math-PlanePath is distributed in the hope that it will be useful, but
\&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0
or \s-1FITNESS FOR A PARTICULAR PURPOSE. \s0 See the \s-1GNU\s0 General Public License for
more details.
.PP
You should have received a copy of the \s-1GNU\s0 General Public License along with
Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.