# 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/>.
# math-image --path=PythagoreanTree --all --scale=3
# http://sunilchebolu.wordpress.com/pythagorean-triples-and-the-integer-points-on-a-hyperboloid/
# http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/pythagtriple.pdf
#
# http://www.math.ou.edu/~dmccullough/teaching/pythagoras1.pdf
# http://www.math.ou.edu/~dmccullough/teaching/pythagoras2.pdf
#
# http://www.microscitech.com/pythag_eigenvectors_invariants.pdf
#
package Math::PlanePath::PythagoreanTree;
use 5.004;
use strict;
use Carp 'croak';
use vars '$VERSION', '@ISA';
$VERSION = 118;
use Math::PlanePath;
*_divrem = \&Math::PlanePath::_divrem;
@ISA = ('Math::PlanePath');
#use List::Util 'min','max';
*min = \&Math::PlanePath::_min;
*max = \&Math::PlanePath::_max;
use Math::PlanePath::Base::Generic
'is_infinite',
'round_nearest';
use Math::PlanePath::Base::Digits
'round_down_pow',
'digit_split_lowtohigh',
'digit_join_lowtohigh';
use Math::PlanePath::GrayCode;
# uncomment this to run the ### lines
# use Smart::Comments;
use constant class_x_negative => 0;
use constant class_y_negative => 0;
use constant tree_num_children_list => (3); # complete ternary tree
use constant tree_n_to_subheight => undef; # complete tree, all infinity
use constant parameter_info_array =>
[ { name => 'tree_type',
share_key => 'tree_type_uadfb',
display => 'Tree Type',
type => 'enum',
default => 'UAD',
choices => ['UAD','UArD','FB','UMT'],
},
{ name => 'coordinates',
share_key => 'coordinates_abcpqsm',
display => 'Coordinates',
type => 'enum',
default => 'AB',
choices => ['AB','AC','BC','PQ', 'SM','SC','MC',
# 'BA'
# 'UV', # q from x=y diagonal down at 45-deg
# 'RS','ST', # experimental
],
},
{ name => 'digit_order',
display => 'Digit Order',
type => 'enum',
default => 'HtoL',
choices => ['HtoL','LtoH'],
},
];
#------------------------------------------------------------------------------
{
my %coordinate_minimum = (A => 3,
B => 4,
C => 5,
P => 2,
Q => 1,
S => 3,
M => 4,
);
sub x_minimum {
my ($self) = @_;
return $coordinate_minimum{substr($self->{'coordinates'},0,1)};
}
sub y_minimum {
my ($self) = @_;
return $coordinate_minimum{substr($self->{'coordinates'},1)};
}
}
{
my %diffxy_minimum = (PQ => 1, # octant X>=Y+1 so X-Y>=1
);
sub diffxy_minimum {
my ($self) = @_;
return $diffxy_minimum{$self->{'coordinates'}};
}
}
{
my %diffxy_maximum = (AC => -2, # C>=A+2 so X-Y<=-2
BC => -1, # C>=B+1 so X-Y<=-1
SM => -1, # S<M so X-Y<=-1
SC => -2, # S<M<C so S-C<=-2
MC => -1, # M<C so M-C<=-1
);
sub diffxy_maximum {
my ($self) = @_;
return $diffxy_maximum{$self->{'coordinates'}};
}
}
{
my %absdiffxy_minimum = (PQ => 1,
AB => 1, # X=Y never occurs
BA => 1, # X=Y never occurs
AC => 2, # C>=A+2 so abs(X-Y)>=2
BC => 1,
SM => 1, # X=Y never occurs
SC => 2, # X<=Y-2
MC => 1, # X=Y never occurs
);
sub absdiffxy_minimum {
my ($self) = @_;
return $absdiffxy_minimum{$self->{'coordinates'}};
}
}
use constant gcdxy_maximum => 1; # no common factor
{
my %absdx_minimum = ('AB,UAD' => 2,
'AB,FB' => 2,
'AB,UMT' => 2,
'AC,UAD' => 2,
'AC,FB' => 2,
'AC,UMT' => 2,
'BC,UAD' => 4, # at N=37
'BC,FB' => 4, # at N=2 X=12,Y=13
'BC,UMT' => 4, # at N=2 X=12,Y=13
'PQ,UAD' => 0,
'PQ,FB' => 0,
'PQ,UMT' => 0,
'SM,UAD' => 1,
'SM,FB' => 1,
'SM,UMT' => 2,
'SC,UAD' => 1,
'SC,FB' => 1,
'SC,UMT' => 1,
'MC,UAD' => 3,
'MC,FB' => 3,
'MC,UMT' => 1,
);
sub absdx_minimum {
my ($self) = @_;
return $absdx_minimum{"$self->{'coordinates'},$self->{'tree_type'}"} || 0;
}
}
{
my %absdy_minimum = ('AB,UAD' => 4,
'AB,FB' => 4,
'AB,UMT' => 4,
'AC,UAD' => 4,
'AC,FB' => 4,
'BC,UAD' => 4,
'BC,FB' => 4,
'PQ,UAD' => 0,
'PQ,FB' => 1,
'SM,UAD' => 3,
'SM,FB' => 3,
'SM,UMT' => 1,
'SC,UAD' => 4,
'SC,FB' => 4,
'MC,UAD' => 4,
'MC,FB' => 4,
);
sub absdy_minimum {
my ($self) = @_;
return $absdy_minimum{"$self->{'coordinates'},$self->{'tree_type'}"} || 0;
}
}
{
my %dir_minimum_dxdy = (# AB apparent minimum dX=16,dY=8
'AB,UAD' => [16,8],
'AC,UAD' => [1,1], # it seems
# 'BC,UAD' => [1,0], # infimum
# 'SM,UAD' => [1,0], # infimum
# 'SC,UAD' => [1,0], # N=255 dX=7,dY=0
# 'MC,UAD' => [1,0], # infimum
# 'SM,FB' => [1,0], # infimum
# 'SC,FB' => [1,0], # infimum
# 'SM,FB' => [1,0], # infimum
'AB,UMT' => [6,12], # it seems
# N=ternary 1111111122 dx=118,dy=40
# in general dx=3*4k-2 dy=4k
'AC,UMT' => [3,1], # infimum
#
# 'BC,UMT' => [1,0], # N=31 dX=72,dY=0
'PQ,UMT' => [1,1], # N=1
'SM,UMT' => [1,0], # infiumum dX=big,dY=3
'SC,UMT' => [3,1], # like AC
# 'MC,UMT' => [1,0], # at N=31
);
sub dir_minimum_dxdy {
my ($self) = @_;
return @{$dir_minimum_dxdy{"$self->{'coordinates'},$self->{'tree_type'}"}
|| [1,0] };
}
}
{
# AB apparent maximum dX=-6,dY=-12 at N=3
# AC apparent maximum dX=-6,dY=-12 at N=3 same
# PQ apparent maximum dX=-1,dY=-1
my %dir_maximum_dxdy = ('AB,UAD' => [-6,-12],
'AC,UAD' => [-6,-12],
# 'BC,UAD' => [0,0],
'PQ,UAD' => [-1,-1],
# 'SM,UAD' => [0,0], # supremum
# 'SC,UAD' => [0,0], # supremum
# 'MC,UAD' => [0,0], # supremum
# 'AB,FB' => [0,0],
# 'AC,FB' => [0,0],
'BC,FB' => [1,-1],
# 'PQ,FB' => [0,0],
# 'SM,FB' => [0,0], # supremum
# 'SC,FB' => [0,0], # supremum
# 'MC,FB' => [0,0], # supremum
# N=ternary 1111111122 dx=118,dy=-40
# in general dx=3*4k-2 dy=-4k
'AB,UMT' => [3,-1], # supremum
#
'AC,UMT' => [-10,-20], # at N=9 apparent maximum
# 'BC,UMT' => [0,0], # apparent approach
'PQ,UMT' => [1,-1], # N=2
# 'SM,UMT' => [0,0], # supremum dX=big,dY=-1
'SC,UMT' => [-3,-5], # apparent approach
# 'MC,UMT' => [0,0], # supremum dX=big,dY=-small
);
sub dir_maximum_dxdy {
my ($self) = @_;
return @{$dir_maximum_dxdy{"$self->{'coordinates'},$self->{'tree_type'}"}
|| [0,0]};
}
}
#------------------------------------------------------------------------------
sub _noop {
return @_;
}
my %xy_to_pq = (AB => \&_ab_to_pq,
AC => \&_ac_to_pq,
BC => \&_bc_to_pqa, # ignoring extra $a return
PQ => \&_noop,
SM => \&_sm_to_pq,
SC => \&_sc_to_pq,
MC => \&_mc_to_pq,
UV => \&_uv_to_pq,
RS => \&_rs_to_pq,
ST => \&_st_to_pq,
);
my %pq_to_xy = (AB => \&_pq_to_ab,
AC => \&_pq_to_ac,
BC => \&_pq_to_bc,
PQ => \&_noop,
SM => \&_pq_to_sm,
SC => \&_pq_to_sc,
MC => \&_pq_to_mc,
UV => \&_pq_to_uv,
RS => \&_pq_to_rs,
ST => \&_pq_to_st,
);
my %tree_types = (UAD => 1,
UArD => 1,
FB => 1,
UMT => 1);
my %digit_orders = (HtoL => 1,
LtoH => 1);
sub new {
my $self = shift->SUPER::new (@_);
{
my $digit_order = ($self->{'digit_order'} ||= 'HtoL');
$digit_orders{$digit_order}
|| croak "Unrecognised digit_order option: ",$digit_order;
}
{
my $tree_type = ($self->{'tree_type'} ||= 'UAD');
$tree_types{$tree_type}
|| croak "Unrecognised tree_type option: ",$tree_type;
}
{
my $coordinates = ($self->{'coordinates'} ||= 'AB');
$self->{'xy_to_pq'} = $xy_to_pq{$coordinates}
|| croak "Unrecognised coordinates option: ",$coordinates;
$self->{'pq_to_xy'} = $pq_to_xy{$coordinates};
}
return $self;
}
sub n_to_xy {
my ($self, $n) = @_;
### PythagoreanTree n_to_xy(): $n
if ($n < 1) { return; }
if (is_infinite($n)) { return ($n,$n); }
{
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 + $y1);
}
}
return &{$self->{'pq_to_xy'}}(_n_to_pq($self,$n));
}
# maybe similar n_to_rsquared() as C^2=(P^2+Q^2)^2
sub n_to_radius {
my ($self, $n) = @_;
if (($self->{'coordinates'} eq 'AB'
|| $self->{'coordinates'} eq 'BA'
|| $self->{'coordinates'} eq 'SM')
&& $n == int($n)) {
if ($n < 1) { return undef; }
if (is_infinite($n)) { return $n; }
my ($p,$q) = _n_to_pq($self,$n);
return $p*$p + $q*$q; # C=P^2+Q^2
}
return $self->SUPER::n_to_radius($n);
}
sub _n_to_pq {
my ($self, $n) = @_;
my $ndigits = _n_to_digits_lowtohigh($n);
### $ndigits
if ($self->{'tree_type'} eq 'UArD') {
Math::PlanePath::GrayCode::_digits_to_gray_reflected($ndigits,3);
### gray: $ndigits
}
if ($self->{'digit_order'} eq 'HtoL') {
@$ndigits = reverse @$ndigits;
### reverse: $ndigits
}
my $zero = $n * 0;
my $p = 2 + $zero;
my $q = 1 + $zero;
if ($self->{'tree_type'} eq 'FB') {
### FB ...
foreach my $digit (@$ndigits) { # high to low, possibly $digit=undef
### $p
### $q
### $digit
if ($digit) {
if ($digit == 1) {
$q = $p-$q; # (2p, p-q) M2
$p *= 2;
} else {
# ($p,$q) = (2*$p, $p+$q);
$q += $p; # (p+q, 2q) M3
$p *= 2;
}
} else { # $digit == 0
# ($p,$q) = ($p+$q, 2*$q);
$p += $q; # (p+q, 2q) M1
$q *= 2;
}
}
} elsif ($self->{'tree_type'} eq 'UMT') {
### UMT ...
foreach my $digit (@$ndigits) { # high to low, possibly $digit=undef
### $p
### $q
### $digit
if ($digit) {
if ($digit == 1) {
$q = $p-$q; # (2p, p-q) M2
$p *= 2;
} else { # $digit == 2
$p += 3*$q; # T
$q *= 2;
}
} else { # $digit == 0
# ($p,$q) = ($p+$q, 2*$q);
($p,$q) = (2*$p-$q, $p); # "U" = (2p-q, p)
}
}
} else {
### UAD or UArD ...
### assert: $self->{'tree_type'} eq 'UAD' || $self->{'tree_type'} eq 'UArD'
# # Could optimize high zeros as repeated U
# # high zeros as repeated U: $depth-scalar(@$ndigits)
# # U^0 = p, q
# # U^1 = 2p-q, p eg. P=2,Q=1 is 2*2-1,2 = 3,2
# # U^2 = 3p-2q, 2p-q eg. P=2,Q=1 is 3*2-2*1,2*2-1 = 4,3
# # U^3 = 4p-3q, 3p-2q
# # U^k = (k+1)p-kq, kp-(k-1)q for k>=2
# # = p + k*(p-q), k*(p-q)+q
# # and with initial p=2,q=1
# # U^k = 2+k, 1+k
# #
# $q = $depth - $#ndigits + $zero; # count high zeros + 1
# $p = $q + 1 + $zero;
foreach my $digit (@$ndigits) { # high to low, possibly $digit=undef
### $p
### $q
### $digit
if ($digit) {
if ($digit == 1) {
($p,$q) = (2*$p+$q, $p); # "A" = (2p+q, p)
} else {
$p += 2*$q; # "D" = (p+2q, q)
}
} else { # $digit==0
($p,$q) = (2*$p-$q, $p); # "U" = (2p-q, p)
}
}
}
### final pq: "$p, $q"
return ($p, $q);
}
# _n_to_digits_lowtohigh() returns an arrayref $ndigits which is a list of
# ternary digits 0,1,2 from low to high which are the position of $n within
# its row of the tree.
# The length of the array is the depth.
#
# depth N N%3 2*N-1 (N-2)/3*2+1
# 0 1 1 1 1/3
# 1 2 2 3 1
# 2 5 2 9 3
# 3 14 2 27 9
# 4 41 2 81 27 28 + (28/2-1) = 41
#
# (N-2)/3*2+1 rounded down to pow=3^k gives depth=k+1 and base=pow+(pow+1)/2
# is the start of the row base=1,2,5,14,41 etc.
#
# An easier calculation is 2*N-1 rounded down to pow=3^d gives depth=d and
# base=2*pow-1, but 2*N-1 and 2*pow-1 might overflow an integer. Though
# just yet round_down_pow() goes into floats and so doesn't preserve 64-bit
# integer. So the technique here helps 53-bit float integers, but not right
# up to 64-bits.
#
sub _n_to_digits_lowtohigh {
my ($n) = @_;
### _n_to_digits_lowtohigh(): $n
my @ndigits;
if ($n >= 2) {
my ($pow) = _divrem($n-2, 3);
($pow, my $depth) = round_down_pow (2*$pow+1, 3);
### $depth
### base: $pow + ($pow+1)/2
### offset: $n - $pow - ($pow+1)/2
@ndigits = digit_split_lowtohigh ($n - $pow - ($pow+1)/2, 3);
push @ndigits, (0) x ($depth - $#ndigits); # pad to $depth with 0s
}
### @ndigits
return \@ndigits;
# {
# my ($pow, $depth) = round_down_pow (2*$n-1, 3);
#
# ### h: 2*$n-1
# ### $depth
# ### $pow
# ### base: ($pow + 1)/2
# ### rem n: $n - ($pow + 1)/2
#
# my @ndigits = digit_split_lowtohigh ($n - ($pow+1)/2, 3);
# $#ndigits = $depth-1; # pad to $depth with undefs
# ### @ndigits
#
# return \@ndigits;
# }
}
#------------------------------------------------------------------------------
# xy_to_n()
# Nrow(depth+1) - Nrow(depth)
# = (3*pow+1)/2 - (pow+1)/2
# = (3*pow + 1 - pow - 1)/2
# = (2*pow)/2
# = pow
#
sub xy_to_n {
my ($self, $x, $y) = @_;
$x = round_nearest ($x);
$y = round_nearest ($y);
### PythagoreanTree xy_to_n(): "$x, $y"
my ($p,$q) = &{$self->{'xy_to_pq'}}($x,$y)
or return undef; # not a primitive A,B,C
unless ($p >= 2 && $q >= 1) { # must be P > Q >= 1
return undef;
}
if (is_infinite($p)) {
return $p; # infinity
}
if (is_infinite($q)) {
return $q; # infinity
}
if ($p%2 == $q%2) { # must be opposite parity, not same parity
return undef;
}
my @ndigits; # low to high
if ($self->{'tree_type'} eq 'FB') {
for (;;) {
unless ($p > $q && $q >= 1) {
return undef;
}
last if $q <= 1 && $p <= 2;
if ($q % 2) {
### q odd, p even, digit 1 or 2 ...
$p /= 2;
if ($q > $p) {
### digit 2, M3 ...
push @ndigits, 2;
$q -= $p; # opp parity of p, and < new p
} else {
### digit 1, M2 ...
push @ndigits, 1;
$q = $p - $q; # opp parity of p, and < p
}
} else {
### q even, p odd, digit 0, M1 ...
push @ndigits, 0;
$q /= 2;
$p -= $q; # opp parity of q
}
### descend: "$q / $p"
}
} elsif ($self->{'tree_type'} eq 'UMT') {
for (;;) {
### at: "p=$p q=$q"
my $qmod2 = $q % 2;
unless ($p > $q && $q >= 1) {
return undef;
}
last if $q <= 1 && $p <= 2;
if ($p < 2*$q) {
($p,$q) = ($q, 2*$q-$p); # U
push @ndigits, 0;
} elsif ($qmod2) {
$p /= 2; # M2
$q = $p - $q;
push @ndigits, 1;
} else {
$q /= 2; # T
$p -= 3*$q;
push @ndigits, 2;
}
}
} else {
### UAD or UArD ...
### assert: $self->{'tree_type'} eq 'UAD' || $self->{'tree_type'} eq 'UArD'
for (;;) {
### $p
### $q
if ($q <= 0 || $p <= 0 || $p <= $q) {
return undef;
}
last if $q <= 1 && $p <= 2;
if ($p > 2*$q) {
if ($p > 3*$q) {
### digit 2 ...
push @ndigits, 2;
$p -= 2*$q;
} else {
### digit 1
push @ndigits, 1;
($p,$q) = ($q, $p - 2*$q);
}
} else {
### digit 0 ...
push @ndigits, 0;
($p,$q) = ($q, 2*$q-$p);
}
### descend: "$q / $p"
}
}
### @ndigits
if ($self->{'digit_order'} eq 'LtoH') {
@ndigits = reverse @ndigits;
### unreverse: @ndigits
}
if ($self->{'tree_type'} eq 'UArD') {
Math::PlanePath::GrayCode::_digits_from_gray_reflected(\@ndigits,3);
### ungray: @ndigits
}
my $zero = $x*0*$y;
### offset: digit_join_lowtohigh(\@ndigits,3,$zero)
### depth: scalar(@ndigits)
### Nrow: $self->tree_depth_to_n($zero + scalar(@ndigits))
return ($self->tree_depth_to_n($zero + scalar(@ndigits))
+ digit_join_lowtohigh(\@ndigits,3,$zero)); # offset into row
}
# numprims(H) = how many with hypot < H
# limit H->inf numprims(H) / H -> 1/2pi
#
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
### PythagoreanTree rect_to_n_range(): "$x1,$y1 $x2,$y2"
$x1 = round_nearest ($x1);
$y1 = round_nearest ($y1);
$x2 = round_nearest ($x2);
$y2 = round_nearest ($y2);
my $zero = ($x1 * 0 * $y1 * $x2 * $y2); # inherit bignum
($x1,$x2) = ($x2,$x1) if $x1 > $x2;
($y1,$y2) = ($y2,$y1) if $y1 > $y2;
### x2: "$x2"
### y2: "$y2"
if ($self->{'coordinates'} eq 'BA') {
($x2,$y2) = ($y2,$x2);
}
if ($self->{'coordinates'} eq 'SM') {
if ($x2 > $y2) { # both max
$y2 = $x2;
} else {
$x2 = $y2;
}
}
if ($self->{'coordinates'} eq 'PQ') {
if ($x2 < 2 || $y2 < 1) {
return (1,0);
}
# P > Q so reduce y2 to at most x2-1
if ($y2 >= $x2) {
$y2 = $x2-1; # $y2 = min ($y2, $x2-1);
}
if ($y2 < $y1) {
### PQ y range all above X=Y diagonal ...
return (1,0);
}
} else {
# AB,AC,BC, SM,SC,MC
if ($x2 < 3 || $y2 < 0) {
return (1,0);
}
}
my $depth;
if ($self->{'tree_type'} eq 'FB') {
### FB ...
if ($self->{'coordinates'} eq 'PQ') {
$x2 *= 3;
}
my ($pow, $exp) = round_down_pow ($x2, 2);
$depth = 2*$exp;
} else {
### UAD or UArD, and UMT ...
if ($self->{'coordinates'} eq 'PQ') {
### PQ ...
# P=k+1,Q=k diagonal N=100..000 first of row is depth=P-2
# anything else in that X=P column is smaller depth
$depth = $x2 - 2;
} else {
my $xdepth = int (($x2+1) / 2);
my $ydepth = int (($y2+31) / 4);
$depth = min($xdepth,$ydepth);
}
}
### depth: "$depth"
return (1, $self->tree_depth_to_n_end($zero+$depth));
}
#------------------------------------------------------------------------------
use constant tree_num_roots => 1;
sub tree_n_children {
my ($self, $n) = @_;
unless ($n >= 1) {
return;
}
$n *= 3;
return ($n-1, $n, $n+1);
}
sub tree_n_num_children {
my ($self, $n) = @_;
return ($n >= 1 ? 3 : undef);
}
sub tree_n_parent {
my ($self, $n) = @_;
unless ($n >= 2) {
return undef;
}
return int(($n+1)/3);
}
sub tree_n_to_depth {
my ($self, $n) = @_;
### PythagoreanTree tree_n_to_depth(): $n
unless ($n >= 1) {
return undef;
}
my ($pow, $depth) = round_down_pow (2*$n-1, 3);
return $depth;
}
sub tree_depth_to_n {
my ($self, $depth) = @_;
return ($depth >= 0
? (3**$depth + 1)/2
: undef);
}
# (3^(d+1)+1)/2-1 = (3^(d+1)-1)/2
sub tree_depth_to_n_end {
my ($self, $depth) = @_;
return ($depth >= 0
? (3**($depth+1) - 1)/2
: undef);
}
sub tree_depth_to_n_range {
my ($self, $depth) = @_;
if ($depth >= 0) {
my $n_lo = (3**$depth + 1) / 2; # same as tree_depth_to_n()
return ($n_lo, 3*$n_lo-2);
} else {
return;
}
}
sub tree_depth_to_width {
my ($self, $depth) = @_;
return ($depth >= 0
? 3**$depth
: undef);
}
#------------------------------------------------------------------------------
# Maybe, or abc_to_pq() perhaps with two of three values.
#
# @EXPORT_OK = ('ab_to_pq','pq_to_ab');
#
# =item C<($p,$q) = Math::PlanePath::PythagoreanTree::ab_to_pq($a,$b)>
#
# Return the P,Q coordinates for C<$a,$b>. As described above this is
#
# P = sqrt((C+A)/2) where C=sqrt(A^2+B^2)
# Q = sqrt((C-A)/2)
#
# The returned P,Q are integers PE<gt>=0,QE<gt>=0, but the further
# conditions for the path (namely PE<gt>QE<gt>=1 and no common factor) are
# not enforced.
#
# If P,Q are not integers or if BE<lt>0 then return an empty list. This
# ensures A,B is a Pythagorean triple, ie. that C=sqrt(A^2+B^2) is an
# integer, but it might not be a primitive triple and might not have A odd B
# even.
#
# =item C<($a,$b) = Math::PlanePath::PythagoreanTree::pq_to_ab($p,$q)>
#
# Return the A,B coordinates for C<$p,$q>. This is simply
#
# $a = $p*$p - $q*$q
# $b = 2*$p*$q
#
# This is intended for use with C<$p,$q> satisfying PE<gt>QE<gt>=1 and no
# common factor, but that's not enforced.
# a=p^2-q^2, b=2pq, c=p^2+q^2
# Done as a=(p-q)*(p+q) for one multiply instead of two squares, and to work
# close to a=UINT_MAX.
#
sub _pq_to_ab {
my ($p, $q) = @_;
return (($p-$q)*($p+$q), 2*$p*$q);
}
# C=(p-q)^2+B for one squaring instead of two.
# Also possible is C=(p+q)^2-B, but prefer "+B" so as not to round-off in
# floating point if (p+q)^2 overflows an integer.
sub _pq_to_bc {
my ($p, $q) = @_;
my $b = 2*$p*$q;
$p -= $q;
return ($b, $p*$p+$b);
}
# a=p^2-q^2, b=2pq, c=p^2+q^2
# Could a=(p-q)*(p+q) to avoid overflow if p^2 exceeds an integer as per
# _pq_to_ab(), but c overflows in that case anyway.
sub _pq_to_ac {
my ($p, $q) = @_;
$p *= $p;
$q *= $q;
return ($p-$q, $p+$q);
}
# a=p^2-q^2, b=2pq, c=p^2+q^2
# a<b
# p^2-q^2 < 2pq
# p^2 + 2pq - q^2 < 0
# (p+q)^2 - 2*q^2 < 0
# (p+q + sqrt(2)*q)*(p+q - sqrt(2)*q) < 0
# (p+q - sqrt(2)*q) < 0
# p + (1-sqrt(2))*q < 0
# p < (sqrt(2)-1)*q
#
sub _pq_to_sc {
my ($p, $q) = @_;
my $b = 2*$p*$q;
my $p_plus_q = $p + $q;
$p -= $q;
return (min($p_plus_q*$p, $b), # A = P^2-Q^2 = (P+Q)*(P-Q)
$p*$p+$b); # C = P^2+Q^2 = (P-Q)^2 + 2*P*Q
}
sub _pq_to_mc {
my ($p, $q) = @_;
my $b = 2*$p*$q;
my $p_plus_q = $p + $q;
$p -= $q;
return (max($p_plus_q*$p, $b), # A = P^2-Q^2 = (P+Q)*(P-Q)
$p*$p+$b); # C = P^2+Q^2 = (P-Q)^2 + 2*P*Q
}
sub _pq_to_sm {
my ($p, $q) = @_;
my ($a, $b) = _pq_to_ab($p,$q);
return ($a < $b ? ($a, $b) : ($b, $a));
}
# u = p+q, v=p-q
# at given p, vertical q
# u=p,v=p on diagonal then p+q,p-q is diagonal down
# so mirror p axis to x=y diagonal and measure down diagonal from there
sub _pq_to_uv {
my ($p, $q) = @_;
return ($p+$q, $p-$q);
}
# r = b+c = 2pq+p^2+q^2 = (p+q)^2
# s = c-a = p^2+q^2 - (p^2-q^2) = 2*q^2
sub _pq_to_rs {
my ($p, $q) = @_;
return (($p+$q)**2, 2*$q*$q);
}
# s = c-a = p^2+q^2 - (p^2-q^2) = 2*q^2
# t = a+b-c = p^2-q^2 + 2pq - (p^2+q^2) = 2pq-2q^2 = 2(p-q)q
sub _pq_to_st {
my ($p, $q) = @_;
my $q2 = 2*$q;
return ($q2*$q, ($p-$q)*$q2);
}
#------------------------------------------------------------------------------
# a = p^2 - q^2
# b = 2pq
# c = p^2 + q^2
#
# q = b/2p
# a = p^2 - (b/2p)^2
# = p^2 - b^2/4p^2
# 4ap^2 = 4p^4 - b^2
# 4(p^2)^2 - 4a*p^2 - b^2 = 0
# p^2 = [ 4a +/- sqrt(16a^2 + 16*b^2) ] / 2*4
# = [ a +/- sqrt(a^2 + b^2) ] / 2
# = (a +/- c) / 2 where c=sqrt(a^2+b^2)
# p = sqrt((a+c)/2) since c>a
#
# a = (a+c)/2 - q^2
# q^2 = (a+c)/2 - a
# = (c-a)/2
# q = sqrt((c-a)/2)
#
# if c^2 = a^2+b^2 is a perfect square then a,b,c is a pythagorean triple
# p^2 = (a+c)/2
# = (a + sqrt(a^2+b^2))/2
# 2p^2 = a + sqrt(a^2+b^2)
#
# p>q so a>0
# a+c even is a odd, c odd or a even, c even
# if a odd then c=a^2+b^2 is opp of b parity, must have b even to make c+a even
# if a even then c=a^2+b^2 is same as b parity, must have b even to c+a even
#
# a=6,b=8 is c=sqrt(6^2+8^2)=10
# a=0,b=4 is c=sqrt(0+4^4)=4 p^2=(a+c)/2 = 2 not a square
# a+c even, then (a+c)/2 == 0,1 mod 4 so a+c==0,2 mod 4
#
sub _ab_to_pq {
my ($a, $b) = @_;
### _ab_to_pq(): "A=$a, B=$b"
unless ($b >= 4 && ($a%2) && !($b%2)) { # A odd, B even
return;
}
# This used to be $c=hypot($a,$b) and check $c==int($c), but libm hypot()
# on Darwin 8.11.0 is somehow a couple of bits off being an integer, for
# example hypot(57,176)==185 but a couple of bits out so $c!=int($c).
# Would have thought hypot() ought to be exact on integer inputs and a
# perfect square sum :-(. Check for a perfect square by multiplying back
# instead.
#
# The condition is "$csquared != $c*$c" with operands that way around
# since the other way is bad for Math::BigInt::Lite 0.14.
#
my $c;
{
my $csquared = $a*$a + $b*$b;
$c = int(sqrt($csquared));
### $csquared
### $c
# since A odd and B even should have C odd, but floating point rounding
# might prevent that
unless ($csquared == $c*$c) {
### A^2+B^2 not a perfect square ...
return;
}
}
return _ac_to_pq($a,$c);
}
sub _bc_to_pqa {
my ($b, $c) = @_;
### _bc_to_pqa(): "B=$b C=$c"
unless ($c > $b && $b >= 4 && !($b%2) && ($c%2)) { # B even, C odd
return;
}
my $a;
{
my $asquared = $c*$c - $b*$b;
unless ($asquared > 0) {
return;
}
$a = int(sqrt($asquared));
### $asquared
### $a
unless ($asquared == $a*$a) {
return;
}
}
# If $c is near DBL_MAX can have $a overflow to infinity, leaving A>C.
# _ac_to_pq() will detect that.
my ($p,$q) = _ac_to_pq($a,$c) or return;
return ($p,$q,$a);
}
sub _ac_to_pq {
my ($a, $c) = @_;
### _ac_to_pq(): "A=$a C=$c"
unless ($c > $a && $a >= 3 && ($a%2) && ($c%2)) { # A odd, C odd
return;
}
$a = ($a-1)/2;
$c = ($c-1)/2;
### halved to: "a=$a c=$c"
my $p;
{
# If a,b,c is a triple but not primitive then can have psquared not an
# integer. Eg. a=9,b=12 has c=15 giving psquared=(9+15)/2=12 is not a
# perfect square. So notice that here.
#
my $psquared = $c+$a+1;
$p = int(sqrt($psquared));
### $psquared
### $p
unless ($psquared == $p*$p) {
### P^2=A+C not a perfect square ...
return;
}
}
my $q;
{
# If a,b,c is a triple but not primitive then can have qsquared not an
# integer. Eg. a=15,b=36 has c=39 giving qsquared=(39-15)/2=12 is not a
# perfect square. So notice that here.
#
my $qsquared = $c-$a;
$q = int(sqrt($qsquared));
### $qsquared
### $q
unless ($qsquared == $q*$q) {
return;
}
}
# Might have a common factor between P,Q here. Eg.
# A=27 = 3*3*3, B=36 = 4*3*3
# A=45 = 3*3*5, B=108 = 4*3*3*3
# A=63, B=216
# A=75 =3*5*5 B=100 = 4*5*5
# A=81, B=360
#
return ($p, $q);
}
sub _sm_to_pq {
my ($s, $m) = @_;
unless ($s < $m) {
return;
}
return _ab_to_pq($s % 2
? ($s,$m) # s odd is A
: ($m,$s)); # s even is B
}
# s^2+m^2=c^2
# if s odd then a=s
# ac_to_pq
# b = 2pq check isn't smaller than s
#
# p^2=(c+a)/2
# q^2=(c-a)/2
sub _sc_to_pq {
my ($s, $c) = @_;
my ($p,$q);
if ($s % 2) {
($p,$q) = _ac_to_pq($s,$c) # s odd is A
or return;
if ($s > 2*$p*$q) { return; } # if s>B then s is not the smaller one
} else {
($p,$q,$a) = _bc_to_pqa($s,$c) # s even is B
or return;
if ($s > $a) { return; } # if s>A then s is not the smaller one
}
return ($p,$q);
}
sub _mc_to_pq {
my ($m, $c) = @_;
### _mc_to_pq() ...
my ($p,$q);
if ($m % 2) {
### m odd is A ...
($p,$q) = _ac_to_pq($m,$c)
or return;
if ($m < 2*$p*$q) { return; } # if m<B then m is not the bigger one
} else {
### m even is B ...
($p,$q,$a) = _bc_to_pqa($m,$c)
or return;
### $a
if ($m < $a) { return; } # if m<A then m is not the bigger one
}
return ($p,$q);
}
# u = p+q, v=p-q
# u+v=2p p = (u+v)/2
# u-v=2q q = (u-v)/2
sub _uv_to_pq {
my ($u, $v) = @_;
return (($u+$v)/2, ($u-$v)/2);
}
# r = (p+q)^2
# s = 2*q^2 so q = sqrt(r/2)
sub _rs_to_pq {
my ($r, $s) = @_;
return if $s % 2;
$s /= 2;
return unless $s >= 1;
my $q = int(sqrt($s));
return unless $q*$q == $s;
return unless $r >= 1;
my $p_plus_q = int(sqrt($r));
return unless $p_plus_q*$p_plus_q == $r;
return ($p_plus_q - $q, $q);
}
# s = 2*q^2
# t = a+b-c = p^2-q^2 + 2pq - (p^2+q^2) = 2pq-2q^2 = 2(p-q)q
#
# p=2,q=1 s=2 t=2.1.1=2
#
sub _st_to_pq {
my ($s, $t) = @_;
### _st_to_pq(): "$s, $t"
return if $s % 2;
$s /= 2;
return unless $s >= 1;
my $q = int(sqrt($s));
### $q
return unless $q*$q == $s;
return if $t % 2;
$t /= 2;
### rem: $t % $q
return if $t % $q;
$t /= $q; # p-q
### pq: ($t+$q).", $q"
return ($t+$q, $q);
}
1;
__END__
# my $a = 1;
# my $b = 1;
# my $c = 2;
# my $d = 3;
# ### at: "$a,$b,$c,$d digit $digit"
# if ($digit == 0) {
# ($a,$b,$c) = ($a,2*$b,$d);
# } elsif ($digit == 1) {
# ($a,$b,$c) = ($d,$a,2*$c);
# } else {
# ($a,$b,$c) = ($a,$d,2*$c);
# }
# $d = $b+$c;
# ### final: "$a,$b,$c,$d"
# # print "$a,$b,$c,$d\n";
# my $x = $c*$c-$b*$b;
# my $y = 2*$b*$c;
# return (max($x,$y), min($x,$y));
# return $x,$y;
=for stopwords eg Ryde UAD FB Berggren Barning ie PQ parameterized parameterization Math-PlanePath someP someQ Q's coprime mixed-radix Nrow N-Nrow Liber Quadratorum gnomon gnomons Diophantus Nrem OEIS UArD mirrorings Firstov Semigroup Matematicheskie Zametki semigroup UMT LtoH
=head1 NAME
Math::PlanePath::PythagoreanTree -- primitive Pythagorean triples by tree
=head1 SYNOPSIS
use Math::PlanePath::PythagoreanTree;
my $path = Math::PlanePath::PythagoreanTree->new
(tree_type => 'UAD',
coordinates => 'AB');
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This path enumerates primitive Pythagorean triples by a breadth-first
traversal of one of three ternary trees,
"UAD" Berggren, Barning, Hall, et al
"FB" Price
"UMT" a third form
Each X,Y point is a pair of integer A,B sides of a right triangle. The
points are "primitive" in the sense that the sense that A and B have no
common factor.
A^2 + B^2 = C^2 gcd(A,B)=1, no common factor
X=A, Y=B
^ * ^
/ /| | right triangle
C / | B A side odd
/ / | | B side even
v *---* v C hypotenuse all integers
<-A->
A primitive triple always has one of A,B odd and the other even. The trees
here have triples ordered as A odd and B even.
The trees are traversed breadth-first and tend to go out to rather large A,B
values while yet to complete smaller ones. The UAD tree goes out further
than the FB. See the author's mathematical write-up for more properties.
=over
L<http://user42.tuxfamily.org/triples/index.html>
=back
=head2 UAD Tree
The UAD tree by Berggren (1934) and later independently by Barning (1963),
Hall (1970), and other authors, uses three matrices U, A and D which can be
multiplied onto an existing primitive triple to form three further new
primitive triples.
tree_type => "UAD" (the default)
Y=40 | 14
|
|
|
| 7
Y=24 | 5
|
Y=20 | 3
|
Y=12 | 2 13
|
| 4
Y=4 | 1
|
+--------------------------------------------------
X=3 X=15 X=20 X=35 X=45
The UAD matrices are
1 -2 2 1 2 2 -1 2 2
U = 2 -1 2 A = 2 1 2 D = -2 1 2
2 -2 3 2 2 3 -2 2 3
They're multiplied on the right of an (A,B,C) column vector, for example
/ 3 \ / 5 \
U * | 4 | = | 12 |
\ 5 / \ 13 /
The starting point is N=1 at X=3,Y=4 which is the well-known triple
3^2 + 4^2 = 5^2
From it three further points N=2, N=3 and N=4 are derived, then three more
from each of those, etc,
=cut
# printed by tools/pythagorean-tree.pl
=pod
depth=0 depth=1 depth=2 depth=3
N=1 N=2..4 N=5..13 N=14...
+-> 7,24 A,B coordinates
+-> 5,12 --+-> 55,48
| +-> 45,28
|
| +-> 39,80
3,4 --+-> 21,20 --+-> 119,120
| +-> 77,36
|
| +-> 33,56
+-> 15,8 --+-> 65,72
+-> 35,12
Counting N=1 as depth=0, each level has 3^depth many points and the first N
of a level (C<tree_depth_to_n()>) is at
Nrow = 1 + (1 + 3 + 3^2 + ... + 3^(depth-1))
= (3^depth + 1) / 2
The level numbering is like a mixed-radix representation of N where the high
digit is binary (so always 1) and the digits below are ternary.
+--------+---------+---------+-- --+---------+
N = | binary | ternary | ternary | ... | ternary |
+--------+---------+---------+-- --+---------+
1 0,1,2 0,1,2 0,1,2
The number of ternary digits is the "depth" and their value without the high
binary 1 is the position in the row.
=head2 A Repeatedly
Taking the middle "A" matrix repeatedly gives
3,4 -> 21,20 -> 119,120 -> 697,696 -> etc
which are the triples with legs A,B differing by 1 and therefore just above
or below the X=Y leading diagonal. The N values are 1,3,9,27,etc = 3^depth.
=cut
# FIXME: were these known to Fermat?
# PQ coordinates A000129 Pell numbers
=pod
=head2 D Repeatedly
Taking the lower "D" matrix repeatedly gives
3,4 -> 15,8 -> 35,12 -> 63,16 -> etc
which is the primitives among a sequence of triples known to the ancients
(Dickson's I<History of the Theory of Numbers>, start of chapter IV),
A = k^2-1 k even for primitives
B = 2*k
C = k^2+1 so C=A+2
When k is even these are primitive. If k is odd then A and B are both even,
ie. a common factor of 2, so not primitive. These points are the last of
each level, so at N=(3^(depth+1)-1)/2 which is C<tree_depth_to_n_end()>.
=head2 U Repeatedly
Taking the upper "U" matrix repeatedly gives
3.4 -> 5,12 -> 7,24 -> 9,40 -> etc
with C=B+1 and A the odd numbers. These are the first of each level so at
Nrow described above. The resulting triples are a sequence known to
Pythagoras (Dickson's I<History of the Theory of Numbers>, start of chapter
IV).
A = any odd integer, so A^2 any odd square
B = (A^2-1)/2
C = (A^2+1)/2
/ A^2-1 \ / A^2+1 \
A^2 + | ------ |^2 = | ----- |^2
\ 2 / \ 2 /
This is also described by X<Fibonacci>Fibonacci in his
X<Liber Quadratorum>I<Liber Quadratorum> (X<Book of Squares>I<Book of
Squares>) in terms of sums of odd numbers
s = any odd square = A^2
B^2 = 1 + 3 + 5 + ... + s-2 = ((s-1)/2)^2
C^2 = 1 + 3 + 5 + ... + s-2 + s = ((s+1)/2)^2
so C^2 = A^2 + B^2
eg. s=25=A^2 B^2=((25-1)/2)^2=144 so A=5,B=12
X<Gnomon>The geometric interpretation is that an existing square of side B
is extended by a X<Gnomon>"gnomon" around two sides making a new larger
square of side C=B+1. The length of the gnomon is odd and when it's an odd
square then the new total area is the sum of two squares.
*****gnomon******* gnomon length an odd square = A^2
+--------------+ *
| | * so new bigger square area
| square | * C^2 = A^2 + B^2
| with side B | *
| | *
+--------------+ *
See L<Math::PlanePath::Corner> for a path following such gnomons.
=head2 UArD Tree
X<Gray code>Option C<tree_type =E<gt> "UArD"> varies the UAD tree by
applying a left-right reflection under each "A" matrix. The result is
ternary reflected Gray code order. The 3 children under each node are
unchanged, just their order.
+-> 7,24 tree_type => "UArD"
+-> 5,12 --+-> 55,48 A,B coordinates
| +-> 45,28
|
| +-> 77,36 <-+- U,D legs swapped
3,4 --+-> 21,20 --+-> 119,120 |
| +-> 39,80 <-+
|
| +-> 33,56
+-> 15,8 --+-> 65,72
+-> 35,12
Notice the middle points 77,36 and 39,80 are swapped relative to the UAD
shown above. In general the whole tree underneath an "A" is mirrored. If
there's an even number of "A"s above then those mirrorings cancel out to be
plain again.
This tree form is primarily of interest for L</Digit Order Low to High>
described below since it gives points in order clockwise down from the Y
axis.
In L</PQ Coordinates> below, with the default digits high to low, UArD also
makes successive steps across the row either horizontal or 45-degrees NE-SW.
In all cases the Gray coding is applied to N first, then the resulting
digits are interpreted either high to low (the default) or low to high
(C<LtoH> option).
=head2 FB Tree
X<Firstov, V. E.>X<Price, H. Lee>Option C<tree_type =E<gt> "FB"> selects a
tree independently by
=over
V. E. Firstov, "A Special Matrix Transformation
Semigroup of Primitive Pairs and the Genealogy of Pythagorean Triples",
Matematicheskie Zametki, 2008, volume 84, number 2, pages 281-299 (in
Russian), and Mathematical Notes, 2008, volume 84, number 2, pages 263-279
(in English)
H. Lee Price, "The Pythagorean Tree: A New Species", 2008,
L<http://arxiv.org/abs/0809.4324> (version 2)
=back
Firstov finds this tree by semigroup transformations. Price finds it by
expressing triples in certain "Fibonacci boxes" with a box of four values
q',q,p,p' having p=q+q' and p'=p+q so each is the sum of the preceding two
in a fashion similar to the Fibonacci sequence. A box where p and q have no
common factor corresponds to a primitive triple. See L</PQ Coordinates> and
L</FB Transformations> below.
tree_type => "FB"
Y=40 | 5
|
|
|
| 17
Y=24 | 4
|
| 8
|
Y=12 | 2 6
|
| 3
Y=4 | 1
|
+----------------------------------------------
X=3 X=15 x=21 X=35
For a given box three transformations can be applied to go to new boxes
corresponding to new primitive triples. This visits all and only primitive
triples, but in a different order to the UAD above.
The first point N=1 is again at X=3,Y=4, from which three further points
N=2,3,4 are derived, then three more from each of those, etc.
=cut
# printed by tools/pythagorean-tree.pl
=pod
N=1 N=2..4 N=5..13 N=14...
+-> 9,40 A,B coordinates
+-> 5,12 --+-> 35,12
| +-> 11,60
|
| +-> 21,20
3,4 --+-> 15,8 --+-> 55,48
| +-> 39,80
|
| +-> 13,84
+-> 7,24 --+-> 63,16
+-> 15,112
=head2 UMT Tree
X<Firstov, V. E.>Option C<tree_type => "UMT"> is a third tree type by
Firstov (reference above). It's a combination of "U", "M2" and a third
matrix T = M1*D.
=cut
# printed by tools/pythagorean-tree.pl
=pod
U +-> 7,24 A,B coordinates
+-> 5,12 --+-> 35,12
| +-> 65,72
|
| M2 +-> 33,56
3,4 --+-> 15,8 --+-> 55,48
| +-> 45,28
|
| T +-> 39,80
+-> 21,20 --+-> 91,60
+-> 105,88
The first "T" child 21,20 is the same as the "A" matrix, but it differs at
further levels down. For example "T" twice is 105,88 which is not the same
as "A" twice 119,120.
=head2 Digit Order Low to High
Option C<digit_order =E<gt> 'LtoH'> applies matrices using the ternary
digits of N taken from low to high. The points in each row are unchanged,
as is the parent-child N numbering, but the X,Y values are rearranged within
the row.
The UAD matrices send points to disjoint regions and the effect of LtoH is
to keep the tree growing into those separate wedge regions. The arms grow
roughly as follows
=cut
# math-image --path=PythagoreanTree,digit_order=LtoH --all --output=numbers_xy --size=75x14
=pod
tree_type => "UAD", digit_order => "LtoH"
Y=80 | 6 UAD LtoH
| /
| /
Y=56 | / 7 10 9
| / / / /
| / / | / 8
| / _/ / / /
| / / / / /
Y=24 | 5 / / | / _/ __--11
| / / _/ |/_/ __--
Y=20 | / / / __3 __-- _____----12
| |/_/ __-- __--- ____-----
Y=12 | 2 __-- _/___---- ____13
| / __-- __-- _____-----
| /_--_____---4-----
Y=4 | 1---
|
+--------------------------------------------------
X=3 X=15 X=20 X=35 X=76
Notice the points of the second row N=5 to N=13 are almost clockwise down
from the Y axis, except N=8,9,10 go upwards. Those N=8,9,10 go upwards
because the A matrix has a reflection (its determinant is -1).
Option C<tree_type =E<gt> "UArD"> reverses the tree underneath each A, and
that plus LtoH gives A,B points going clockwise in each row. P,Q
coordinates go clockwise too.
=head2 AC Coordinates
Option C<coordinates =E<gt> 'AC'> gives the A and C legs of each triple as
X=A,Y=C.
coordinates => "AC"
85 | 122 10
|
|
73 | 6
|
65 | 11 40
61 | 41
|
| 7
|
|
41 | 14
| 13
35 |
| 3
25 | 5
|
17 | 4
13 | 2
|
Y=5 | 1
|
+-------------------------------------------
X=3 7 9 21 35 45 55 63 77
Since AE<lt>C the coordinates are XE<lt>Y all above the X=Y diagonal. The
L</D Repeatedly> triples described above have C=A+2 so they are the points
Y=X+2 just above the diagonal.
For the FB tree the set of points visited is the same, but with a different
N numbering.
tree_type => "FB", coordinates => "AC"
85 | 11 35
|
|
73 | 9
|
65 | 23 12
61 | 7
|
| 17
|
|
41 | 5
| 6
35 |
| 8
25 | 4
|
17 | 3
13 | 2
|
Y=5 | 1
|
+-------------------------------------------
X=3 7 9 21 35 45 55 63 77
=head2 BC Coordinates
Option C<coordinates =E<gt> 'BC'> gives the B and C legs of each triple as
X=B,Y=C. This is the B=even and C=long legs of all primitive triples. This
combination has points on 45-degree straight lines.
coordinates => "BC"
101 | 121
97 | 12
|
89 | 8
85 | 10 122
|
|
73 | 6
|
65 | 40 11
61 | 41
|
| 7
|
|
41 | 14
| 13
35 |
| 3
25 | 5
|
17 | 4
13 | 2
|
Y=5 | 1
|
+--------------------------------------------------
X=4 12 24 40 60 84
Since BE<lt>C the coordinates are XE<lt>Y above the X=Y leading diagonal.
N=1,2,5,14,41,etc along the X=Y-1 diagonal are the L</U Repeatedly> triples
described above which have C=B+1 and are at the start of each tree row.
For the FB tree the set of points visited is the same, but with a different
N numbering.
tree_type => "FB", coordinates => "BC"
101 | 15
97 | 50
|
89 | 10
85 | 35 11
|
|
73 | 9
|
65 | 12 23
61 | 7
|
| 17
|
|
41 | 5
| 6
35 |
| 8
25 | 4
|
17 | 3
13 | 2
|
Y=5 | 1
|
+----------------------------------------------
X=4 12 24 40 60 84
As seen from the diagrams, the B,C points fall on 45-degree straight lines
going up from X=Y-1. This occurs because a primitive triple A,B,C with A
odd and B even can be written
A^2 = C^2 - B^2
= (C+B)*(C-B)
gcd(A,B)=1 means also gcd(C+B,C-B)=1 and so since C+B and C-B have no common
factor they must each be squares to give A^2. Call them s^2 and t^2,
C+B = s^2 and conversely C = (s^2 + t^2)/2
C-B = t^2 B = (s^2 - t^2)/2
s = odd integer s >= 3
t = odd integer s > t >= 1
with gcd(s,t)=1 so that gcd(C+B,C-B)=1
When t=1 this is C=(s^2+1)/2 and B=(s^2-1)/2 which is the "U"-repeated
points at Y=X+1 for each s. As t increases the B,C coordinate combination
makes a line upwards at 45-degrees from those t=1 positions,
C + B = s^2 anti-diagonal 45-degrees,
position along diagonal determined by t
All primitive triples start from a C=B+1 with C=(s^2+1)/2 half an odd
square, and go up from there. To ensure the triple is primitive must have
gcd(s,t)=1. Values of t where gcd(s,t)!=1 are gaps in the anti-diagonal
lines.
=head2 PQ Coordinates
Primitive Pythagorean triples can be parameterized as follows for A odd and
B even. This is per Diophantus, and anonymous Arabic manuscript for
constraining it to primitive triples (Dickson's I<History of the Theory of
Numbers>).
A = P^2 - Q^2
B = 2*P*Q
C = P^2 + Q^2
with P > Q >= 1, one odd, one even, and no common factor
P = sqrt((C+A)/2)
Q = sqrt((C-A)/2)
The first P=2,Q=1 is the triple A=3,B=4,C=5.
Option C<coordinates =E<gt> 'PQ'> gives these as X=P,Y=Q, for either
C<tree_type>. Because PE<gt>QE<gt>=1 the values fall in the eighth of the
plane below the X=Y diagonal,
=cut
# math-image --path=PythagoreanTree,coordinates=PQ --all --output=numbers_xy --size=75x14
=pod
tree_type => "UAD", coordinates => "PQ"
10 | 9842
9 | 3281
8 | 1094 23
7 | 365 32
6 | 122 38
5 | 41 8
4 | 14 11 12 15
3 | 5 6 16
2 | 2 3 7 10 22
1 | 1 4 13 40 121
Y=0 |
+--------------------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11
The diagonal N=1,2,5,14,41,etc is P=Q+1 as per L</U Repeatedly> above.
The one-to-one correspondence between P,Q and A,B means both tree types
visit all P,Q pairs, so all X,Y with no common factor and one odd one even.
There's other ways to iterate through such coprime pairs and any such method
would generate Pythagorean triples too, in a different order from the trees
here.
The letters P and Q here are a little bit arbitrary. This parameterization
is often written m,n or u,v but don't want "n" to be confused that with N
point numbering or "u" to be confused with the U matrix.
=head2 SM Coordinates
Option C<coordinates =E<gt> 'SM'> gives the small and medium legs from each
triple as X=small,Y=medium. This is like "AB" except that if AE<gt>B then
they're swapped to X=B,Y=A so XE<lt>Y always. The effect is to mirror the
AB points below the X=Y diagonal up to the upper eighth,
coordinates => "SM"
91 | 16
84 | 122
| 8
| 10
72 | 12
|
|
60 | 41 40
| 11
55 | 6
|
| 7
40 | 14
|
35 | 13
|
24 | 5
21 | 3
|
12 | 2 4
|
Y=4 | 1
|
+----------------------------------------
X=3 8 20 33 48 60 65
=head2 SC Coordinates
Option C<coordinates =E<gt> 'SC'> gives the small leg and hypotenuse from
each triple,
coordinates => "SC"
85 | 122 10
|
|
73 | 6
|
| 40 11
61 | 41
|
53 | 7
|
|
41 | 14
37 | 13
|
| 3
25 | 5
|
| 4
13 | 2
|
Y=5 | 1
|
+-----------------------------
X=3 8 20 33 48
The points are all X E<lt> 0.7*Y since with X as the smaller leg must have
S<X^2 E<lt> Y^2/2> so S<X E<lt> Y*1/sqrt(2)>.
=head2 MC Coordinates
Option C<coordinates =E<gt> 'MC'> gives the medium leg and hypotenuse from
each triple,
coordinates => "MC"
65 | 11 40
61 | 41
|
53 | 7
|
|
41 | 14
37 | 13
|
29 | 3
25 | 5
|
17 | 4
13 | 2
|
Y=5 | 1
|
+-----------------------------------
X=4 15 24 35 40 56 63
The points are in a wedge 0.7*Y E<lt> X E<lt> Y. X is the bigger leg and
S<X^2 E<gt> Y^2/2> so S<X E<gt> Y*1/sqrt(2)>.
=cut
# if A=B=C/sqrt(2)
# A^2+B^2 = C^2/2+C^2/2 = C^2
# so X=Y/sqrt(2) = Y*0.7071
=pod
=head2 UAD Coordinates AB, AC, PQ -- Turn Right
In the UAD tree with coordinates AB, AC or PQ the path always turns to the
right. For example in AB coordinates at N=2 the path turns to the right to
go towards N=3.
coordinates => "AB"
20 | 3 N X,Y
| -- ------
12 | 2 1 3,4
| 2 5,12
| 3 21,20
4 | 1
| turn towards the
+------------------------- right at N=2
3 5 21
This can be proved from the transformations applied to seven cases, a
triplet U,A,D, then four crossing a gap within a level, then two wrapping
around at the end of a level. The initial N=1,2,3 can be treated as a
wrap-around from the end of depth=0 (the last case D to U,A).
U triplet U,A,D
A
D
U.D^k.A crossing A,D to U
U.D^k.D across U->A gap
A.U^k.U k>=0
A.D^k.A crossing A,D to U
A.D^k.D across A->D gap
D.U^k.U k>=0
U.D^k.D crossing D to U,A
U.U^k.U across U->A gap
A.U^k.A k>=0
A.D^k.D crossing D to U,A
A.U^k.U across A->D gap
D.U^k.A k>=0
D^k .A wraparound A,D to U
D^k .D k>=0
U^(k+1).U
D^k wraparound D to U,A
U^k.U k>=0
U^k.A (k=0 is initial N=1,N=2,N=3 for none,U,A)
The powers U^k and D^k are an arbitrary number of descents U or D. In P,Q
coordinates these powers are
U^k P,Q -> (k+1)*P-k*Q, k*P-(k-1)*Q
D^k P,Q -> P+2k*Q, Q
For AC coordinates squaring to stretch to P^2,Q^2 doesn't change the turns.
Then a rotate by -45 degrees to A=P^2-Q^2, C=P^2+Q^2 also doesn't change the
turns.
=head2 UAD Coordinates BC -- Turn Left
In the UAD tree with coordinates BC the path always turns to the left. For
example in BC coordinates at N=2 the path turns to the right to go towards
N=3.
coordinates => "BC"
29 | 3 N X,Y
| -- ------
| 1 4,5
| 2 12,13
13 | 2 3 20,29
|
5 | 1 turn towards the
| left at N=2
+---------------
4 12 20
As per above A,C turns to the right, which squared is A^2,C^2 to the right
too, which equals C^2-B^2,C^2. Negating the X coordinate to B^2-C^2,C^2
mirrors to be a left turn always, and addition shearing to X+Y,Y doesn't
change that, giving B^2,C^2 always left and so B,C always left.
=cut
# U P -> 2P-Q
# Q -> P
#
# A P -> 2P+Q
# Q -> P
#
# D P -> P+2Q
# Q -> Q unchanged
#
# ------------------------------------
# none (P,Q)
# U (2P-Q,P) dx1=P-Q dy1=P-Q
# A (2P+Q,P) dx2=P+Q dy2=P-Q
# dx2*dy1 - dx1*dy2
# = (P+Q)*(P-Q) - (P-Q)*(P-Q)
# = (P-Q) * (P+Q - (P-Q))
# = (P-Q) * 2Q > 0 so Right
#
# ------------------------------------
# U (2P-Q,P)
# A (2P+Q,P) dx1=2Q dy1=0
# D (P+2Q,Q) dx2=-P+3Q dy2=Q-P
# dx2*dy1 - dx1*dy2
# = (-P+3Q)*0 - 2Q * (Q-P)
# = 2Q*(P-Q) > 0 so Right
#
# ------------------------------------
# crossing A,D to U from gap U,A
# U.D^k.A = (2*P-Q,P) . D^k . A
# = (2*P-Q + 2*k*P, P) . A
# = ((2*k+2)*P-Q, P) . A
# = 2*((2*k+2)*P-Q) + P, (2*k+2)*P-Q
# = (4*k+4)*P - 2*Q + P, (2*k+2)*P-Q
# = (4*k+5)*P - 2*Q, (2*k+2)*P-Q
# U.D^k.D = ((2*k+2)*P-Q, P) . D
# = (2*k+2)*P-Q + 2*P, P
# = (2*k+4)*P-Q, P
# A.U^k.U = (2*P+Q, P) . U^(k+1)
# = (k+2)*(2*P+Q) - (k+1)*P, (k+1)*(2*P+Q) - k*P
# = (k+3)*P + (k+2)*Q, (k+2)*P + (k+1)*Q
# dx1 = (2*k+4)*P-Q - ((4*k+5)*P - 2*Q)
# dy1 = P - ((2*k+2)*P-Q)
# dx2 = (k+3)*P + (k+2)*Q - ((4*k+5)*P - 2*Q)
# dy2 = (k+2)*P + (k+1)*Q - ((2*k+2)*P-Q)
# dx2*dy1 - dx1*dy2
# = 4*P^2*k^2 + (6*P^2 - 6*Q*P)*k + (2*P^2 - 4*Q*P + 2*Q^2)
# = 4*P^2*k^2 + 6*P*(P-Q)*k + 2*(P-Q)^2
# > 0 turn right
#
# ------------------------------------
# wraparound A,D to U
# D^k .A = (P+2kQ, Q) . A
# = 2*(P+2*k*Q)+Q, P+2*k*Q
# = 2*P+(4*k+1)*Q, P+2*k*Q
# D^k .D = D^(k+1) = P+(2*k+2)*Q, Q
# U^(k+1).U = U^(k+1) = (k+3)*P-(k+2)*Q, (k+2)*P-(k+1)*Q
# dx1 = P+(2*k+2)*Q - (2*P+(4*k+1)*Q)
# = -P + (-2*k+1)*Q
# dy1 = Q - (P+2*k*Q)
# = -P + (-2k+1)Q
# dx2 = (k+3)*P-(k+2)*Q - (2*P+(4*k+1)*Q)
# = (k+1)*P + (-5*k-3)*Q
# dy2 = (k+2)*P-(k+1)*Q - (P+2*k*Q)
# = (k+1)P + (-k-1 -2k)Q
# = (k+1)*P + (-3k-1)*Q
# dx2*dy1 - dx1*dy2
# = ((k+1)P + (-5k-3)Q) * (-P + (-2k+1)Q) - (-P + (-2k+1)) * ((k+1)P + (-3k-1)Q)
# = (2*Q*k + 2*Q)*P + (4*Q^2*k^2 + 2*Q^2*k - 2*Q^2)
# = (2*k + 2)*P*Q + (4*k^2 + 2*k - 2)*Q^2
# > 0 turn Right
#
# eg. P=2,Q=1 k=0
# D^k .A = 5,2
# D^k .D = 4,1
# U^k+1.U = 4,3
# dx1 = -1
# dy1 = -1
# dx2 = -1
# dy2 = 1
# dx2*dy1 - dx1*dy2 = 2
#
# ------------------------------------
# wraparound D to U,A
# D^k = P+2*k*Q, Q
# U^k.U = U^(k+1)
# = (k+2)*P-(k+1)*Q, (k+1)*P-k*Q
# U^k.A = (k+1)*P-k*Q, k*P-(k-1)*Q . A
# = 2*((k+1)*P-k*Q) + k*P-(k-1)*Q, (k+1)*P-k*Q
# = (3*k+2)*P + (-3*k+1)*Q, (k+1)*P-k*Q
# dx1 = (k+2)*P-(k+1)*Q - (P+2*k*Q)
# = (k+1)*P + (-3*k-1)*Q
# dy1 = (k+1)*P-k*Q - Q
# = (k+1)*P-(k+1)*Q
# dx2 = (3*k+2)*P + (-3*k+1)*Q - (P+2*k*Q)
# = (3*k+1)*P + (-5*k+1)*Q
# dy2 = (k+1)*P-k*Q - Q
# = (k+1)*P-(k+1)*Q
# dx2*dy1 - dx1*dy2
# = (2*P^2 - 4*Q*P + 2*Q^2)*k^2 + (2*P^2 - 2*Q*P)*k + (2*Q*P - 2*Q^2)
# = 2*(P-Q)^2*k^2 + 2*P*(P-Q)*k + 2*Q*(P-Q)
# > 0 turn Right
#
# eg. P=2;Q=1;k=1
# 4,1
# 4,3
# 8,3
# 2P-Q,P to 2P+Q,P to P+2Q,Q P>Q>=1
#
# right at first "U"
# 3P-2Q,2P-Q ----- 5P-2Q,2P-Q
# |
# |
# 2P-Q,P ---- 2P+Q,P right at "A"
# | /
# | /
# P,Q P+2Q,Q
#
# 3P+2Q,2P+Q
#
#
# "U" 3P-2Q,2P-Q ----- 5P-2Q,2P-Q "A"
# /
# /
# 4P-Q,P "D"
#
#
# P,Q
#
# / U 4P-2Q-P,2P-Q = 3P-2Q,2P-Q
# U 2P-Q,P -- A 4P-2Q+P,2P-Q = 5P-2Q,2P-Q
# / \ D 2P-Q+2P,P = 4P-Q, P
# / / U 4P+2Q-P,2P+Q = 3P+2Q,2P+Q
# P,Q -- A 2P+Q,P -- A
# \ \ D
# \ / U
# D P+2Q,Q -- A
# \ D
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::PythagoreanTree-E<gt>new ()>
=item C<$path = Math::PlanePath::PythagoreanTree-E<gt>new (tree_type =E<gt> $str, coordinates =E<gt> $str)>
Create and return a new path object. The C<tree_type> option can be
"UAD" (the default)
"UArD" UAD with Gray code reflections
"FB"
"UMT"
The C<coordinates> option can be
"AB" odd, even legs (the default)
"AC" odd leg, hypotenuse
"BC" even leg, hypotenuse
"PQ"
"SM" small, medium legs
"SC" small leg, hypotenuse
"MC" medium leg, hypotenuse
The C<digit_order> option can be
"HtoL" high to low (the default)
"LtoH" low to high (the default)
=item C<$n = $path-E<gt>n_start()>
Return 1, the first N in the path.
=item C<($x,$y) = $path-E<gt>n_to_xy ($n)>
Return the X,Y coordinates of point number C<$n> on the path. Points begin
at 1 and if C<$nE<lt>1> then the return is an empty list.
=item C<$n = $path-E<gt>xy_to_n ($x,$y)>
Return the point number for coordinates C<$x,$y>. If there's nothing at
C<$x,$y> then return C<undef>.
The return is C<undef> if C<$x,$y> is not a primitive Pythagorean triple,
per the C<coordinates> option.
=item C<$rsquared = $path-E<gt>n_to_radius ($n)>
Return the radial distance R=sqrt(X^2+Y^2) of point C<$n>. If there's no
point C<$n> then return C<undef>.
For coordinates=AB or SM this is the hypotenuse C and therefore an integer,
for integer C<$n>.
=item C<($n_lo, $n_hi) = $path-E<gt>rect_to_n_range ($x1,$y1, $x2,$y2)>
Return a range of N values which occur in a rectangle with corners at
C<$x1>,C<$y1> and C<$x2>,C<$y2>. The range is inclusive.
Both trees go off into large X,Y coordinates while yet to finish values
close to the origin which means the N range for a rectangle can be quite
large. For UAD C<$n_hi> is roughly C<3**max(x/2)>, or for FB smaller at
roughly C<3**log2(x)>.
=back
=head2 Descriptive Methods
=over
=item C<$x = $path-E<gt>x_minimum()>
=item C<$y = $path-E<gt>y_minimum()>
Return the minimum X or Y occurring in the path. The value goes according
to the C<coordinates> option,
coordinate minimum
---------- -------
A,S 3
B,M 4
C 5
P 2
Q 1
=back
=head2 Tree Methods
X<Complete ternary tree>Each point has 3 children, so the path is a complete
ternary tree.
=over
=item C<@n_children = $path-E<gt>tree_n_children($n)>
Return the three children of C<$n>, or an empty list if C<$n E<lt> 1>
(ie. before the start of the path).
This is simply C<3*$n-1, 3*$n, 3*$n+1>. This is appending an extra ternary
digit 0, 1 or 2 to the mixed-radix form for N described above. Or staying
all in ternary then appending to N+1 rather than N and adjusting back.
=item C<$num = $path-E<gt>tree_n_num_children($n)>
Return 3, since every node has three children, or return C<undef> if
C<$nE<lt>1> (ie. before the start of the path).
=item C<$n_parent = $path-E<gt>tree_n_parent($n)>
Return the parent node of C<$n>, or C<undef> if C<$n E<lt>= 1> (the top of
the tree).
This is simply C<floor(($n+1)/3)>, reversing the C<tree_n_children()>
calculation above.
=item C<$depth = $path-E<gt>tree_n_to_depth($n)>
Return the depth of node C<$n>, or C<undef> if there's no point C<$n>. The
top of the tree at N=1 is depth=0, then its children depth=1, etc.
The structure of the tree with 3 nodes per point means the depth is
floor(log3(2N-1)), so for example N=5 through N=13 all have depth=2.
=item C<$n = $path-E<gt>tree_depth_to_n($depth)>
=item C<$n = $path-E<gt>tree_depth_to_n_end($depth)>
Return the first or last N at tree level C<$depth> in the path, or C<undef>
if nothing at that depth or not a tree. The top of the tree is depth=0.
=back
=head2 Tree Descriptive Methods
=over
=item C<$num = $path-E<gt>tree_num_children_minimum()>
=item C<$num = $path-E<gt>tree_num_children_maximum()>
Return 3 since every node has 3 children, making that both the minimum and
maximum.
=item C<$bool = $path-E<gt>tree_any_leaf()>
Return false, since there are no leaf nodes in the tree.
=back
=head1 FORMULAS
=head2 UAD Matrices
Internally the code uses P,Q and calculates A,B at the end as necessary.
The UAD transformations in P,Q coordinates are
U P -> 2P-Q ( 2 -1 )
Q -> P ( 1 0 )
A P -> 2P+Q ( 2 1 )
Q -> P ( 1 0 )
D P -> P+2Q ( 1 2 )
Q -> Q unchanged ( 0 1 )
The advantage of P,Q for the calculation is that it's 2 values instead of 3.
The transformations can be written with the 2x2 matrices shown, but explicit
steps are enough for the code.
Repeatedly applying "U" gives
U 2P-Q, P
U^2 3P-2Q, 2P-Q
U^3 4P-3Q, 3P-2Q
...
U^k (k+1)P-kQ, kP-(k-1)Q
= P+k(P-Q), Q+k*(P-Q)
If there's a run of k many high zeros in the Nrem = N-Nrow position in the
level then they can be applied to the initial P=2,Q=1 as
U^k P=k+2, Q=k+1 start for k high zeros
=head2 FB Transformations
The FB tree is calculated in P,Q and converted to A,B at the end as
necessary. Its three transformations are
M1 P -> P+Q ( 1 1 )
Q -> 2Q ( 0 2 )
M2 P -> 2P ( 2 0 )
Q -> P-Q ( 1 -1 )
M3 P -> 2P ( 2 0 )
Q -> P+Q ( 1 1 )
Price's paper shows rearrangements of a set of four values q',q,p,p'. Just
the p and q are enough for the calculation. The set of four has some
attractive geometric interpretations though.
=head2 X,Y to N -- UAD
C<xy_to_n()> works in P,Q coordinates. An A,B or other input is converted
to P,Q per the formulas in L</PQ Coordinates> above. A P,Q point can be
reversed up the UAD tree to its parent point
if P > 3Q reverse "D" P -> P-2Q
digit=2 Q -> unchanged
if P > 2Q reverse "A" P -> Q
digit=1 Q -> P-2Q
otherwise reverse "U" P -> Q
digit=0 Q -> 2Q-P
This gives a ternary digit 2, 1, 0 respectively from low to high. Those
plus a high "1" bit make N. The number of steps is the "depth" level.
If at any stage P,Q doesn't satisfy PE<gt>QE<gt>=1, one odd, the other even,
then it means the original point, however it was converted, was not a
primitive triple. For a primitive triple the endpoint is always P=2,Q=1.
The conditions PE<gt>3Q or PE<gt>2Q work because each matrix sends its
parent P,Q to one of three disjoint regions,
Q P=Q P=2Q P=3Q
| * U ---- A ++++++
| * ---- ++++++
| * ---- ++++++
| * ---- ++++++
| * ---- ++++++
| * ---- ++++++
| * ---- ++++++ D
| * ----++++++
| * ----++++
| ----++
|
+------------------------------------------------- P
So U is the upper wedge, A the middle, and D the lower. The parent P,Q can
be anywhere in PE<gt>QE<gt>=1, the matrices always map to these regions.
=head2 X,Y to N -- FB
After converting to P,Q as necessary, a P,Q point can be reversed up the FB
tree to its parent
if P odd reverse M1 P -> P-Q
(Q even) Q -> Q/2
if P > 2Q reverse M2 P -> P/2
(P even) Q -> P/2 - Q
otherwise reverse M3 P -> P/2
(P even) Q -> Q - P/2
This is a little like the binary greatest common divisor algorithm, but
designed for one value odd and the other even. Like the UAD ascent above if
at any stage P,Q doesn't satisfy PE<gt>QE<gt>=1, one odd, the other even,
then the initial point wasn't a primitive triple.
The M1 reversal works because M1 sends any parent P,Q to a child which has P
odd. All odd P,Q comes from M1. The M2 and M3 always make children with P
even. Those children are divided between two disjoint regions above and
below the line P=2Q.
Q P=Q P=2Q
| * M3 P=even ----
| * ----
| * ----
| * ----
| * ---- M2 P=even
| * ----
| * ----
| * ----
| * ---- M1 P=odd anywhere
| ----
|
+------------------------------------------------- P
=head2 X,Y to N -- UMT
After converting to P,Q as necessary, a P,Q point can be reversed up the UMT
tree to its parent
if P > 2Q reverse "U" P -> Q
digit=0 Q -> 2Q-P
if P even reverse "M2" P -> P/2
(Q odd) Q -> P/2 - Q
otherwise reverse "T" P -> P - 3 * Q/2
(Q even) Q -> Q/2
These reversals work because U sends any parent P,Q to a child PE<gt>2Q
whereas the M2 and T go below that line. M2 and T are distinguished by M2
giving P even whereas T gives P odd.
Q P=Q P=2Q
| * U ----
| * ----
| * ----
| * ----
| * ---- M2 for P=even
| * ---- T for P=odd
| * ----
| * ----
| * ----
| ----
|
+------------------------------------------------- P
=head2 Rectangle to N Range -- UAD
For the UAD tree, the smallest A,B within each level is found at the topmost
"U" steps for the smallest A or the bottom-most "D" steps for the smallest
B. For example in the table above of level=2 N=5..13 the smallest A is
the top A=7,B=24, and the smallest B is in the bottom A=35,B=12. In general
Amin = 2*level + 1
Bmin = 4*level
In P,Q coordinates the same topmost line is the smallest P and bottom-most
the smallest Q. The values are
Pmin = level+1
Qmin = 1
The fixed Q=1 arises from the way the "D" transformation sends Q-E<gt>Q
unchanged, so every level includes a Q=1. This means if you ask what range
of N is needed to cover all Q E<lt> someQ then there isn't one, only a P
E<lt> someP has an N to go up to.
=head2 Rectangle to N Range -- FB
For the FB tree, the smallest A,B within each level is found in the topmost
two final positions. For example in the table above of level=2 N=5..13 the
smallest A is in the top A=9,B=40, and the smallest B is in the next row
A=35,B=12. In general,
Amin = 2^level + 1
Bmin = 2^level + 4
In P,Q coordinates a Q=1 is found in that second row which is the minimum B,
and the smallest P is found by taking M1 steps half-way then a M2 step, then
M1 steps for the balance. This is a slightly complicated
Pmin = / 3*2^(k-1) + 1 if even level = 2*k
\ 2^(k+1) + 1 if odd level = 2*k+1
Q = 1
The fixed Q=1 arises from the M1 steps giving
P = 2 + 1+2+4+8+...+2^(level-2)
= 2 + 2^(level-1) - 1
= 2^(level-1) + 1
Q = 2^(level-1)
followed by M2 step
Q -> P-Q
= 1
As for the UAD above this means small Q's always remain no matter how big N
gets, only a P range determines an N range.
=head1 OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this
path include,
=over
L<http://oeis.org/A007051> (etc)
=back
A007051 N start of depth=n, (3^n+1)/2, ie. tree_depth_to_n()
A003462 N end of depth=n-1, (3^n-1)/2, ie. tree_depth_to_n_end()
A000244 N of row middle line, 3^n
A058529 possible values taken by abs(A-B),
being integers with all prime factors == +/-1 mod 8
"U" repeatedly
A046092 coordinate B, 2n(n+1) = 4*triangular numbers
A099776 \ coordinate C, being 2n(n+1)+1
A001844 / which is the "centred squares"
"A" repeatedly
A046727 \ coordinate A
A084159 / "Pell oblongs"
A046729 coordinate B
A001653 coordinate C, numbers n where 2*n^2-1 is square
A000129 coordinate P and Q, the Pell numbers
A001652 coordinate S, the smaller leg
A046090 coordinate M, the bigger leg
"D" repeatedly
A000466 coordinate A, being 4*n^2-1 for n>=1
"M1" repeatedly
A028403 coordinate B, binary 10..010..000
A007582 coordinate B/4, binary 10..010..0
A085601 coordinate C, binary 10..010..001
"M2" repeatedly
A015249 \ coordinate A, binary 111000111000...
A084152 |
A084175 /
A054881 coordinate B, binary 1010..1010000..00
"M3" repeatedly
A106624 coordinate P,Q pairs, 2^k-1,2^k
"T" repeatedly
A134057 coordinate A, binomial(2^n-1,2)
binary 111..11101000..0001
A093357 coordinate B, binary 10111..111000..000
A052940 \
A055010 | coordinate P, 3*2^n-1
A083329 | binary 10111..111
A153893 /
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::Hypot>,
L<Math::PlanePath::RationalsTree>,
L<Math::PlanePath::CoprimeColumns>
=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