# Copyright 2011, 2012, 2013, 2014, 2015 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=ComplexPlus --all --scale=5
#
# math-image --path=ComplexPlus --expression='i<128?i:0' --output=numbers --size=132x40
#
# Realpart:
# math-image --path=ComplexPlus,realpart=2 --expression='i<50?i:0' --output=numbers --size=180
#
# Arms:
# math-image --path=ComplexPlus,arms=2 --expression='i<64?i:0' --output=numbers --size=79
package Math::PlanePath::ComplexPlus;
use 5.004;
use strict;
#use List::Util 'max';
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 119;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
*_divrem_mutate = \&Math::PlanePath::_divrem_mutate;
use Math::PlanePath::Base::Generic
'is_infinite',
'round_nearest';
use Math::PlanePath::Base::Digits
'round_up_pow',
'digit_split_lowtohigh',
'digit_join_lowtohigh';
# uncomment this to run the ### lines
#use Smart::Comments;
use constant n_start => 0;
use constant parameter_info_array =>
[ { name => 'realpart',
display => 'Real Part',
type => 'integer',
default => 1,
minimum => 1,
width => 2,
description => 'Real part r in the i+r complex base.',
},
{ name => 'arms',
share_key => 'arms_2',
display => 'Arms',
type => 'integer',
minimum => 1,
maximum => 2,
default => 1,
width => 1,
description => 'Arms',
when_name => 'realpart',
when_value => '1',
},
];
# b=i+r
# theta = atan(1/r)
sub x_negative_at_n {
my ($self) = @_;
if ($self->{'realpart'} == 1) { return 8; }
return $self->{'norm'} ** _ceil((2*atan2(1,1)) / atan2(1,$self->{'realpart'}));
}
sub y_negative_at_n {
my ($self) = @_;
if ($self->{'realpart'} == 1) { return 32; }
return $self->{'norm'} ** _ceil((4*atan2(1,1)) / atan2(1,$self->{'realpart'}));
}
sub _ceil {
my ($x) = @_;
my $int = int($x);
return ($x > $int ? $int+1 : $int);
}
sub absdx_minimum {
my ($self) = @_;
return ($self->{'realpart'} == 1
? 0 # i+1 N=1 dX=0,dY=1
: 1); # i+r otherwise always diff
}
# use constant dir_maximum_dxdy => (0,0); # supremum, almost full way
#------------------------------------------------------------------------------
sub new {
my $self = shift->SUPER::new(@_);
my $realpart = $self->{'realpart'};
if (! defined $realpart || $realpart < 1) {
$self->{'realpart'} = $realpart = 1;
}
$self->{'norm'} = $realpart*$realpart + 1;
my $arms = $self->{'arms'};
if (! defined $arms || $arms <= 0 || $realpart != 1) { $arms = 1; }
elsif ($arms > 2) { $arms = 2; }
$self->{'arms'} = $arms;
return $self;
}
sub n_to_xy {
my ($self, $n) = @_;
### ComplexPlus n_to_xy(): $n
if ($n < 0) { return; }
if (is_infinite($n)) { return ($n,$n); }
{
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+$self->{'arms'});
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
}
my $realpart = $self->{'realpart'};
my $norm = $self->{'norm'};
### $norm
### $realpart
my $x = my $y = ($n * 0); # inherit bignum
my $dx;
my $dy = 0;
{
my $arm = _divrem_mutate ($n, $self->{'arms'});
if ($arm) {
$y += 1; # start X=0,Y=1
$dx = -1;
} else {
$dx = 1;
}
}
foreach my $digit (digit_split_lowtohigh($n,$norm)) {
### at: "$x,$y digit=$digit dxdy=$dx,$dy"
$x += $dx * $digit;
$y += $dy * $digit;
# multiply i+r, ie. (dx,dy) = (dx + i*dy)*(i+$realpart)
($dx,$dy) = ($realpart*$dx - $dy, $dx + $realpart*$dy);
}
### final: "$x,$y"
return ($x,$y);
}
sub xy_to_n {
my ($self, $x, $y) = @_;
### ComplexPlus xy_to_n(): "$x, $y"
$x = round_nearest ($x);
$y = round_nearest ($y);
my $realpart = $self->{'realpart'};
{
my $rx = $realpart*$x;
my $ry = $realpart*$y;
foreach my $overflow ($rx+$ry, $rx-$ry) {
if (is_infinite($overflow)) { return $overflow; }
}
}
my $orig_x = $x;
my $orig_y = $y;
my $norm = $self->{'norm'};
my $zero = ($x * 0 * $y); # inherit bignum 0
my @n; # digits low to high
my $prev_x = 0;
my $prev_y = 0;
while ($x || $y) {
my $neg_y = $x - $y*$realpart;
my $digit = $neg_y % $norm;
### at: "$x,$y n=$n digit $digit"
push @n, $digit;
$x -= $digit;
$neg_y -= $digit;
### assert: ($neg_y % $norm) == 0
### assert: (($x * $realpart + $y) % $norm) == 0
# divide i+r = mul (i-r)/(i^2 - r^2)
# = mul (i-r)/-norm
# is (i*y + x) * (i-realpart)/-norm
# x = [ x*-realpart - y ] / -norm
# = [ x*realpart + y ] / norm
# y = [ y*-realpart + x ] / -norm
# = [ y*realpart - x ] / norm
#
($x,$y) = (($x*$realpart+$y)/$norm, -$neg_y/$norm);
if ($x == $prev_x && $y == $prev_y) {
last;
}
$prev_x = $x;
$prev_y = $y;
}
### final: "$x,$y n=$n cf arms $self->{'arms'}"
if ($y) {
if ($self->{'arms'} > 1) {
### not on first arm, re-run as: -$orig_x, 1-$orig_y
local $self->{'arms'} = 1;
my $n = $self->xy_to_n(-$orig_x,1-$orig_y);
if (defined $n) {
return 1 + 2*$n; # 180 degrees
}
}
### X,Y not visited
return undef;
}
my $n = digit_join_lowtohigh (\@n, $norm, $zero);
if ($self->{'arms'} > 1) {
$n *= 2;
}
return $n;
}
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
### ComplexPlus rect_to_n_range(): "$x1,$y1 $x2,$y2"
my $xm = max(abs($x1),abs($x2));
my $ym = max(abs($y1),abs($y2));
my $n_hi = ($xm*$xm + $ym*$ym) * $self->{'arms'};
if ($self->{'realpart'} == 1) {
$n_hi *= 16; # 2**4
}
return (0, int($n_hi));
}
#------------------------------------------------------------------------------
# levels
sub level_to_n_range {
my ($self, $level) = @_;
return (0, $self->{'norm'}**$level * $self->{'arms'} - 1);
}
sub n_to_level {
my ($self, $n) = @_;
if ($n < 0) { return undef; }
if (is_infinite($n)) { return $n; }
$n = round_nearest($n);
_divrem_mutate ($n, $self->{'arms'});
my ($pow, $exp) = round_up_pow ($n+1, $self->{'norm'});
return $exp;
}
#------------------------------------------------------------------------------
1;
__END__
=for stopwords eg Ryde Math-PlanePath ie Nstart Nlevel
=head1 NAME
Math::PlanePath::ComplexPlus -- points in complex base i+r
=head1 SYNOPSIS
use Math::PlanePath::ComplexPlus;
my $path = Math::PlanePath::ComplexPlus->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This path traverses points by a complex number base i+r with integer
rE<gt>=1. The default is base i+1 which gives a shape similar to the
C<DragonCurve>,
30 31 14 15 5
28 29 12 13 4
26 27 22 23 10 11 6 7 3
24 25 20 21 8 9 4 5 2
62 63 46 47 18 19 2 3 1
60 61 44 45 16 17 0 1 <- Y=0
58 59 54 55 42 43 38 39 -1
56 57 52 53 40 41 36 37 -2
50 51 94 95 34 35 78 79 -3
48 49 92 93 32 33 76 77 -4
90 91 86 87 74 75 70 71 -5
88 89 84 85 72 73 68 69 -6
126 127 110 111 82 83 66 67 -7
124 125 108 109 80 81 64 65 -8
122 123 118 119 106 107 102 103 -9
120 121 116 117 104 105 100 101 -10
114 115 98 99 -11
112 113 96 97 -12
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2
=head2 Real Part
Option C<realpart =E<gt> $r> selects another r for complex base b=i+r. For
example
realpart => 2
45 46 47 48 49 8
40 41 42 43 44 7
35 36 37 38 39 6
30 31 32 33 34 5
25 26 27 28 29 20 21 22 23 24 4
15 16 17 18 19 3
10 11 12 13 14 2
5 6 7 8 9 1
0 1 2 3 4 <- Y=0
^
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
N is broken into digits of a base norm=r*r+1, ie. digits 0 to r*r inclusive.
norm = r*r + 1
Nstart = 0
Nlevel = norm^level - 1
The low digit of N makes horizontal runs of r*r+1 many points, such as above
N=0 to N=4, then N=5 to N=9, etc. In the default r=1 these runs are 2 long.
For r=2 shown above they're 2*2+1=5 long, or r=3 would be 3*3+1=10, etc.
The offset for each successive run is i+r, ie. Y=1,X=r such as the N=5 shown
above. Then the offset for the next level is (i+r)^2 = (2r*i + r^2-1) so
N=25 begins at Y=2*r=4, X=r*r-1=3. In general each level adds an angle
angle = atan(1/r)
Nlevel_angle = level * angle
So the points spiral around anti-clockwise. For r=1 the angle is
atan(1/1)=45 degrees, so that for example level=4 is angle 4*45=180 degrees,
putting N=2^4=16 on the negative X axis as shown in the first sample above.
As r becomes bigger the angle becomes smaller, making it spiral more slowly.
The points never fill the plane, but the set of points N=0 to Nlevel are all
touching.
=head2 Arms
For C<realpart =E<gt> 1>, an optional C<arms =E<gt> 2> adds a second copy of
the curve rotated 180 degrees and starting from X=0,Y=1. It meshes
perfectly to fill the plane. Each arm advances successively so N=0,2,4,etc
is the plain path and N=1,3,5,7,etc is the copy
arms=>2
60 62 28 30 5
56 58 24 26 4
52 54 44 46 20 22 12 14 3
48 50 40 42 16 18 8 10 2
36 38 3 1 4 6 35 33 1
32 34 7 5 0 2 39 37 <- Y=0
11 9 19 17 43 41 51 49 -1
15 13 23 21 47 45 55 53 -2
27 25 59 57 -3
31 29 63 61 -4
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
There's no C<arms> parameter for other C<realpart> values as yet, only i+1.
Is there a good rotated arrangement for others? Do "norm" many copies fill
the plane in general?
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::ComplexPlus-E<gt>new ()>
Create and return a new path object.
=item C<($x,$y) = $path-E<gt>n_to_xy ($n)>
Return the X,Y coordinates of point number C<$n> on the path. Points begin
at 0 and if C<$n E<lt> 0> then the return is an empty list.
=back
=head2 Level Methods
=over
=item C<($n_lo, $n_hi) = $path-E<gt>level_to_n_range($level)>
Return C<(0, 2**$level - 1)>, or for 2 arms return C<(0, 2 * 2**$level -
1)>. With the C<realpart> option return C<(0, $norm**$level - 1)> where
norm=realpart^2+1.
=back
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::ComplexMinus>,
L<Math::PlanePath::ComplexRevolving>,
L<Math::PlanePath::DragonCurve>
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-planepath/index.html>
=head1 LICENSE
Copyright 2011, 2012, 2013, 2014, 2015 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