/usr/share/perl5/Math/PlanePath/TerdragonRounded.pm is in libmath-planepath-perl 123-1.
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# 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/>.
package Math::PlanePath::TerdragonRounded;
use 5.004;
use strict;
#use List::Util 'max';
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 123;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
*_divrem_mutate = \&Math::PlanePath::_divrem_mutate;
use Math::PlanePath::Base::Generic
'is_infinite',
'round_nearest';
use Math::PlanePath::Base::Digits
'round_up_pow';
use Math::PlanePath::TerdragonCurve;
# uncomment this to run the ### lines
#use Smart::Comments;
use constant n_start => 0;
*parameter_info_array # arms
= \&Math::PlanePath::TerdragonCurve::parameter_info_array;
*new = \&Math::PlanePath::TerdragonCurve::new;
{
my @x_negative_at_n = (undef, 24, 7, 2, 2, 2, 2);
sub x_negative_at_n {
my ($self) = @_;
return $x_negative_at_n[$self->{'arms'}];
}
}
{
my @y_negative_at_n = (undef, 316, 145, 32, 11, 4, 4);
sub y_negative_at_n {
my ($self) = @_;
return $y_negative_at_n[$self->{'arms'}];
}
}
use constant sumabsxy_minimum => 2; # X=2,Y=0
sub rsquared_minimum {
my ($self) = @_;
return ($self->arms_count < 2
? 4 # 1 arm, minimum X=2,Y=0
: 2); # 2 or more arms, minimum X=1,Y=1
}
use constant dx_minimum => -2;
use constant dx_maximum => 2;
use constant dy_minimum => -1;
use constant dy_maximum => 1;
*_UNDOCUMENTED__dxdy_list = \&Math::PlanePath::_UNDOCUMENTED__dxdy_list_six;
use constant absdx_minimum => 1;
use constant dsumxy_minimum => -2; # diagonals
use constant dsumxy_maximum => 2;
use constant ddiffxy_minimum => -2;
use constant ddiffxy_maximum => 2;
use constant dir_maximum_dxdy => (1,-1); # South-East
use constant turn_any_straight => 0; # never straight
#------------------------------------------------------------------------------
sub n_to_xy {
my ($self, $n) = @_;
### TerdragonRounded n_to_xy(): $n
if ($n < 0) { # negative
return;
}
if (is_infinite($n)) {
return ($n,$n);
}
{
# ENHANCE-ME: the ends join and the direction can be had without a full
# N+1 calculation
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 $arms_count = $self->{'arms'};
my $arm = _divrem_mutate ($n, $arms_count);
my $pair = _divrem_mutate ($n, 2);
my ($x, $y) = $self->Math::PlanePath::TerdragonCurve::n_to_xy
((9*$n + ($pair ? 4 : 2)) * $arms_count + $arm);
### is: (($x+3*$y)/2).", ".(($y-$x)/2)
return (($x+3*$y)/2, ($y-$x)/2); # rotate -60
}
sub xy_to_n {
my ($self, $x, $y) = @_;
### TerdragonRounded xy_to_n(): "$x, $y"
$x = round_nearest($x);
$y = round_nearest($y);
if (($x+$y) % 2) {
return undef;
}
($x,$y) = (($x-3*$y)/2, # rotate +60
($x+$y)/2);
### rot: "$x,$y"
my @n_list = $self->Math::PlanePath::TerdragonCurve::xy_to_n_list ($x, $y);
### @n_list
my $arms_count = $self->{'arms'};
foreach my $n (@n_list) {
my $arm = _divrem_mutate ($n, $arms_count);
my $mod = $n % 9;
if ($mod == 2) {
return (2*int(($n-2)/9))*$arms_count + $arm;
}
if ($mod == 4) {
return (2*int(($n-4)/9) + 1)*$arms_count + $arm;
}
}
return undef;
}
# arms==6 is all "hex_centred" points X+3Y mod 6 == 2 or 4
sub xy_is_visited {
my ($self, $x, $y) = @_;
if ($self->{'arms'} == 6) {
my $mod = (3*$y + $x) % 6;
return ($mod == 2 || $mod == 4);
}
return defined($self->xy_to_n($x,$y));
}
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
# my $xmax = int(max(abs($x1),abs($x2))) + 1;
# my $ymax = int(max(abs($y1),abs($y2))) + 1;
# return (0,
# ($xmax*$xmax + 3*$ymax*$ymax)
# * 1
# * $self->{'arms'});
$x1 = round_nearest ($x1);
$y1 = round_nearest ($y1);
$x2 = round_nearest ($x2);
$y2 = round_nearest ($y2);
($x1,$x2) = ($x2,$x1) if $x1 > $x2;
($y1,$y2) = ($y2,$y1) if $y1 > $y2;
# FIXME: How much wider ?
# Might matter when TerdragonCurve becomes exact.
$x1 = int($x1/3) - 2;
$y1 = int($y1/3) - 2;
$x2 = int($x2/3) + 2;
$y2 = int($y2/3) + 2;
my ($n_lo, $n_hi) = $self->Math::PlanePath::TerdragonCurve::rect_to_n_range
($x1,$y1, $x2,$y2);
if ($n_hi >= $n_hi) {
$n_lo *= 2;
$n_hi = 2*$n_hi + 1;
}
return ($n_lo, $n_hi);
}
#-----------------------------------------------------------------------------
# level_to_n_range()
# like TerdragonMidpoint but 2* points on each arm, numbered starting 0
#
sub level_to_n_range {
my ($self, $level) = @_;
return (0, (2*$self->{'arms'}) * 3**$level - 1);
}
sub n_to_level {
my ($self, $n) = @_;
if ($n < 0) { return undef; }
if (is_infinite($n)) { return $n; }
$n = round_nearest($n);
_divrem_mutate ($n, 2 * $self->{'arms'});
my ($pow, $exp) = round_up_pow ($n+1, 3);
return $exp;
}
#-----------------------------------------------------------------------------
1;
__END__
=for stopwords Guiseppe Terdragon terdragon eg Sur une courbe qui remplit toute aire Mathematische Annalen Ryde OEIS ie Math-PlanePath versa Online Radix radix Jorg Arndt Hexdragon hexdragon
=head1 NAME
Math::PlanePath::TerdragonRounded -- triangular dragon curve, with rounded corners
=head1 SYNOPSIS
use Math::PlanePath::TerdragonRounded;
my $path = Math::PlanePath::TerdragonRounded->new;
my ($x, $y) = $path->n_to_xy (123);
# or another radix digits ...
my $path5 = Math::PlanePath::TerdragonRounded->new (radix => 5);
=head1 DESCRIPTION
This is a version of the terdragon curve with rounded-off corners,
=cut
# math-image --path=TerdragonRounded --all --output=numbers_dash --size=132x70
=pod
... 44----43 14
\ / \
46----45 . 42 13
/
. 40----41 12
/
39 . 24----23 20----19 11
\ / \ / \
. 38 25 . 22----21 . 18 10
/ \ /
36----37 . 26----27 . 16----17 9
/ \ /
35 . 32----31 . 28 15 . 8
\ / \ / \
34----33 30----29 . 14 7
/
. 12----13 . 6
/
11 . 8-----7 5
\ / \
10-----9 . 6 4
/
. 4-----5 3
/
3 2
\
. 2 1
/
. 0-----1 . <- Y=0
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
-8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8
The plain C<TerdragonCurve> is tripled in size and two points on each 3-long
edge are visited by the C<TerdragonRounded> here.
=head2 Arms
Multiple copies of the curve can be selected, each advancing successively.
The curve is 1/6 of the plane (like the plain terdragon) and 6 arms rotated
by 60, 120, 180, 240 and 300 degrees mesh together perfectly.
C<arms =E<gt> 6> begins as follows. N=0,6,12,18,etc is the first arm (the
curve shown above), then N=1,7,13,19 the second copy rotated 60 degrees,
N=2,8,14,20 the third rotated 120, etc.
=cut
# math-image --path=TerdragonRounded,arms=6 --all --output=numbers_dash --size=80x30
=pod
arms=>6 43----37 72--...
/ \ /
... 49 31 66 48----42
/ \ / \ / \
73 55 25 60----54 36
\ / \ /
67----61 19----13 24----30
\ /
38----32 14-----8 7 18 71---...
/ \ / \ / \ /
44 26----20 2 1 12 65
\ / \
50----56 9-----3 . 0-----6 59----53
\ / \
... 62 15 4 5 23----29 47
\ / \ / \ / \ /
74----68 21 10 11----17 35----41
/ \
33----27 16----22 64----70
/ \ / \
39 57----63 28 58 76
\ / \ / \ /
45----51 69 34 52 ...
/ \ /
...--75 40----46
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
-11-10-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9 10 11
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for the behaviour common to all path
classes.
=over 4
=item C<$path = Math::PlanePath::TerdragonRounded-E<gt>new ()>
=item C<$path = Math::PlanePath::TerdragonRounded-E<gt>new (arms =E<gt> $count)>
Create and return a new path object.
=item C<($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.
Fractional positions give an X,Y position along a straight line between the
integer positions.
=back
=head2 Level Methods
=over
=item C<($n_lo, $n_hi) = $path-E<gt>level_to_n_range($level)>
Return C<(0, 2 * 3**$level - 1)>, or for multiple arms return C<(0, 2 *
$arms * 3**$level - 1)>.
These level ranges are like C<TerdragonMidpoint> but with 2 points on each
line segment terdragon line segment instead of 1.
=back
=head1 FORMULAS
=head2 X,Y Visited
When arms=6 all "hex centred" points of the plane are visited, being those
points with
X+3Y mod 6 == 2 or 4 "hex_centred"
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::TerdragonCurve>,
L<Math::PlanePath::TerdragonMidpoint>,
L<Math::PlanePath::DragonRounded>
X<Arndt, Jorg>X<fxtbook>Jorg Arndt C<http://www.jjj.de/fxt/#fxtbook> section
1.31.4 "Terdragon and Hexdragon", where this rounded terdragon is called
hexdragon.
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-planepath/index.html>
=head1 LICENSE
Copyright 2012, 2013, 2014, 2015, 2016 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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