# Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017 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/>.
# points singles A052548 2^n + 2
# points doubles A000918 2^n - 2
# points triples A028243 3^(n-1) - 2*2^(n-1) + 1 cf A[k] = 2*3^(k-1) - 2*2^(k-1)
# T(3*N) = (w+1)*T(N) dir(N)=w^(2*count1digits)
# T(3*N+1) = (w+1)*T(N) + 1*dir(N)
# T(3*N+2) = (w+1)*T(N) + w*dir(N)
# T(0*3^k + N) = T(N)
# T(1*3^k + N) = 2^k + w^2*T(N) # rotate and offset
# T(2*3^k + N) = w*2^k + T(N) # offset only
package Math::PlanePath::TerdragonCurve;
use 5.004;
use strict;
use List::Util 'first';
use List::Util 'min'; # 'max'
*max = \&Math::PlanePath::_max;
use Math::PlanePath;
*_divrem_mutate = \&Math::PlanePath::_divrem_mutate;
use Math::PlanePath::Base::Generic
'is_infinite',
'round_nearest',
'xy_is_even';
use Math::PlanePath::Base::Digits
'digit_split_lowtohigh',
'digit_join_lowtohigh',
'round_up_pow';
use vars '$VERSION', '@ISA';
$VERSION = 125;
@ISA = ('Math::PlanePath');
use Math::PlanePath::TerdragonMidpoint;
# uncomment this to run the ### lines
# use Smart::Comments;
use constant n_start => 0;
use constant parameter_info_array =>
[ { name => 'arms',
share_key => 'arms_6',
display => 'Arms',
type => 'integer',
minimum => 1,
maximum => 6,
default => 1,
width => 1,
description => 'Arms',
} ];
{
my @x_negative_at_n = (undef, 13, 5, 5, 6, 7, 8);
sub x_negative_at_n {
my ($self) = @_;
return $x_negative_at_n[$self->{'arms'}];
}
}
{
my @y_negative_at_n = (undef, 159, 75, 20, 11, 9, 10);
sub y_negative_at_n {
my ($self) = @_;
return $y_negative_at_n[$self->{'arms'}];
}
}
sub dx_minimum {
my ($self) = @_;
return ($self->{'arms'} == 1 ? -1 : -2);
}
use constant dx_maximum => 2;
use constant dy_minimum => -1;
use constant dy_maximum => 1;
sub _UNDOCUMENTED__dxdy_list {
my ($self) = @_;
return ($self->{'arms'} == 1
? Math::PlanePath::_UNDOCUMENTED__dxdy_list_three()
: Math::PlanePath::_UNDOCUMENTED__dxdy_list_six());
}
{
my @_UNDOCUMENTED__dxdy_list_at_n = (undef, 4, 9, 13, 7, 8, 5);
sub _UNDOCUMENTED__dxdy_list_at_n {
my ($self) = @_;
return $_UNDOCUMENTED__dxdy_list_at_n[$self->{'arms'}];
}
}
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;
# arms=1 curve goes at 0,120,240 degrees
# arms=2 second +60 to 60,180,300 degrees
# so when arms==1 dir maximum is 240 degrees
sub dir_maximum_dxdy {
my ($self) = @_;
return ($self->{'arms'} == 1
? (-1,-1) # 0,2,4 only South-West
: ( 1,-1)); # rotated to 1,3,5 too South-East
}
use constant turn_any_straight => 0; # never straight
#------------------------------------------------------------------------------
sub new {
my $self = shift->SUPER::new(@_);
$self->{'arms'} = max(1, min(6, $self->{'arms'} || 1));
return $self;
}
my @dir6_to_si = (1,0,0, -1,0,0);
my @dir6_to_sj = (0,1,0, 0,-1,0);
my @dir6_to_sk = (0,0,1, 0,0,-1);
sub n_to_xy {
my ($self, $n) = @_;
### TerdragonCurve n_to_xy(): $n
if ($n < 0) { return; }
if (is_infinite($n)) { return ($n, $n); }
my $zero = ($n * 0); # inherit bignum 0
my $i = 0;
my $j = 0;
my $k = 0;
my $si = $zero;
my $sj = $zero;
my $sk = $zero;
# initial rotation from arm number
{
my $int = int($n);
my $frac = $n - $int; # inherit possible BigFloat
$n = $int; # BigFloat int() gives BigInt, use that
my $rot = _divrem_mutate ($n, $self->{'arms'});
my $s = $zero + 1; # inherit bignum 1
if ($rot >= 3) {
$s = -$s; # rotate 180
$frac = -$frac;
$rot -= 3;
}
if ($rot == 0) { $i = $frac; $si = $s; } # rotate 0
elsif ($rot == 1) { $j = $frac; $sj = $s; } # rotate +60
else { $k = $frac; $sk = $s; } # rotate +120
}
foreach my $digit (digit_split_lowtohigh($n,3)) {
### at: "$i,$j,$k side $si,$sj,$sk"
### $digit
if ($digit == 1) {
($i,$j,$k) = ($si-$j, $sj-$k, $sk+$i); # rotate +120 and add
} elsif ($digit == 2) {
$i -= $sk; # add rotated +60
$j += $si;
$k += $sj;
}
# add rotated +60
($si,$sj,$sk) = ($si - $sk,
$sj + $si,
$sk + $sj);
}
### final: "$i,$j,$k side $si,$sj,$sk"
### is: (2*$i + $j - $k).",".($j+$k)
return (2*$i + $j - $k, $j+$k);
}
# all even points when arms==6
sub xy_is_visited {
my ($self, $x, $y) = @_;
if ($self->{'arms'} == 6) {
return xy_is_even($self,$x,$y);
} else {
return defined($self->xy_to_n($x,$y));
}
}
# maximum extent -- no, not quite right
#
# .----*
# \
# *----.
#
# Two triangle heights, so
# rnext = 2 * r * sqrt(3)/2
# = r * sqrt(3)
# rsquared_next = 3 * rsquared
# Initial X=2,Y=0 is rsquared=4
# then X=3,Y=1 is 3*3+3*1*1 = 9+3 = 12 = 4*3
# then X=3,Y=3 is 3*3+3*3*3 = 9+3 = 36 = 4*3^2
#
my @try_dx = (2, 1, -1, -2, -1, 1);
my @try_dy = (0, 1, 1, 0, -1, -1);
sub xy_to_n {
return scalar((shift->xy_to_n_list(@_))[0]);
}
sub xy_to_n_list {
my ($self, $x, $y) = @_;
### TerdragonCurve xy_to_n_list(): "$x, $y"
$x = round_nearest($x);
$y = round_nearest($y);
if (is_infinite($x)) {
return $x; # infinity
}
if (is_infinite($y)) {
return $y; # infinity
}
my @n_list;
my $xm = 2*$x; # doubled out
my $ym = 2*$y;
foreach my $i (0 .. $#try_dx) {
my $t = $self->Math::PlanePath::TerdragonMidpoint::xy_to_n
($xm+$try_dx[$i], $ym+$try_dy[$i]);
### try: ($xm+$try_dx[$i]).",".($ym+$try_dy[$i])
### $t
next unless defined $t;
# function call here to get our n_to_xy(), not the overridden method
# when in TerdragonRounded or other subclass
my ($tx,$ty) = n_to_xy($self,$t)
or next;
if ($tx == $x && $ty == $y) {
### found: $t
if (@n_list && $t < $n_list[0]) {
unshift @n_list, $t;
} elsif (@n_list && $t < $n_list[-1]) {
splice @n_list, -1,0, $t;
} else {
push @n_list, $t;
}
if (@n_list == 3) {
return @n_list;
}
}
}
return @n_list;
}
# minimum -- no, not quite right
#
# *----------*
# \
# \ *
# * \
# \
# *----------*
#
# width = side/2
# minimum = side*sqrt(3)/2 - width
# = side*(sqrt(3)/2 - 1)
#
# minimum 4/9 * 2.9^level roughly
# h = 4/9 * 2.9^level
# 2.9^level = h*9/4
# level = log(h*9/4)/log(2.9)
# 3^level = 3^(log(h*9/4)/log(2.9))
# = h*9/4, but big bigger for log
#
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
### TerdragonCurve rect_to_n_range(): "$x1,$y1 $x2,$y2"
my $xmax = int(max(abs($x1),abs($x2)));
my $ymax = int(max(abs($y1),abs($y2)));
return (0,
($xmax*$xmax + 3*$ymax*$ymax + 1)
* 2
* $self->{'arms'});
}
my @dir6_to_dx = (2, 1,-1,-2, -1, 1);
my @dir6_to_dy = (0, 1, 1, 0, -1,-1);
my @digit_to_nextturn = (2,-2);
sub n_to_dxdy {
my ($self, $n) = @_;
### n_to_dxdy(): $n
if ($n < 0) {
return; # first direction at N=0
}
if (is_infinite($n)) {
return ($n,$n);
}
my $int = int($n); # integer part
$n -= $int; # fraction part
# initial direction from arm
my $dir6 = _divrem_mutate ($int, $self->{'arms'});
my @ndigits = digit_split_lowtohigh($int,3);
$dir6 += 2 * scalar(grep {$_==1} @ndigits); # count 1s for total turn
$dir6 %= 6;
my $dx = $dir6_to_dx[$dir6];
my $dy = $dir6_to_dy[$dir6];
if ($n) {
# fraction part
# find lowest non-2 digit, or zero if all 2s or no digits at all
$dir6 += $digit_to_nextturn[ first {$_!=2} @ndigits, 0];
$dir6 %= 6;
$dx += $n*($dir6_to_dx[$dir6] - $dx);
$dy += $n*($dir6_to_dy[$dir6] - $dy);
}
return ($dx, $dy);
}
#-----------------------------------------------------------------------------
# eg. arms=5 0 .. 5*3^k step by 5s
# 1 .. 5*3^k+1 step by 5s
# 4 .. 5*3^k+4 step by 5s
#
sub level_to_n_range {
my ($self, $level) = @_;
return (0, (3**$level + 1) * $self->{'arms'} - 1);
}
sub n_to_level {
my ($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, 3);
return $exp;
}
#-----------------------------------------------------------------------------
# right boundary N
# mixed radix binary, ternary
# no 11, 12, 20
# 11 -> 21, including low digit
# run of 11111 becomes 22221
# low to high 1 or 0 <- 0 cannot 20 can 10 00
# 2 or 0 <- 1 cannot 11 can 21 01
# 2 or 0 <- 2 cannot 12 can 02 22
sub _UNDOCUMENTED__right_boundary_i_to_n {
my ($self, $i) = @_;
my @digits = _digit_split_mix23_lowtohigh($i);
for (my $i = $#digits; $i >= 1; $i--) { # high to low
if ($digits[$i] == 1 && $digits[$i-1] != 0) {
$digits[$i] = 2;
}
}
return digit_join_lowtohigh(\@digits, 3, $i*0);
# {
# for (my $i = 0; $i < $#digits; $i++) { # low to high
# if ($digits[$i+1] == 1 && ($digits[$i] == 1 || $digits[$i] == 2)) {
# $digits[$i+1] = 2;
# }
# }
# return digit_join_lowtohigh(\@digits,3);
# }
}
# Return a list of digits, low to high, which is a mixed radix
# representation low digit ternary and the rest binary.
sub _digit_split_mix23_lowtohigh {
my ($n) = @_;
if ($n == 0) {
return ();
}
my $low = _divrem_mutate($n,3);
return ($low, digit_split_lowtohigh($n,2));
}
{
# disallowed digit pairs $disallowed[high][low]
my @disallowed;
$disallowed[1][1] = 1;
$disallowed[1][2] = 1;
$disallowed[2][0] = 1;
sub _UNDOCUMENTED__n_segment_is_right_boundary {
my ($self, $n) = @_;
if (is_infinite($n)) { return 0; }
unless ($n >= 0) { return 0; }
$n = int($n);
# no boundary when arms=6, right boundary is only in arm 0
{
my $arms = $self->{'arms'};
if ($arms == 6) { return 0; }
if (_divrem_mutate($n,$arms)) { return 0; }
}
my $prev = _divrem_mutate($n,3);
while ($n) {
my $digit = _divrem_mutate($n,3);
if ($disallowed[$digit][$prev]) {
return 0;
}
$prev = $digit;
}
return 1;
}
}
#-----------------------------------------------------------------------------
# left boundary N
# mixed 0,1, 2, 10, 11, 12, 100, 101, 102, 110, 111, 112, 1000, 1001, 1002, 1010, 1011, 1012, 1100, 1101, 1102,
# vals 0,1,12,120,121,122,1200,1201,1212,1220,1221,1222,12000,12001,12012,12120,12121,12122,12200,12201,12212,
{
my @_UNDOCUMENTED__left_boundary_i_to_n = ([0,2], # 0
[0,2], # 1
[1,2]); # 2
sub _UNDOCUMENTED__left_boundary_i_to_n {
my ($self, $i, $level) = @_;
### _UNDOCUMENTED__left_boundary_i_to_n(): $i
### $level
if (defined $level && $level < 0) {
if ($i <= 2) {
return $i;
}
$i += 2;
}
my @digits = _digit_split_mix23_lowtohigh($i);
### @digits
if (defined $level) {
if ($level >= 0) {
if (@digits > $level) {
### beyond given level ...
return undef;
}
# pad for $level, total $level many digits
push @digits, (0) x ($level - scalar(@digits));
} else {
### union all levels ...
pop @digits;
if ($digits[-1]) {
push @digits, 0; # high 0,1 or 0,2 when i=3
} else {
$digits[-1] = 1; # high 1
}
}
} else {
### infinite curve, an extra high 0 ...
push @digits, 0;
}
### @digits
my $prev = $digits[0];
foreach my $i (1 .. $#digits) {
$prev = $digits[$i] = $_UNDOCUMENTED__left_boundary_i_to_n[$prev][$digits[$i]];
}
### ternary: @digits
return digit_join_lowtohigh(\@digits, 3, $i*0);
}
}
{
# disallowed digit pairs $disallowed[high][low]
my @disallowed;
$disallowed[0][2] = 1;
$disallowed[1][0] = 1;
$disallowed[1][1] = 1;
sub _UNDOCUMENTED__n_segment_is_left_boundary {
my ($self, $n, $level) = @_;
### _UNDOCUMENTED__n_segment_is_left_boundary(): $n
### $level
if (is_infinite($n)) { return 0; }
unless ($n >= 0) { return 0; }
$n = int($n);
if (defined $level && $level == 0) {
### level 0 curve, N=0 is only segment: ($n == 0)
return ($n == 0);
}
{
my $arms = $self->{'arms'};
if ($arms == 6) {
return 0;
}
my $arm = _divrem_mutate($n,$arms);
if ($arm != $arms-1) {
return 0;
}
}
my $prev = _divrem_mutate($n,3);
if (defined $level) { $level -= 1; }
for (;;) {
if (defined $level && $level == 0) {
### end of level many digits, must be N < 3**$level
return ($n == 0);
}
last unless $n;
my $digit = _divrem_mutate($n,3);
if ($disallowed[$digit][$prev]) {
return 0;
}
if (defined $level) { $level -= 1; }
$prev = $digit;
}
return ((defined $level && $level < 0) # union all levels
|| ($prev != 2)); # not high 2 otherwise
}
sub _UNDOCUMENTED__n_segment_is_any_left_boundary {
my ($self, $n) = @_;
my $prev = _divrem_mutate($n,3);
while ($n) {
my $digit = _divrem_mutate($n,3);
if ($disallowed[$digit][$prev]) {
return 0;
}
$prev = $digit;
}
return 1;
}
# sub left_boundary_n_pred {
# my ($n) = @_;
# my $n3 = '0' . Math::BaseCnv::cnv($n,10,3);
# return ($n3 =~ /02|10|11/ ? 0 : 1);
# }
}
sub _UNDOCUMENTED__n_segment_is_boundary {
my ($self, $n, $level) = @_;
return $self->_UNDOCUMENTED__n_segment_is_right_boundary($n)
|| $self->_UNDOCUMENTED__n_segment_is_left_boundary($n,$level);
}
1;
__END__
# old n_to_xy()
#
# # initial rotation from arm number
# my $arms = $self->{'arms'};
# my $rot = $n % $arms;
# $n = int($n/$arms);
# my @digits;
# my (@si, @sj, @sk); # vectors
# {
# my $si = $zero + 1; # inherit bignum 1
# my $sj = $zero; # inherit bignum 0
# my $sk = $zero; # inherit bignum 0
#
# for (;;) {
# push @digits, ($n % 3);
# push @si, $si;
# push @sj, $sj;
# push @sk, $sk;
# ### push: "digit $digits[-1] $si,$sj,$sk"
#
# $n = int($n/3) || last;
#
# # straight + rot120 + straight
# ($si,$sj,$sk) = (2*$si - $sj,
# 2*$sj - $sk,
# 2*$sk + $si);
# }
# }
# ### @digits
#
# my $i = $zero;
# my $j = $zero;
# my $k = $zero;
# while (defined (my $digit = pop @digits)) { # digits high to low
# my $si = pop @si;
# my $sj = pop @sj;
# my $sk = pop @sk;
# ### at: "$i,$j,$k $digit side $si,$sj,$sk"
# ### $rot
#
# $rot %= 6;
# if ($rot == 1) { ($si,$sj,$sk) = (-$sk,$si,$sj); }
# elsif ($rot == 2) { ($si,$sj,$sk) = (-$sj,-$sk,$si); }
# elsif ($rot == 3) { ($si,$sj,$sk) = (-$si,-$sj,-$sk); }
# elsif ($rot == 4) { ($si,$sj,$sk) = ($sk,-$si,-$sj); }
# elsif ($rot == 5) { ($si,$sj,$sk) = ($sj,$sk,-$si); }
#
# if ($digit) {
# $i += $si; # digit=1 or digit=2
# $j += $sj;
# $k += $sk;
# if ($digit == 2) {
# $i -= $sj; # digit=2, straight+rot120
# $j -= $sk;
# $k += $si;
# } else {
# $rot += 2; # digit=1
# }
# }
# }
#
# $rot %= 6;
# $i = $frac * $dir6_to_si[$rot] + $i;
# $j = $frac * $dir6_to_sj[$rot] + $j;
# $k = $frac * $dir6_to_sk[$rot] + $k;
#
# ### final: "$i,$j,$k"
# return (2*$i + $j - $k, $j+$k);
=for stopwords eg Ryde Dragon Math-PlanePath Nlevel Knuth et al vertices doublings OEIS Online terdragon ie morphism si,sj,sk dX,dY Pari rhombi dX si
=head1 NAME
Math::PlanePath::TerdragonCurve -- triangular dragon curve
=head1 SYNOPSIS
use Math::PlanePath::TerdragonCurve;
my $path = Math::PlanePath::TerdragonCurve->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
X<Davis>X<Knuth, Donald>This is the terdragon curve by Davis and Knuth,
=over
Chandler Davis and Donald Knuth, "Number Representations and Dragon Curves
-- I", Journal Recreational Mathematics, volume 3, number 2 (April 1970),
pages 66-81 and "Number Representations and Dragon Curves -- II", volume 3,
number 3 (July 1970), pages 133-149.
Reprinted with addendum in Knuth "Selected Papers on Fun and Games", 2010,
pages 571--614. L<http://www-cs-faculty.stanford.edu/~uno/fg.html>
=back
Points are a triangular grid using every second integer X,Y as per
L<Math::PlanePath/Triangular Lattice>, beginning
\ / \
--- 26,29,32 ---------- 27 6
/ \
\ / \
-- 24,33,42 ---------- 22,25 5
/ \ / \
\ / \
--- 20,23,44 -------- 12,21 10 4
/ \ / \ / \
\ / \ / \ / \
18,45 --------- 13,16,19 ------ 8,11,14 -------- 9 3
\ / \ / \
\ / \ / \
17 6,15 --------- 4,7 2
\ / \
\ / \
2,5 ---------- 3 1
\
\
0 ----------- 1 <-Y=0
^ ^ ^ ^ ^ ^ ^
-3 -2 -1 X=0 1 2 3
The base figure is an "S" shape
2-----3
\
\
0-----1
which then repeats in self-similar style, so N=3 to N=6 is a copy rotated
+120 degrees, which is the angle of the N=1 to N=2 edge,
6 4 base figure repeats
\ / \ as N=3 to N=6,
\/ \ rotated +120 degrees
5 2----3
\
\
0-----1
Then N=6 to N=9 is a plain horizontal, which is the angle of N=2 to N=3,
8-----9 base figure repeats
\ as N=6 to N=9,
\ no rotation
6----7,4
\ / \
\ / \
5,2----3
\
\
0-----1
Notice X=1,Y=1 is visited twice as N=2 and N=5. Similarly X=2,Y=2 as N=4
and N=7. Each point can repeat up to 3 times. "Inner" points are 3 times
and on the edges up to 2 times. The first tripled point is X=1,Y=3 which as
shown above is N=8, N=11 and N=14.
The curve never crosses itself. The vertices touch as triangular corners
and no edges repeat.
The curve turns are the same as the C<GosperSide>, but here the turns are by
120 degrees each whereas C<GosperSide> is 60 degrees each. The extra angle
here tightens up the shape.
=head2 Spiralling
The first step N=1 is to the right along the X axis and the path then slowly
spirals anti-clockwise and progressively fatter. The end of each
replication is
Nlevel = 3^level
That point is at level*30 degrees around (as reckoned with Y*sqrt(3) for a
triangular grid).
Nlevel X, Y Angle (degrees)
------ ------- -----
1 1, 0 0
3 3, 1 30
9 3, 3 60
27 0, 6 90
81 -9, 9 120
243 -27, 9 150
729 -54, 0 180
The following is points N=0 to N=3^6=729 going half-circle around to 180
degrees. The N=0 origin is marked "0" and the N=729 end is marked "E".
=cut
# the following generated by
# math-image --path=TerdragonCurve --expression='i<=729?i:0' --text --size=132x40
=pod
* * * *
* * * * * * * *
* * * * * * * *
* * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * *
* E * * * * * * * * * * * * * * * * 0 *
* * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * *
* * * * * * * *
* * * * * * * *
* * * *
=head2 Tiling
The little "S" shapes of the base figure N=0 to N=3 can be thought of as a
rhombus
2-----3
. .
. .
0-----1
The "S" shapes of each 3 points make a tiling of the plane with those rhombi
\ \ / / \ \ / /
*-----*-----* *-----*-----*
/ / \ \ / / \ \
\ / / \ \ / / \ \ /
--*-----* *-----*-----* *-----*--
/ \ \ / / \ \ / / \
\ \ / / \ \ / /
*-----*-----* *-----*-----*
/ / \ \ / / \ \
\ / / \ \ / / \ \ /
--*-----* *-----o-----* *-----*--
/ \ \ / / \ \ / / \
\ \ / / \ \ / /
*-----*-----* *-----*-----*
/ / \ \ / / \ \
Which is an ancient pattern,
=over
L<http://tilingsearch.org/HTML/data23/C07A.html>
=back
=head2 Arms
The curve fills a sixth of the plane and six copies rotated by 60, 120, 180,
240 and 300 degrees mesh together perfectly. The C<arms> parameter can
choose 1 to 6 such curve arms successively advancing.
For example C<arms =E<gt> 6> begins as follows. N=0,6,12,18,etc is the
first arm (the same shape as the plain curve above), then N=1,7,13,19 the
second, N=2,8,14,20 the third, etc.
\ / \ /
\ / \ /
--- 8/13/31 ---------------- 7/12/30 ---
/ \ / \
\ / \ / \ /
\ / \ / \ /
--- 9/14/32 ------------- 0/1/2/3/4/5 -------------- 6/17/35 ---
/ \ / \ / \
/ \ / \ / \
\ / \ /
--- 10/15/33 ---------------- 11/16/34 ---
/ \ / \
/ \ / \
With six arms every X,Y point is visited three times, except the origin 0,0
where all six begin. Every edge between points is traversed once.
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::TerdragonCurve-E<gt>new ()>
=item C<$path = Math::PlanePath::TerdragonCurve-E<gt>new (arms =E<gt> 6)>
Create and return a new path object.
The optional C<arms> parameter can make 1 to 6 copies of the curve, each arm
successively advancing.
=item C<($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.
=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,$y> then return C<undef>.
The curve can visit an C<$x,$y> up to three times. C<xy_to_n()> returns the
smallest of the these N values.
=item C<@n_list = $path-E<gt>xy_to_n_list ($x,$y)>
Return a list of N point numbers for coordinates C<$x,$y>.
The origin 0,0 has C<arms_count()> many N since it's the starting point for
each arm. Other points have up to 3 Ns for a given C<$x,$y>. If arms=6
then every C<$x,$y> except the origin has exactly 3 Ns.
=back
=head2 Descriptive Methods
=over
=item C<$n = $path-E<gt>n_start()>
Return 0, the first N in the path.
=item C<$dx = $path-E<gt>dx_minimum()>
=item C<$dx = $path-E<gt>dx_maximum()>
=item C<$dy = $path-E<gt>dy_minimum()>
=item C<$dy = $path-E<gt>dy_maximum()>
The dX,dY values on the first arm take three possible combinations, being
120 degree angles.
dX,dY for arms=1
-----
2, 0 dX minimum = -1, maximum = +2
-1, 1 dY minimum = -1, maximum = +1
1,-1
For 2 or more arms the second arm is rotated by 60 degrees so giving the
following additional combinations, for a total six. This changes the dX
minimum.
dX,dY for arms=2 or more
-----
-2, 0 dX minimum = -2, maximum = +2
1, 1 dY minimum = -1, maximum = +1
-1,-1
=back
=head2 Level Methods
=over
=item C<($n_lo, $n_hi) = $path-E<gt>level_to_n_range($level)>
Return C<(0, 3**$level)>, or for multiple arms return C<(0, $arms *
3**$level + ($arms-1))>.
There are 3^level segments in a curve level, so 3^level+1 points numbered
from 0. For multiple arms there are arms*(3^level+1) points, numbered from
0 so n_hi = arms*(3^level+1)-1.
=back
=head1 FORMULAS
Various formulas for boundary length, area and more can be found in the
author's mathematical write-up
=over
L<http://user42.tuxfamily.org/terdragon/index.html>
=back
=head2 N to X,Y
There's no reversals or reflections in the curve so C<n_to_xy()> can take
the digits of N either low to high or high to low and apply what is
effectively powers of the N=3 position. The current code goes low to high
using i,j,k coordinates as described in L<Math::PlanePath/Triangular
Calculations>.
si = 1 # position of endpoint N=3^level
sj = 0 # where level=number of digits processed
sk = 0
i = 0 # position of N for digits so far processed
j = 0
k = 0
loop base 3 digits of N low to high
if digit == 0
i,j,k no change
if digit == 1
(i,j,k) = (si-j, sj-k, sk+i) # rotate +120, add si,sj,sk
if digit == 2
i -= sk # add (si,sj,sk) rotated +60
j += si
k += sj
(si,sj,sk) = (si - sk, # add rotated +60
sj + si,
sk + sj)
The digit handling is a combination of rotate and offset,
digit==1 digit 2
rotate and offset offset at si,sj,sk rotated
^ 2------>
\
\ \
*--- --1 *-- --*
The calculation can also be thought of in term of w=1/2+I*sqrt(3)/2, a
complex number sixth root of unity. i is the real part, j in the w
direction (60 degrees), and k in the w^2 direction (120 degrees). si,sj,sk
increase as if multiplied by w+1.
=head2 Turn
At each point N the curve always turns 120 degrees either to the left or
right, it never goes straight ahead. If N is written in ternary then the
lowest non-zero digit gives the turn
ternary lowest
non-zero digit turn
-------------- -----
1 left
2 right
At N=3^level or N=2*3^level the turn follows the shape at that 1 or 2 point.
The first and last unit step in each level are in the same direction, so the
next level shape gives the turn.
2*3^k-------3*3^k
\
\
0-------1*3^k
=head2 Next Turn
The next turn, ie. the turn at position N+1, can be calculated from the
ternary digits of N similarly. The lowest non-2 digit gives the turn.
ternary lowest
non-2 digit turn
-------------- -----
0 left
1 right
If N is all 2s then the lowest non-2 is taken to be a 0 above the high end.
For example N=8 is 22 ternary so considered 022 for lowest non-2 digit=0 and
turn left after the segment at N=8, ie. at point N=9 turn left.
This rule works for the same reason as the plain turn above. The next turn
of N is the plain turn of N+1 and adding +1 turns trailing 2s into trailing
0s and increments the 0 or 1 digit above them to be 1 or 2.
=head2 Total Turn
The direction at N, ie. the total cumulative turn, is given by the number of
1 digits when N is written in ternary,
direction = (count 1s in ternary N) * 120 degrees
For example N=12 is ternary 110 which has two 1s so the cumulative turn at
that point is 2*120=240 degrees, ie. the segment N=16 to N=17 is at angle
240.
The segments for digit 0 or 2 are in the "current" direction unchanged. The
segment for digit 1 is rotated +120 degrees.
=head2 X,Y to N
The current code applies C<TerdragonMidpoint> C<xy_to_n()> to calculate six
candidate N from the six edges around a point. Those N values which convert
back to the target X,Y by C<n_to_xy()> are the results for
C<xy_to_n_list()>.
The six edges are three going towards the point and three going away. The
midpoint calculation gives N-1 for the towards and N for the away. Is there
a good way to tell which edge will be the smaller? Or just which 3 edges
lead away? It would be directions 0,2,4 for the even arms and 1,3,5 for the
odd ones, but identifying the boundaries of those arms to know which is
which is difficult.
=head2 X,Y Visited
When arms=6 all "even" points of the plane are visited. As per the
triangular representation of X,Y this means
X+Y mod 2 == 0 "even" points
=head1 OEIS
The terdragon is in Sloane's Online Encyclopedia of Integer Sequences as,
=over
L<http://oeis.org/A080846> (etc)
=back
A080846 next turn 0=left,1=right, by 120 degrees
(n=0 is turn at N=1)
A060236 turn 1=left,2=right, by 120 degrees
(lowest non-zero ternary digit)
A137893 turn 1=left,0=right (morphism)
A189673 turn 1=left,0=right (morphism, extra initial 0)
A189640 turn 0=left,1=right (morphism, extra initial 0)
A038502 strip trailing ternary 0s,
taken mod 3 is turn 1=left,2=right
A133162 1=segment, 2=right turn between
A189673 and A026179 start with extra initial values arising from their
morphism definition. That can be skipped to consider the turns starting
with a left turn at N=1.
A026225 N positions of left turns,
being (3*i+1)*3^j so lowest non-zero digit is a 1
A026179 N positions of right turns (except initial 1)
A060032 bignum turns 1=left,2=right to 3^level
A189674 num left turns 1 to N
A189641 num right turns 1 to N
A189672 same
A026141 \ dN increment between left turns N
A026171 /
A026181 \ dN increment between left turns N
A131989 /
A062756 total turn, count ternary 1s
A005823 N positions where net turn == 0, ternary no 1s
A111286 boundary length, N=0 to N=3^k, skip initial 1
A003945 boundary/2
A002023 boundary odd levels N=0 to N=3^(2k+1),
or even levels one side N=0 to N=3^(2k),
being 6*4^k
A164346 boundary even levels N=0 to N=3^(2k),
or one side, odd levels, N=0 to N=3^(2k+1),
being 3*4^k
A042950 V[k] boundary length
A056182 area enclosed N=0 to N=3^k, being 2*(3^k-2^k)
A081956 same
A118004 1/2 area N=0 to N=3^(2k+1), odd levels, 9^n-4^n
A155559 join area, being 0 then 2^k
A099754 1/2 count distinct visited points N=0 to N=3^k
A092236 count East segments N=0 to N=3^k-1
A135254 count North-West segments N=0 to N=3^k-1, extra 0
A133474 count South-West segments N=0 to N=3^k-1
A057083 count segments diff from 3^(k-1)
A101990 count segments same dir as middle N=0 to N=3^k-1
A097038 num runs of 12 consecutive segments within N=0 to 3^k-1
each segment enclosing a new unit triangle
A057682 level X, at N=3^level
also arms=2 level Y, at N=2*3^level
A057083 level Y, at N=3^level
also arms=6 level X at N=6*3^level
A057681 arms=2 level X, at N=2*3^level
also arms=3 level Y at 3*3^level
A103312 same
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::TerdragonRounded>,
L<Math::PlanePath::TerdragonMidpoint>,
L<Math::PlanePath::GosperSide>
L<Math::PlanePath::DragonCurve>,
L<Math::PlanePath::R5DragonCurve>
Larry Riddle's Terdragon page, for boundary and area calculations of the
terdragon as an infinite fractal
L<http://ecademy.agnesscott.edu/~lriddle/ifs/heighway/terdragon.htm>
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-planepath/index.html>
=head1 LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017 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