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# Copyright 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.
# Edwin L. Godfrey, "Enumeration of the Rational Points Between 0 and 1",
# National Mathematics Magazine, volume 12, number 4, January 1938, pages
# 163-166. http://www.jstor.org/stable/3028080
# cf
# A126572 Array read by antidiagonals: a(n,m) = the m-th integer from among those positive integers coprime to n.
# 1/1 1/2 1/3 1/4 1/5 1/6 1/7 ...
# 2/1 2/3 2/5 2/7 2/9 2/11 2/13 ...
# 3/1 3/2 3/4 3/5 3/7 3/8 3/10 ...
# 4/1 4/3 4/5 4/7 4/9 4/11 4/13 ...
# 5/1 5/2 5/3 5/4 5/6 5/7 5/8 ...
# 6/1 6/5 6/7 6/11 6/13 6/17 6/19 ...
# 7/1 7/2 7/3 7/4 7/5 7/6 7/8 ...
# 1/2 1/3 1/4 1/5 1/6 1/7
# 2/3 2/5 2/7 2/9 2/11 2/13
# 3/4 3/5 3/7 3/8 3/10 3/11
# 4/5 4/7 4/9 4/11 4/13 4/15
package Math::PlanePath::Godfrey;
use 5.004;
use strict;
use vars '$VERSION', '@ISA';
$VERSION = 129;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
*_divrem = \&Math::PlanePath::_divrem;
*_sqrtint = \&Math::PlanePath::_sqrtint;
use Math::PlanePath::Base::Generic
'is_infinite',
'round_nearest';
use Math::PlanePath::CoprimeColumns;
# uncomment this to run the ### lines
# use Smart::Comments;
use constant class_x_negative => 0;
use constant class_y_negative => 0;
use constant x_minimum => 1;
use constant y_minimum => 2;
use constant diffxy_maximum => -1; # upper octant X<=Y-1 so X-Y<=-1
use constant gcdxy_maximum => 1; # no common factor
#------------------------------------------------------------------------------
sub n_to_xy {
my ($self, $n) = @_;
### Godfrey n_to_xy(): $n
if ($n < 1) { return; }
if (is_infinite($n)) { return ($n,$n); }
my $d = int((_sqrtint(8*$n-7) + 1) / 2);
### $d
### base: ($d-1)*$d/2
$n -= ($d-1)*$d/2;
my $y = $n;
my $q = $d - $y;
# ### assert: $n >= 0
# ### assert: $y >= 1
my $tot = Math::PlanePath::CoprimeColumns::_totient($y);
my ($f, $count) = _divrem ($q, $tot);
### $y
### $q
### $tot
my $x = 1;
if ($count) {
for (;;) {
$x++;
if (Math::PlanePath::CoprimeColumns::_coprime($x,$y)) {
--$count or last;
}
}
}
# final den: $x + ($f+1)*$y)
return ($y, $x + ($f+1)*$y);
}
sub xy_to_n {
my ($self, $x, $y) = @_;
### Godfrey xy_to_n(): "$x, $y"
$x = round_nearest ($x);
$y = round_nearest ($y);
if ($x < 1 || $y < 1) {
return undef;
}
if (is_infinite($x)) {
return $x;
}
my ($f, $r) = _divrem ($y, $x);
### $f
### $r
my $w = ($f-1) * Math::PlanePath::CoprimeColumns::_totient($x);
### w from totient: $w
foreach my $i (1 .. $r) {
if (Math::PlanePath::CoprimeColumns::_coprime($i,$x)) {
### coprime: "$i, x=$x, increment"
$w++;
}
}
my $d = $x + $w - 1;
### $x
### $w
### $d
### return: $d*($d-1)/2 + $x
return $d*($d-1)/2 + $x;
}
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
### Godfrey rect_to_n_range(): "$x1,$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 > $y2;
if ($x2 < 1 || $y2 < 1) { return (1,0); }
return (1, $self->xy_to_n($x2,$y2));
}
1;
__END__
=cut
# math-image --path=Godfrey --output=numbers --all --size=60x14
=pod
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