# 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 .
# A000328 Number of points of norm <= n^2 in square lattice.
# 1, 5, 13, 29, 49, 81, 113, 149, 197, 253, 317, 377, 441, 529, 613, 709, 797
# a(n) = 1 + 4 * sum(j=0, n^2 / 4, n^2 / (4*j+1) - n^2 / (4*j+3) )
# A014200 num points norm <= n^2, excluding 0, divided by 4
#
# A046109 num points norm == n^2
#
# A057655 num points x^2+y^2 <= n
# A014198 = A057655 - 1
#
# A004018 num points x^2+y^2 == n
#
# A057962 hypot count x-1/2,y-1/2 <= n
# is last point of each hypot in points=odd
#
# A057961 hypot count as radius increases
#
# points="square_horiz"
# points="square_vert"
# points="square_centre"
# A199015 square_centred partial sums
#
package Math::PlanePath::Hypot;
use 5.004;
use strict;
use Carp;
use vars '$VERSION', '@ISA';
$VERSION = 116;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
use Math::PlanePath::Base::Generic
'is_infinite',
'round_nearest';
# uncomment this to run the ### lines
# use Smart::Comments;
use constant parameter_info_array =>
[ { name => 'points',
share_key => 'points_aeo',
display => 'Points',
type => 'enum',
default => 'all',
choices => ['all','even','odd'],
choices_display => ['All','Even','Odd'],
description => 'Which X,Y points visit, either all of them or just X+Y=even or odd.',
},
Math::PlanePath::Base::Generic::parameter_info_nstart1(),
];
{
my %x_negative_at_n = (all => 3,
even => 2,
odd => 2);
sub x_negative_at_n {
my ($self) = @_;
return $self->n_start + $x_negative_at_n{$self->{'points'}};
}
}
{
my %y_negative_at_n = (all => 4,
even => 3,
odd => 3);
sub y_negative_at_n {
my ($self) = @_;
return $self->n_start + $y_negative_at_n{$self->{'points'}};
}
}
sub rsquared_minimum {
my ($self) = @_;
return ($self->{'points'} eq 'odd'
? 1 # odd at X=1,Y=0
: 0); # even,all at X=0,Y=0
}
# points=even includes X=Y so abs(X-Y)>=0
# points=odd doesn't include X=Y so abs(X-Y)>=1
*absdiffxy_minimum = \&rsquared_minimum;
*sumabsxy_minimum = \&rsquared_minimum;
#------------------------------------------------------------------------------
sub new {
my $self = shift->SUPER::new(@_);
if (! defined $self->{'n_start'}) {
$self->{'n_start'} = $self->default_n_start;
}
my $points = ($self->{'points'} ||= 'all');
if ($points eq 'all') {
$self->{'n_to_x'} = [0];
$self->{'n_to_y'} = [0];
$self->{'hypot_to_n'} = [0];
$self->{'y_next_x'} = [1, 1];
$self->{'y_next_hypot'} = [1, 2];
$self->{'x_inc'} = 1;
$self->{'x_inc_factor'} = 2;
$self->{'x_inc_squared'} = 1;
$self->{'y_factor'} = 2;
$self->{'opposite_parity'} = -1;
} elsif ($points eq 'even') {
$self->{'n_to_x'} = [0];
$self->{'n_to_y'} = [0];
$self->{'hypot_to_n'} = [0];
$self->{'y_next_x'} = [2, 1];
$self->{'y_next_hypot'} = [4, 2];
$self->{'x_inc'} = 2;
$self->{'x_inc_factor'} = 4;
$self->{'x_inc_squared'} = 4;
$self->{'y_factor'} = 2;
$self->{'opposite_parity'} = 1;
} elsif ($points eq 'odd') {
$self->{'n_to_x'} = [];
$self->{'n_to_y'} = [];
$self->{'hypot_to_n'} = [];
$self->{'y_next_x'} = [1];
$self->{'y_next_hypot'} = [1];
$self->{'x_inc'} = 2;
$self->{'x_inc_factor'} = 4;
$self->{'x_inc_squared'} = 4;
$self->{'y_factor'} = 2;
$self->{'opposite_parity'} = 0;
} elsif ($points eq 'square_centred') {
$self->{'n_to_x'} = [];
$self->{'n_to_y'} = [];
$self->{'hypot_to_n'} = [];
$self->{'y_next_x'} = [undef,1];
$self->{'y_next_hypot'} = [undef,2];
$self->{'x_inc'} = 2;
$self->{'x_inc_factor'} = 4; # ((x+2)^2 - x^2) = 4*x+4
$self->{'x_inc_squared'} = 4;
$self->{'y_start'} = 1;
$self->{'y_inc'} = 2;
$self->{'opposite_parity'} = -1;
} else {
croak "Unrecognised points option: ", $points;
}
return $self;
}
sub _extend {
my ($self) = @_;
### _extend() n: scalar(@{$self->{'n_to_x'}})
### y_next_x: $self->{'y_next_x'}
my $n_to_x = $self->{'n_to_x'};
my $n_to_y = $self->{'n_to_y'};
my $hypot_to_n = $self->{'hypot_to_n'};
my $y_next_x = $self->{'y_next_x'};
my $y_next_hypot = $self->{'y_next_hypot'};
my $y_start = $self->{'y_start'} || 0;
my $y_inc = $self->{'y_inc'} || 1;
# set @y to the Y with the smallest $y_next_hypot[$y], and if there's some
# Y's with equal smallest hypot then all those Y's
my @y = ($y_start);
my $hypot = $y_next_hypot->[$y_start] || 99;
for (my $y = $y_start+$y_inc; $y < @$y_next_x; $y += $y_inc) {
if ($hypot == $y_next_hypot->[$y]) {
push @y, $y;
} elsif ($hypot > $y_next_hypot->[$y]) {
@y = ($y);
$hypot = $y_next_hypot->[$y];
}
}
### chosen y list: @y
# if the endmost of the @$y_next_x, @$y_next_hypot arrays are used then
# extend them by one
if ($y[-1] == $#$y_next_x) {
### grow y_next_x ...
my $y = $#$y_next_x + $y_inc;
my $x = $y + ($self->{'points'} eq 'odd');
$y_next_x->[$y] = $x;
$y_next_hypot->[$y] = $x*$x+$y*$y;
### $y_next_x
### $y_next_hypot
### assert: $y_next_hypot->[$y] == $y**2 + $x*$x
}
# @x is the $y_next_x[$y] for each of the @y smallests, and step those
# selected elements next X and hypot for that new X,Y
my @x = map {
my $y = $_;
my $x = $y_next_x->[$y];
$y_next_x->[$y] += $self->{'x_inc'};
$y_next_hypot->[$y]
+= $self->{'x_inc_factor'} * $x + $self->{'x_inc_squared'};
### assert: $y_next_hypot->[$y] == ($x+$self->{'x_inc'})**2 + $y**2
$x
} @y;
### $hypot
### base octant: join(' ',map{"$x[$_],$y[$_]"} 0 .. $#x)
# transpose X,Y to Y,X
{
my @base_x = @x;
my @base_y = @y;
unless ($y[0]) { # no transpose of x,0
shift @base_x;
shift @base_y;
}
if ($x[-1] == $y[-1]) { # no transpose of x,x
pop @base_x;
pop @base_y;
}
push @x, reverse @base_y;
push @y, reverse @base_x;
}
### with transpose q1: join(' ',map{"$x[$_],$y[$_]"} 0 .. $#x)
# rotate +90 quadrant 1 into quadrant 2
{
my @base_y = @y;
push @y, @x;
push @x, map {-$_} @base_y;
}
### with rotate q2: join(' ',map{"$x[$_],$y[$_]"} 0 .. $#x)
# rotate +180 quadrants 1+2 into quadrants 2+3
push @x, map {-$_} @x;
push @y, map {-$_} @y;
### store: join(' ',map{"$x[$_],$y[$_]"} 0 .. $#x)
### at n: scalar(@$n_to_x)
### hypot_to_n: "h=$hypot n=".scalar(@$n_to_x)
$hypot_to_n->[$hypot] = scalar(@$n_to_x);
push @$n_to_x, @x;
push @$n_to_y, @y;
# ### hypot_to_n now: join(' ',map {defined($hypot_to_n->[$_]) && "h=$_,n=$hypot_to_n->[$_]"} 0 .. $#$hypot_to_n)
# my $x = $y_next_x->[0];
#
# $x = $y_next_x->[$y];
# $n_to_x->[$next_n] = $x;
# $n_to_y->[$next_n] = $y;
# $xy_to_n{"$x,$y"} = $next_n++;
#
# $y_next_x->[$y]++;
# $y_next_hypot->[$y] = $y*$y + $y_next_x->[$y]**2;
}
sub n_to_xy {
my ($self, $n) = @_;
### Hypot n_to_xy(): $n
$n = $n - $self->{'n_start'}; # starting $n==0, warn if $n==undef
if ($n < 0) { return; }
if (is_infinite($n)) { return ($n,$n); }
my $int = int($n);
$n -= $int; # fraction part
my $n_to_x = $self->{'n_to_x'};
my $n_to_y = $self->{'n_to_y'};
while ($int >= $#$n_to_x) {
_extend($self);
}
my $x = $n_to_x->[$int];
my $y = $n_to_y->[$int];
return ($x + $n * ($n_to_x->[$int+1] - $x),
$y + $n * ($n_to_y->[$int+1] - $y));
}
sub xy_is_visited {
my ($self, $x, $y) = @_;
if ($self->{'opposite_parity'} >= 0) {
$x = round_nearest ($x);
$y = round_nearest ($y);
if ((($x%2) ^ ($y%2)) == $self->{'opposite_parity'}) {
return 0;
}
}
if ($self->{'points'} eq 'square_centred') {
unless (($y%2) && ($x%2)) {
return 0;
}
}
return 1;
}
sub xy_to_n {
my ($self, $x, $y) = @_;
### Hypot xy_to_n(): "$x, $y"
### hypot_to_n last: $#{$self->{'hypot_to_n'}}
$x = round_nearest ($x);
$y = round_nearest ($y);
if ((($x%2) ^ ($y%2)) == $self->{'opposite_parity'}) {
return undef;
}
if ($self->{'points'} eq 'square_centred') {
unless (($y%2) && ($x%2)) {
return undef;
}
}
my $hypot = $x*$x + $y*$y;
if (is_infinite($hypot)) {
### infinity
return undef;
}
my $n_to_x = $self->{'n_to_x'};
my $n_to_y = $self->{'n_to_y'};
my $hypot_to_n = $self->{'hypot_to_n'};
while ($hypot > $#$hypot_to_n) {
_extend($self);
}
my $n = $hypot_to_n->[$hypot];
for (;;) {
if ($x == $n_to_x->[$n] && $y == $n_to_y->[$n]) {
return $n + $self->{'n_start'};
}
$n += 1;
if ($n_to_x->[$n]**2 + $n_to_y->[$n]**2 != $hypot) {
### oops, hypot_to_n no good ...
return undef;
}
}
# if ($x < 0 || $y < 0) {
# return undef;
# }
# my $h = $x*$x + $y*$y;
#
# while ($y_next_x[$y] <= $x) {
# _extend($self);
# }
# return $xy_to_n{"$x,$y"};
}
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
$x1 = abs (round_nearest ($x1));
$y1 = abs (round_nearest ($y1));
$x2 = abs (round_nearest ($x2));
$y2 = abs (round_nearest ($y2));
if ($x1 > $x2) { ($x1,$x2) = ($x2,$x1); }
if ($y1 > $y2) { ($y1,$y2) = ($y2,$y1); }
# circle area pi*r^2, with r^2 = $x2**2 + $y2**2
return ($self->{'n_start'},
$self->{'n_start'} + int (3.2 * (($x2+1)**2 + ($y2+1)**2)));
}
1;
__END__
# Quadrant style ...
#
# 9 73 75 79 83 85
# 8 58 62 64 67 71 81 ...
# 7 45 48 52 54 61 69 78 86
# 6 35 37 39 43 50 56 65 77 88
# 5 26 28 30 33 41 47 55 68 80
# 4 17 19 22 25 31 40 49 60 70 84
# 3 11 13 15 20 24 32 42 53 66 82
# 2 6 8 9 14 21 29 38 51 63 76
# 1 3 4 7 12 18 27 36 46 59 74
# Y=0 1 2 5 10 16 23 34 44 57 72
#
# X=0 1 2 3 4 5 6 7 8 9 ...
#
# For example N=37 is at X=1,Y=6 which is sqrt(1*1+6*6) = sqrt(37) from the
# origin. The next closest to the origin is X=6,Y=2 at sqrt(40). In general
# it's the sums of two squares X^2+Y^2 taken in order from smallest to biggest.
#
# Points X,Y and swapped Y,X are the same distance from the origin. The one
# with bigger X is taken first, then the swapped Y,X (as long as X!=Y). For
# example N=21 is X=4,Y=2 and N=22 is X=2,Y=4.
=for stopwords Ryde Math-PlanePath ie hypot octant onwards OEIS hypots
=head1 NAME
Math::PlanePath::Hypot -- points in order of hypotenuse distance
=head1 SYNOPSIS
use Math::PlanePath::Hypot;
my $path = Math::PlanePath::Hypot->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This path visits integer points X,Y in order of their distance from the
origin 0,0, or anti-clockwise from the X axis among those of equal distance,
=cut
# math-image --expression='i<=89?i:0' --path=Hypot --output=numbers --size=79
=pod
84 73 83 5
74 64 52 47 51 63 72 4
75 59 40 32 27 31 39 58 71 3
65 41 23 16 11 15 22 38 62 2
85 53 33 17 7 3 6 14 30 50 82 1
76 48 28 12 4 1 2 10 26 46 70 <- Y=0
86 54 34 18 8 5 9 21 37 57 89 -1
66 42 24 19 13 20 25 45 69 -2
77 60 43 35 29 36 44 61 81 -3
78 67 55 49 56 68 80 -4
87 79 88 -5
^
-5 -4 -3 -2 -1 X=0 1 2 3 4 5
For example N=58 is at X=4,Y=-1 is sqrt(4*4+1*1) = sqrt(17) from the origin.
The next furthest from the origin is X=3,Y=3 at sqrt(18).
See C for points in order of X^2+3*Y^2, or C
and C in order of plain sum X+Y.
=head2 Equal Distances
Points with the same distance are taken in anti-clockwise order around from
the X axis. For example X=3,Y=1 is sqrt(10) from the origin, as are the
swapped X=1,Y=3, and X=-1,Y=3 etc in other quadrants, for a total 8 points
N=30 to N=37 all the same distance.
When one of X or Y is 0 there's no negative, so just four negations like
N=10 to 13 points X=2,Y=0 through X=0,Y=-2. Or on the diagonal X==Y there's
no swap, so just four like N=22 to N=25 points X=3,Y=3 through X=3,Y=-3.
There can be more than one way for the same distance to arise.
A Pythagorean triple like 3^2 + 4^2 == 5^2 has 8 points from the 3,4, then 4
points from the 5,0 giving a total 12 points N=70 to N=81. Other
combinations like 20^2 + 15^2 == 24^2 + 7^2 occur too, and also with more
than two different ways to have the same sum.
=head2 Multiples of 4
The first point of a given distance from the origin is either on the X axis
or somewhere in the first octant. The row Y=1 just above the axis is the
first of its equals from XE=2 onwards, and similarly further rows for
big enough X.
There's always a multiple of 4 many points with the same distance so the
first point has N=4*k+2, and similarly on the negative X side N=4*j, for
some k or j. If you plot the prime numbers on the path then those even N's
(composites) are gaps just above the positive X axis, and on or just below
the negative X axis.
=head2 Circle Lattice
Gauss's circle lattice problem asks how many integer X,Y points there are
within a circle of radius R.
The points on the X axis N=2,10,26,46, etc are the first for which
X^2+Y^2==R^2 (integer X==R). Adding option C0> to make them
each 1 less gives the number of points strictly inside, ie. X^2+Y^2 E
R^2.
The last point satisfying X^2+Y^2==R^2 is either in the octant below the X
axis, or is on the negative Y axis. Those N's are the number of points
X^2+Y^2E=R^2, Sloane's A000328.
When that A000328 sequence is plotted on the path a straight line can be
seen in the fourth quadrant extending down just above the diagonal. It
arises from multiples of the Pythagorean 3^2 + 4^2, first X=4,Y=-3, then
X=8,Y=-6, etc X=4*k,Y=-3*k. But sometimes the multiple is not the last
among those of that 5*k radius, so there's gaps in the line. For example
20,-15 is not the last since because 24,-7 is also 25 away from the origin.
=head2 Even Points
Option C "even"> visits just the even points, meaning the sum
X+Y even, so X,Y both even or both odd.
=cut
# math-image --expression='i<70?i:0' --path=Hypot,points=even --output=numbers --size=79
=pod
points => "even"
52 40 39 51 5
47 32 23 31 46 4
53 27 16 15 26 50 3
33 11 7 10 30 2
41 17 3 2 14 38 1
24 8 1 6 22 <- Y=0
42 18 4 5 21 45 -1
34 12 9 13 37 -2
54 28 19 20 29 57 -3
48 35 25 36 49 -4
55 43 44 56 -5
^
-5 -4 -3 -2 -1 X=0 1 2 3 4 5
Even points can be mapped to all points by a 45 degree rotate and flip.
N=1,6,22,etc on the X axis here is on the X=Y diagonal of all-points. And
conversely N=1,2,10,26,etc on the X=Y diagonal here is the X axis of
all-points.
The sets of points with equal hypotenuse are the same in the even and all,
but the flip takes them in a reversed order.
=head2 Odd Points
Option C "odd"> visits just the odd points, meaning sum X+Y
odd, so X,Y one odd the other even.
=cut
# math-image --expression='i<=76?i:0' --path=Hypot,points=odd --output=numbers --size=78x30
=pod
points => "odd"
71 55 54 70 6
63 47 36 46 62 5
64 37 27 26 35 61 4
72 38 19 14 18 34 69 3
48 20 7 6 17 45 2
56 28 8 2 5 25 53 1
39 15 3 + 1 13 33 <- Y=0
57 29 9 4 12 32 60 -1
49 21 10 11 24 52 -2
73 40 22 16 23 44 76 -3
65 41 30 31 43 68 -4
66 50 42 51 67 -5
74 58 59 75 -6
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
Odd points can be mapped to all points by a 45 degree rotate and a shift
X-1,Y+1 to put N=1 at the origin. The effect of that shift is as if the
hypot measure in "all" points was (X-1/2)^2+(Y-1/2)^2 and for that reason
the sets of points with equal hypots are not the same in odd and all.
=head1 FUNCTIONS
See L for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::Hypot-Enew ()>
=item C<$path = Math::PlanePath::Hypot-Enew (points =E $str), n_start =E $n>
Create and return a new hypot path object. The C option can be
"all" all integer X,Y (the default)
"even" only points with X+Y even
"odd" only points with X+Y odd
=item C<($x,$y) = $path-En_to_xy ($n)>
Return the X,Y coordinates of point number C<$n> on the path.
For C<$n E 1> the return is an empty list, it being considered the first
point at X=0,Y=0 is N=1.
Currently it's unspecified what happens if C<$n> is not an integer.
Successive points are a fair way apart, so it may not make much sense to say
give an X,Y position in between the integer C<$n>.
=item C<$n = $path-Exy_to_n ($x,$y)>
Return an integer point number for coordinates C<$x,$y>. Each integer N is
considered the centre of a unit square and an C<$x,$y> within that square
returns N.
For "even" and "odd" options only every second square in the plane has an N
and if C<$x,$y> is a position not covered then the return is C.
=back
=head1 FORMULAS
The calculations are not particularly efficient currently. Private arrays
are built similar to what's described for C, but with
replication for negative and swapped X,Y.
=head1 OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to
this path include
=over
L (etc)
=back
points="all", n_start=0
A051132 N on X axis, being count points norm < X^2
points="odd"
A005883 count of points with norm==4*n+1
=head1 SEE ALSO
L,
L,
L,
L,
L
=head1 HOME PAGE
L
=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 .
=cut