# Copyright 2011, 2012, 2013, 2014 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/>.
# http://sodwana.uni-ak.ac.at/geom/mitarbeiter/wallner/wunderlich/pdf/125.pdf
# [8.5mb]
#
# Walter Wunderlich. Uber Peano-Kurven. Elemente der Mathematik, 28(1):1-10,
# 1973.
#
# Coil order 111 111 111
#
# math-image --path=WunderlichSerpentine --all --output=numbers_dash
# math-image --path=WunderlichSerpentine,radix=5 --all --output=numbers_dash
# math-image --path=WunderlichSerpentine,serpentine_type=170 --all --output=numbers_dash
#
package Math::PlanePath::WunderlichSerpentine;
use 5.004;
use strict;
use Carp;
#use List::Util 'max';
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 115;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
*_divrem = \&Math::PlanePath::_divrem;
use Math::PlanePath::Base::Generic
'is_infinite',
'round_nearest';
use Math::PlanePath::Base::Digits
'round_down_pow',
'digit_split_lowtohigh';
# uncomment this to run the ### lines
#use Smart::Comments;
use constant n_start => 0;
use constant class_x_negative => 0;
use constant class_y_negative => 0;
*xy_is_visited = \&Math::PlanePath::Base::Generic::xy_is_visited_quad1;
use constant parameter_info_array =>
[ { name => 'serpentine_type',
display => 'Serpentine Type',
type => 'string',
default => '010 101 010',
choices => ['alternating','coil','Peano'],
width => 11,
type_hint => 'bit_string',
description => 'Serpentine type, as a bit string or one of the predefined choices.',
},
{ name => 'radix',
share_key => 'radix_3',
display => 'Radix',
type => 'integer',
minimum => 2,
default => 3,
width => 3,
},
];
# same as PeanoCurve
use Math::PlanePath::PeanoCurve;
*dx_minimum = Math::PlanePath::PeanoCurve->can('dx_minimum');
*dx_maximum = Math::PlanePath::PeanoCurve->can('dx_maximum');
*dy_minimum = Math::PlanePath::PeanoCurve->can('dy_minimum');
*dy_maximum = Math::PlanePath::PeanoCurve->can('dy_maximum');
*_UNDOCUMENTED__dxdy_list = Math::PlanePath::PeanoCurve->can('_UNDOCUMENTED__dxdy_list');
#
# bit=0 bit=1
# *--- b^2-1 -- b^2
# | |
# *------- |
# | |
# 0 ----- b
#
# bit=0 bit=0
# *--- b^2-1 -- b^2 ---- b^2+b-1
# | |
# *-------
# |
# 0 ----- b
#
# *--------
# |
# *-------* bit=1
# |
# *--- b^2-1
# |
# *------- bit=1
# |
# 0 ----- b
#
sub _UNDOCUMENTED__dxdy_list_at_n {
my ($self) = @_;
my $radix = $self->{'radix'};
my $n = $radix*$radix;
if ($self->{'serpentine_array'}->[0]) {
foreach my $i (1 .. $self->{'radix'}-1) {
if ($self->{'serpentine_array'}->[$i] == 0) {
return $n*$i + $radix;
}
}
if ($self->{'serpentine_array'}->[$radix] == 0) {
return $n*$radix;
} else {
return $n*$radix + $radix-1;
}
} else {
if ($self->{'serpentine_array'}->[1]) {
return $n;
} else {
return $n + $radix-1;
}
}
}
*dsumxy_minimum = \&dx_minimum;
*dsumxy_maximum = \&dx_maximum;
*ddiffxy_minimum = \&dy_minimum;
*ddiffxy_maximum = \&dy_maximum;
# radix=2 0101 is straight NSEW parts, other evens are diagonal
sub dir_maximum_dxdy {
my ($self) = @_;
return (($self->{'radix'} % 2)
|| join('',@{$self->{'serpentine_array'}}) eq '0101'
? (0,-1) # odd, South
: (0,0)); # even, supremum
}
#------------------------------------------------------------------------------
sub new {
my $self = shift->SUPER::new(@_);
my $radix = $self->{'radix'};
if (! $self->{'radix'} || $self->{'radix'} < 2) {
$radix = $self->{'radix'} = 3;
}
my @serpentine_array;
my $serpentine_type = $self->{'serpentine_type'};
if (! defined $serpentine_type) {
$serpentine_type = 'alternating';
}
$serpentine_type = lc($serpentine_type);
### $serpentine_type
if ($serpentine_type eq 'alternating') {
@serpentine_array = map {$_&1} 0 .. $radix*$radix - 1;
} elsif ($serpentine_type eq 'coil') {
@serpentine_array = (1) x ($radix*$radix);
} elsif (lc($serpentine_type) eq 'peano') {
@serpentine_array = (0) x ($radix*$radix);
} elsif ($serpentine_type =~ /^([01_,.]|\s)*$/) {
# bits 010,101,010 etc
$serpentine_type =~ tr/01//cd; # keep only 0,1 chars
@serpentine_array
= map {$_+0} # numize for xor
split //, $serpentine_type; # each char, 0,1
push @serpentine_array,
(0) x max(0,$radix*$radix-scalar(@serpentine_array));
# foreach my $char {
# if ($char eq '0' || $char eq '1') {
# push , $char;
# }
# my @parts = split /[^01]+/, $serpentine_type;
# ### @parts
# @parts = grep {$_ ne ''} @parts; # no empty parts
# foreach my $part (@parts) {
# my @bits = split //, $part;
# push @bits, (0) x max(0,scalar(@bits)-$radix);
# $#bits = $radix-1;
# push @serpentine_array, @bits; # radix many row
# }
} else {
croak "Unrecognised serpentine_type \"$serpentine_type\"";
}
### @serpentine_array
$self->{'serpentine_array'} = \@serpentine_array;
return $self;
}
sub n_to_xy {
my ($self, $n) = @_;
### WunderlichSerpentine n_to_xy(): $n
if ($n < 0) {
return;
}
if (is_infinite($n)) {
return ($n,$n);
}
{
# ENHANCE-ME: for odd radix the ends join and the direction can be had
# without a full N+1 calculation
my $int = int($n);
### $int
### $n
if ($n != $int) {
my ($x1,$y1) = $self->n_to_xy($int);
my ($x2,$y2) = $self->n_to_xy($int+1);
my $frac = $n - $int; # inherit possible BigFloat
my $dx = $x2-$x1;
my $dy = $y2-$y1;
return ($frac*$dx + $x1, $frac*$dy + $y1);
}
$n = $int; # BigFloat int() gives BigInt, use that
}
# high to low
my $radix = $self->{'radix'};
my $rsquared = $radix * $radix;
my $radix_minus_1 = $radix - 1;
my $serpentine_array = $self->{'serpentine_array'};
my @digits = digit_split_lowtohigh($n,$rsquared);
my $x = 0;
my $y = 0;
my $transpose = ($#digits & 1) && $serpentine_array->[0];
my $xk = my $yk = 0;
while (@digits) {
my $ndigit = pop @digits; # high to low
my ($highdigit, $lowdigit) = _divrem ($ndigit, $radix);
### $lowdigit
### $highdigit
if ($transpose) {
$yk ^= $highdigit;
$x *= $radix;
$x += ($xk & 1 ? $radix_minus_1-$highdigit : $highdigit);
$xk ^= $lowdigit;
$y *= $radix;
$y += ($yk & 1 ? $radix_minus_1-$lowdigit : $lowdigit);
} else {
$xk ^= $highdigit;
$y *= $radix;
$y += ($yk & 1 ? $radix_minus_1-$highdigit : $highdigit);
$yk ^= $lowdigit;
$x *= $radix;
$x += ($xk & 1 ? $radix_minus_1-$lowdigit : $lowdigit);
}
### $serpentine_array
### $ndigit
$transpose ^= $serpentine_array->[$ndigit];
}
return ($x, $y);
# my $x = 0;
# my $y = 0;
# my $power = $x + 1; # inherit bignum 1
# my $odd = 1;
# my $transpose = 0;
# while ($n) {
# $odd ^= 1;
# ### $n
# ### $power
#
# my $xdigit = $n % $radix;
# $n = int($n/$radix);
# my $ydigit = $n % $radix;
# $n = int($n/$radix);
#
# if ($transpose) {
# ($xdigit,$ydigit) = ($ydigit,$xdigit);
# }
# $transpose ^= $serpentine_array->[$xdigit + $radix*$ydigit];
#
# ### $xdigit
# ### $ydigit
#
# if ($xdigit & 1) {
# $y = $power-1 - $y; # 99..99 - Y
# }
# $x += $power * $xdigit;
#
# $y += $power * $ydigit;
# $power *= $radix;
# if ($ydigit & 1) {
# $x = $power-1 - $x;
# }
# }
#
# #
# # if ($odd) {
# # ($x,$y) = ($y,$x);
# # ### final transpose to: "$x,$y"
# # }
#
# return ($x, $y);
}
sub xy_to_n {
my ($self, $x, $y) = @_;
### WunderlichSerpentine xy_to_n(): "$x, $y"
$x = round_nearest ($x);
$y = round_nearest ($y);
if ($x < 0 || $y < 0
|| is_infinite($x)
|| is_infinite($y)) {
return undef;
}
my $radix = $self->{'radix'};
my $radix_minus_1 = $radix - 1;
my @xdigits = digit_split_lowtohigh($x,$radix);
my @ydigits = digit_split_lowtohigh($y,$radix);
### @xdigits
### @ydigits
my $serpentine_array = $self->{'serpentine_array'};
my $xk = 0;
my $yk = 0;
my $highpos = max($#xdigits,$#ydigits);
my $transpose = $serpentine_array->[0] && ($highpos & 1);
my $n = ($x * 0 * $y); # inherit bignum 0
foreach my $i (reverse 0 .. $highpos) { # high to low
my $xdigit = $xdigits[$i] || 0;
my $ydigit = $ydigits[$i] || 0;
my $ndigit;
### $n
### $xk
### $yk
### $transpose
### $xdigit
### $ydigit
if ($transpose) {
if ($xk & 1) { $xdigit = $radix_minus_1 - $xdigit; }
$n *= $radix;
$n += $xdigit;
$yk ^= $xdigit;
if ($yk & 1) { $ydigit = $radix_minus_1 - $ydigit; }
$n *= $radix;
$n += $ydigit;
$xk ^= $ydigit;
$ndigit = $radix*$xdigit + $ydigit;
} else {
if ($yk & 1) { $ydigit = $radix_minus_1 - $ydigit; }
$n *= $radix;
$n += $ydigit;
$xk ^= $ydigit;
if ($xk & 1) { $xdigit = $radix_minus_1 - $xdigit; }
$n *= $radix;
$n += $xdigit;
$yk ^= $xdigit;
$ndigit = $radix*$ydigit + $xdigit;
}
### ndigits: "$ydigit, $xdigit"
$transpose ^= $serpentine_array->[$ndigit];
}
return $n;
# my $n = 0;
# while (@x) {
# if (($xk ^ $yk) & 1) {
# {
# my $digit = pop @x;
# if ($xk & 1) {
# $digit = $radix_minus_1 - $digit;
# }
# $n = ($n * $radix) + $digit;
# $yk ^= $digit;
# }
# {
# my $digit = pop @y;
# if ($yk & 1) {
# $digit = $radix_minus_1 - $digit;
# }
# $n = ($n * $radix) + $digit;
# $xk ^= $digit;
# }
# } else {
# {
# my $digit = pop @y;
# if ($yk & 1) {
# $digit = $radix_minus_1 - $digit;
# }
# $n = ($n * $radix) + $digit;
# $xk ^= $digit;
# }
# {
# my $digit = pop @x;
# if ($xk & 1) {
# $digit = $radix_minus_1 - $digit;
# }
# $n = ($n * $radix) + $digit;
# $yk ^= $digit;
# }
# }
# }
#
# return $n;
# return $n;
}
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
$x1 = round_nearest ($x1);
$y1 = round_nearest ($y1);
$x2 = round_nearest ($x2);
$y2 = round_nearest ($y2);
($x1,$x2) = ($x2,$x1) if $x1 > $x2;
($y1,$y2) = ($y2,$y1) if $y1 > $y2;
### rect_to_n_range(): "$x1,$y1 to $x2,$y2"
if ($x2 < 0 || $y2 < 0) {
return (1, 0);
}
my $radix = $self->{'radix'};
my ($power, $level) = round_down_pow (max($x2,$y2), $radix);
if (is_infinite($level)) {
return (0, $level);
}
$power *= $radix;
return (0, $power * $power - 1);
# my $n_power = $power * $power;
# my $max_x = 0;
# my $max_y = 0;
# my $max_n = 0;
# my $max_xk = 0;
# my $max_yk = 0;
#
# my $min_x = 0;
# my $min_y = 0;
# my $min_n = 0;
# my $min_xk = 0;
# my $min_yk = 0;
#
# my $serpentine_array = $self->{'serpentine_array'};
# if ($serpentine_array->[0] && ($level&1)) {
# $max_xk = $max_yk = $min_xk = $min_yk = 1;
# }
#
#
# # l<=c<h doesn't overlap c1<=c<=c2 if
# # l>c2 or h-1<c1
# # l>c2 or h<=c1
# # so does overlap if
# # l<=c2 and h>c1
# #
# my $radix_minus_1 = $radix - 1;
# my $overlap = sub {
# my ($c,$ck,$digit, $c1,$c2) = @_;
# if ($ck & 1) {
# $digit = $radix_minus_1 - $digit;
# }
# ### overlap consider: "inv@{[$ck&1]}digit=$digit ".($c+$digit*$power)."<=c<".($c+($digit+1)*$power)." cf $c1 to $c2 incl"
# return ($c + $digit*$power <= $c2
# && $c + ($digit+1)*$power > $c1);
# };
#
# while ($power > 1) {
# $power = int($power/$radix);
# $n_power = int($n_power/$radix);
#
# my $min_transpose = ($min_xk ^ $min_yk) & 1;
# my $max_transpose = ($min_xk ^ $min_yk) & 1;
#
# ### $power
# ### $n_power
# ### $max_n
# ### $min_n
# if ($max_transpose) {
# my $digit;
# for ($digit = $radix_minus_1; $digit > 0; $digit--) {
# last if &$overlap ($max_x,$max_xk,$digit, $x1,$x2);
# }
# $max_n += $n_power * $digit;
# $max_yk ^= $digit;
# if ($max_xk&1) { $digit = $radix_minus_1 - $digit; }
# $max_x += $power * $digit;
# } else {
# my $digit;
# for ($digit = $radix_minus_1; $digit > 0; $digit--) {
# last if &$overlap ($max_y,$max_yk,$digit, $y1,$y2);
# }
# $max_n += $n_power * $digit;
# $max_xk ^= $digit;
# if ($max_yk&1) { $digit = $radix_minus_1 - $digit; }
# $max_y += $power * $digit;
# }
#
# if ($min_transpose) {
# my $digit;
# for ($digit = 0; $digit < $radix_minus_1; $digit++) {
# last if &$overlap ($min_x,$min_xk,$digit, $x1,$x2);
# }
# $min_n += $n_power * $digit;
# $min_yk ^= $digit;
# if ($min_xk&1) { $digit = $radix_minus_1 - $digit; }
# $min_x += $power * $digit;
# } else {
# my $digit;
# for ($digit = 0; $digit < $radix_minus_1; $digit++) {
# last if &$overlap ($min_y,$min_yk,$digit, $y1,$y2);
# }
# $min_n += $n_power * $digit;
# $min_xk ^= $digit;
# if ($min_yk&1) { $digit = $radix_minus_1 - $digit; }
# $min_y += $power * $digit;
# }
#
# $n_power = int($n_power/$radix);
# if ($max_transpose) {
# my $digit;
# for ($digit = $radix_minus_1; $digit > 0; $digit--) {
# last if &$overlap ($max_y,$max_yk,$digit, $y1,$y2);
# }
# $max_n += $n_power * $digit;
# $max_xk ^= $digit;
# if ($max_yk&1) { $digit = $radix_minus_1 - $digit; }
# $max_y += $power * $digit;
# } else {
# my $digit;
# for ($digit = $radix_minus_1; $digit > 0; $digit--) {
# last if &$overlap ($max_x,$max_xk,$digit, $x1,$x2);
# }
# $max_n += $n_power * $digit;
# $max_yk ^= $digit;
# if ($max_xk&1) { $digit = $radix_minus_1 - $digit; }
# $max_x += $power * $digit;
# }
#
# if ($min_transpose) {
# my $digit;
# for ($digit = 0; $digit < $radix_minus_1; $digit++) {
# last if &$overlap ($min_y,$min_yk,$digit, $y1,$y2);
# }
# $min_n += $n_power * $digit;
# $min_xk ^= $digit;
# if ($min_yk&1) { $digit = $radix_minus_1 - $digit; }
# $min_y += $power * $digit;
# } else {
# my $digit;
# for ($digit = 0; $digit < $radix_minus_1; $digit++) {
# last if &$overlap ($min_x,$min_xk,$digit, $x1,$x2);
# }
# $min_n += $n_power * $digit;
# $min_yk ^= $digit;
# if ($min_xk&1) { $digit = $radix_minus_1 - $digit; }
# $min_x += $power * $digit;
# }
# }
# ### is: "$min_n at $min_x,$min_y to $max_n at $max_x,$max_y"
# return ($min_n, $max_n);
}
1;
__END__
=for stopwords Walter Wunderlich Wunderlich's there'll eg Ryde OEIS trit-twiddling ie bignums prepending trit Math-PlanePath versa Online radix Uber Peano-Kurven Elemente der Mathematik
=head1 NAME
Math::PlanePath::WunderlichSerpentine -- transpose parts of Peano curve, including coil order
=head1 SYNOPSIS
use Math::PlanePath::WunderlichSerpentine;
my $path = Math::PlanePath::WunderlichSerpentine->new (serpentine_type => '111_000_111');
my ($x, $y) = $path->n_to_xy (123);
# or another radix digits ...
my $path5 = Math::PlanePath::WunderlichSerpentine->new (radix => 5);
=head1 DESCRIPTION
X<Wunderlich, Walter>This is an integer version of Walter Wunderlich's
variations on the C<PeanoCurve>. A "serpentine type" controls transposing
of selected 3x3 sub-parts. The default is "alternating" 010,101,010 which
transposes every second sub-part,
8 | 60--61--62--63 68--69 78--79--80--81
| | | | | | |
7 | 59--58--57 64 67 70 77--76--75 ...
| | | | | |
6 | 54--55--56 65--66 71--72--73--74
| |
5 | 53 48--47 38--37--36--35 30--29
| | | | | | | |
4 | 52 49 46 39--40--41 34 31 28
| | | | | | | |
3 | 51--50 45--44--43--42 33--32 27
| |
2 | 6-- 7-- 8-- 9 14--15 24--25--26
| | | | | |
1 | 5-- 4-- 3 10 13 16 23--22--21
| | | | | |
Y=0 | 0-- 1-- 2 11--12 17--18--19--20
|
+-------------------------------------
X=0 1 2 3 4 5 6 7 8
C<serpentine_type> can be a string of 0s and 1s, with optional space, comma
or _ separators at each group of 3,
"011111011" 0/1 string
"011,111,011"
"011_111_011"
"011 111 011"
or special values
"alternating" 01010101.. the default
"coil" 11111... all 1s described below
"Peano" 00000... all 0s, gives PeanoCurve
Each "1" sub-part is transposed. The string is applied in order of the N
parts, irrespective of what net reversals and transposes are in force on a
particular part.
When no parts are transposed, which is a string of all 0s, the result is the
same as the C<PeanoCurve>. The special C<serpentine_type =E<gt> "Peano">
gives that.
=head2 Coil Order
C<serpentine_type =E<gt> "coil"> means "111 111 111" to transpose all parts.
The result is like a coil viewed side-on,
8 24--25--26--27--28--29 78--79--80--81--...
| | |
7 23--22--21 32--31--30 77--76--75
| | |
6 18--19--20 33--34--35 72--73--74
| | |
5 17--16--15 38--37--36 71--70--69
| | |
4 12--13--14 39--40--41 66--67--68
| | |
3 11--10-- 9 44--43--42 65--64--63
| | |
2 6-- 7-- 8 45--46--47 60--61--62
| | |
1 5-- 4-- 3 50--49--48 59--58--57
| | |
Y=0 0-- 1-- 2 51--52--53--54--55--56
X=0 1 2 3 4 5 6 7 8
Whenever C<serpentine_type> begins with a "1" the initial sub-part is
transposed at each level. The first step N=0 to N=1 is kept fixed along the
X axis, then the higher levels are transposed. For example in the coil
above The N=9 to N=17 part is upwards, and then the N=81 to N=161 part is to
the right, and so on.
=head2 Radix
The optional C<radix> parameter gives the size of the sub-parts, similar to
the C<PeanoCurve> C<radix> parameter (see
L<Math::PlanePath::PeanoCurve/Radix>). For example radix 5 gives
radix => 5
4 | 20-21-22-23-24-25 34-35 44-45 70-71-72-73-74-75 84-85
| | | | | | | | | | |
3 | 19-18-17-16-15 26 33 36 43 46 69-68-67-66-65 76 83 86
| | | | | | | | | | |
2 | 10-11-12-13-14 27 32 37 42 47 60-61-62-63-64 77 82 87
| | | | | | | | | | |
1 | 9--8--7--6--5 28 31 38 41 48 59-58-57-56-55 78 81 88
| | | | | | | | | | |
Y=0 | 0--1--2--3--4 29-30 39-40 49-50-51-52-53-54 79-80 89-..
+---------------------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Like the C<PeanoCurve> if the radix is even then the ends of each sub-part
don't join up. For example in radix 4 N=15 isn't next to N=16, nor N=31 to
N=32, etc.
| |
3 | 15--14--13--12 16 23--24 31 47--46--45--44 48 55--56 63
| | | | | | | | | | |
2 | 8-- 9--10--11 17 22 25 30 40--41--42--43 49 54 57 62
| | | | | | | | | | |
1 | 7-- 6-- 5-- 4 18 21 26 29 39--38--37--36 50 53 58 61
| | | | | | | | | | |
Y=0 | 0-- 1-- 2-- 3 19--20 27--28 32--33--34--35 51--52 59--60
+----------------------------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
In the C<serpentine_type> 0,1 form, any space, comma, etc, separators should
group C<radix> many values, so for example
serpentine_type => "00000_11111_00000_00000_11111"
The intention is to do something friendly if the separators are not on such
boundaries, so that say 000_111_000 can have a sensible meaning in a radix
higher than 3. But exactly what is not settled, so always give a full
string of desired 0,1 for now.
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for the behaviour common to all path
classes.
=over 4
=item C<$path = Math::PlanePath::WunderlichSerpentine-E<gt>new ()>
=item C<$path = Math::PlanePath::WunderlichSerpentine-E<gt>new (serpentine_type =E<gt> $str, radix =E<gt> $r)>
Create and return a new path object.
The optional C<radix> parameter gives the base for digit splitting. The
default is ternary, radix 3. The radix should be an odd number, 3, 5, 7, 9
etc.
=back
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::PeanoCurve>
Walter Wunderlich "Uber Peano-Kurven", Elemente der Mathematik, 28(1):1-10,
1973. L<http://sodwana.uni-ak.ac.at/geom/mitarbeiter/wallner/wunderlich/>Z<>
L<http://sodwana.uni-ak.ac.at/geom/mitarbeiter/wallner/wunderlich/pdf/125.pdf>
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-planepath/index.html>
=head1 LICENSE
Copyright 2011, 2012, 2013, 2014 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/>.
=cut