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# Copyright 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/>.
# x=45,y=10 x=59,y=19 dx=14,dy=9 14/9=1.55
#
# x=42,y=8 x=113,y=52 dx=71,dy=44 71/44=1.613
#
# below
# 32,12 to 36,4 sqrt((32-36)^2+(12-4)^2) = 9
# 84,34 to 99,14 sqrt((84-99)^2+(34-14)^2) = 25
# 180,64 to 216,11 sqrt((180-216)^2+(64-11)^2) = 64
#
# above
# 14,20 to 5,32 sqrt((14-5)^2+(20-32)^2) = 15 = 9*1.618 3
# 34,50 to 14,85 sqrt((34-14)^2+(50-85)^2) = 40 = 25*1.618 5
# 132,158 to 77,247 sqrt((132-77)^2+(158-247)^2) = 104 = 64*1.618 8
# 8,525 to 133,280 sqrt((8-133)^2+(525-280)^2) = 275 = 169*1.618 13
package Math::PlanePath::WythoffLines;
use 5.004;
use strict;
use List::Util 'max';
use vars '$VERSION', '@ISA';
$VERSION = 117;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
use Math::PlanePath::Base::Generic
'is_infinite',
'round_nearest';
use Math::PlanePath::Base::Digits
'bit_split_lowtohigh';
# uncomment this to run the ### lines
# use Smart::Comments;
use constant parameter_info_array =>
[ { name => 'shift',
display => 'Shift',
type => 'integer',
default => 0,
width => 3,
},
];
# shift x_minimum() y_minimum()
# -4 13 8
# -3 8 5
# -2 5 3
# -1 3 2
# 1 2 1
# 0 2 1 ...
# 1 1 1 fib(1)
# 2 1 /---> 0 -----^ fib(0)
# 3 0 <--/ 1 a
# 4 1 -1 b
# 5 -1 2 c
# 6 2 -4 d -4=2*-1-2
# 7 -4 4 e 4=2*2-0
# 8 4 -12 -12=2*-4-4
# 9 -12 9 9=2*4-(-1)
# 10 9 -33
# 11 -33 22 22=3*9-4-1 a(n)=3a(n-2)-a(n-4)-1
# 12 22 -88 -88=2*-33-22 2*a(n-2)-a(n-1)
# 13 -88 56 56=2*22+12 2*a(n-2)-a(n-5)
# 14 56 -232 -232=2*-88-56 2*a(n-2)-a(n-1)
# 15 -232 145 145=2*56+33 2*a(n-2)-a(n-5)
# 16 -609 -609=2*-232-145
# 17 -609 378 378=2*145-(-88)
#
# shift -4,-12,-33,-88,-232 = 1-Fib(2*s+1)
# shift 9,22,56,145,378,988
# a(n)=3*a(n-1)-a(n-2)-1
# with $shift reckoned for y_minimum()
sub _calc_minimum {
my ($shift) = @_;
if ($shift <= 2) {
return _fibonacci(2-$shift);
}
if ($shift & 1) {
# shift odd >= 3, so (shift-1)/2 >= 1
my $a = 1;
my $b = 2;
foreach (2 .. ($shift-1)/2) {
($a,$b) = ($b, 3*$b-$a-1);
}
return $a;
} else {
# shift even >= 4
return 1 - _fibonacci($shift-1);
}
# $a = 1;
# $b = -1;
# my $c = 2;
# my $d = -4;
# my $e = 4;
# for (my $i = 2; $i < $shift; $i++) {
# ($a,$b,$c,$d,$e) = ($b,$c,$d,$e, 2*$d-$e);
# $i++;
# last unless $i < $shift;
# ($a,$b,$c,$d,$e) = ($b,$c,$d,$e, 2*$d-$a);
# }
# return $a;
}
sub _fibonacci {
my ($n) = @_;
$a = 0;
$b = 1;
foreach (1 .. $n) {
($a,$b) = ($b,$a+$b);
}
return $a;
}
sub x_minimum {
my ($self) = @_;
return _calc_minimum($self->{'shift'}-1);
}
sub y_minimum {
my ($self) = @_;
return _calc_minimum($self->{'shift'});
}
#------------------------------------------------------------------------------
use Math::PlanePath::WythoffArray;
my $wythoff = Math::PlanePath::WythoffArray->new;
sub new {
my $self = shift->SUPER::new(@_);
$self->{'shift'} ||= 0;
return $self;
}
sub n_to_xy {
my ($self, $n) = @_;
### WythoffLines n_to_xy(): $n
if ($n < 1) { return; }
if (is_infinite($n) || $n == 0) { return ($n,$n); }
{
# fractions on straight line
my $int = int($n);
if ($n != $int) {
my $frac = $n - $int; # inherit possible BigFloat/BigRat
my ($x1,$y1) = $self->n_to_xy($int);
my ($x2,$y2) = $self->n_to_xy($int+1);
my $dx = $x2-$x1;
my $dy = $y2-$y1;
return ($frac*$dx + $x1, $frac*$dy + $y1);
}
$n = $int;
}
# $n -= 1;
# my $y = $wythoff->xy_to_n(0,$n);
# my $x = $wythoff->xy_to_n(1,$n);
# 1 2.000, 1.000 1 1_100000 5.000,3.000(5.831)
# 2 7.000, 4.000 2 1_100000 3.000,2.000(3.606)
# 3 10.000, 6.000 3 1_100000 5.000,3.000(5.831)
# 4 15.000, 9.000 4 1_100000 5.000,3.000(5.831)
# 5 20.000, 12.000 5 1_100000 3.000,2.000(3.606)
# 6 23.000, 14.000 6 1_100000 5.000,3.000(5.831)
# 7 28.000, 17.000 7 1_100000 3.000,2.000(3.606)
my $zero = $n*0;
# spectrum(Y+1) so Y,Ybefore are notional two values at X=-2 and X=-1
my $y = $n-1;
my $x = int((sqrt(5*$n*$n) + $n) / 2);
# ($y,$x) = (1*$x + 1*$y,
# 2*$x + 1*$y);
# shift s to -1
# 1 to s
# but forward by 2 extra
# s to -1+2=1
# 1+2=3 to s
foreach ($self->{'shift'} .. 1) {
($y,$x) = ($x,$x+$y);
}
foreach (3 .. $self->{'shift'}) {
# prev+y=x
# prev = x-y
($y,$x) = ($x-$y,$y);
}
return ($x,$y);
}
sub xy_to_n {
my ($self, $x, $y) = @_;
### WythoffLines xy_to_n(): "$x, $y"
$x = round_nearest ($x);
$y = round_nearest ($y);
# if (is_infinite($y)) { return $y; }
# unshift
#
foreach ($self->{'shift'} .. -1) {
($y,$x) = ($x-$y,$y);
}
foreach (1 .. $self->{'shift'}) {
($y,$x) = ($x,$x+$y);
}
### unshifted to: "$x,$y"
if (my ($cy,$ny) = $wythoff->n_to_xy($y)) {
### y: "cy=$cy ny=$ny"
if ($cy == 0) {
if (my ($cx,$nx) = $wythoff->n_to_xy($x)) {
if ($cx == 1 && $nx == $ny) {
return $nx+1;
}
}
}
}
return undef;
# my $y = $wythoff->xy_to_n(0,$n);
# my $x = $wythoff->xy_to_n(1,$n);
}
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
### WythoffLines rect_to_n_range(): "$x1,$y1 $x2,$y2"
my $zero = $x1 * 0 * $y1 * $x2 * $y2;
$x1 = round_nearest ($x1);
$y1 = round_nearest ($y1);
$x2 = round_nearest ($x2);
$y2 = round_nearest ($y2);
# FIXME: probably not quite right
my $phi = (1 + sqrt(5+$zero)) / 2;
return (1,
max (1,
int($phi**($self->{'shift'}-2)
* max ($x1,$x2, max($y1,$y2)*$phi))));
}
1;
__END__
=for stopwords eg Ryde Math-PlanePath Moore Wythoff Zeckendorf concecutive fibbinary OEIS
=head1 NAME
Math::PlanePath::WythoffLines -- table of Fibonacci recurrences
=head1 SYNOPSIS
use Math::PlanePath::WythoffLines;
my $path = Math::PlanePath::WythoffLines->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
X<Kimberling, Clark>This path is the Wythoff preliminary triangle by Clark
Kimberling,
=cut
# math-image --path=WythoffLines --output=numbers --all --size=60x14
=pod
13 | 105 118 131 144 60 65 70 75 80 85 90 95 100
12 | 97 110 47 52 57 62 67 72 77 82 87 92
11 | 34 39 44 49 54 59 64 69 74 79 84
10 | 31 36 41 46 51 56 61 66 71 76
9 | 28 33 38 43 48 53 58 63 26
8 | 25 30 35 40 45 50 55 23
7 | 22 27 32 37 42 18 20
6 | 19 24 29 13 15 17
5 | 16 21 10 12 14
4 | 5 7 9 11
3 | 4 6 8
2 | 3 2
1 | 1
Y=0 |
+-----------------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12
A coordinate pair Y and X are the start of a Fibonacci style recurrence,
F[1]=Y, F[2]=X F[i+i] = F[i] + F[i-1]
Any such sequence eventually becomes a row of the Wythoff array
(L<Math::PlanePath::WythoffArray>) after some number of initial iterations.
The N value at X,Y is the row number of the Wythoff array containing
sequence beginning Y and X. Rows are numbered starting from 1. Eg.
Y=4,X=1 sequence: 4, 1, 5, 6, 11, 17, 28, 45, ...
row 7 of WythoffArray: 17, 28, 45, ...
so N=7 at Y=4,X=1
Conversely a given N is positioned in the triangle according to where row
number N of the Wythoff array "precurses" by running the recurrence in
reverse,
F[i-1] = F[i+i] - F[i]
It can be shown that such a precurse always reaches a pair Y and X with
YE<gt>=1 and 0E<lt>=XE<lt>Y, hence making the triangular X,Y arrangement
above.
N=7 WythoffArray row 7 is 17,28,45,73,...
go backwards from 17,28 by subtraction
11 = 28 - 17
6 = 17 - 11
5 = 11 - 6
1 = 6 - 5
4 = 5 - 1
stop on reaching 4,1 which is Y=4,X=1 satisfying Y>=1 and 0<=X<Y
=head2 Phi Slope Blocks
The effect of each step backwards is to move to successive blocks of values,
with slope golden ratio phi=(sqrt(5)+1)/2.
Suppose no backwards steps were applied, so Y,X were the first two values of
Wythoff row N. In the example above that would be N=7 at Y=17,X=28. The
first two values of the Wythoff array are
Y = W[0,r] = r-1 + floor(r*phi) # r = row numbered from 1
X = W[1,r] = r-1 + 2*floor(r*phi)
So this would put N values on a line of slope Y/X = 1/phi = 0.618. The
portion of that line which falls within 0E<lt>=XE<lt>Y
=cut
# (r-1 + floor(r*phi)) / (r-1 + 2*floor(r*phi))
# ~= (r-1+r*phi)/(r-1+2*r*phi)
# = (r*(phi+1) - 1) / (r*(2phi+1) - 1)
# -> r*(phi+1) / r*(2*phi+1)
# = (phi+1) / (2*phi+1)
# = 1/phi = 0.618
=pod
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for the behaviour common to all path
classes.
=over 4
=item C<$path = Math::PlanePath::WythoffLines-E<gt>new ()>
Create and return a new path object.
=back
=head1 OEIS
The Wythoff array is in Sloane's Online Encyclopedia of Integer Sequences
in various forms,
=over
L<http://oeis.org/A035614> (etc)
=back
A165360 X
A165359 Y
A166309 N by rows
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::WythoffArray>
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-planepath/index.html>
=head1 LICENSE
Copyright 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
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