# 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=DragonMidpoint --lines --scale=20
# math-image --path=DragonMidpoint --all --output=numbers_dash
# A006466 contfrac 2*sum( 1/2^(2^n)), 1 and 2 only
# a(5n) recurrence ...
# 1,1,1,1, 2,
# 1,1,1,1,1,1,1, 2,
# 1,1,1,1, 2,
# 1,1,1,1, 2,
# 1, 2,
# 1,1,1,1, 2,
# 1,1,1,1,1,1,1, 2,
# 1,1,1,1, 2,
# 1, 2,
# 1,1,1,1,1,1,1, 2,
# 1,1,1,1, 2,
# 1, 2,
# 1,1,1,1, 2,
# 1,1,1,1, 2,
# 1,1,1,1,1,1,1, 2,
# 1,1,1,1, 2,
# 1, 2,
# 1,1,1,1,1,1,1, 2,
# 1,1,1,1, 2,
# 1,1,1,1, 2,
# 1, 2
# A076214 in decimal
#
# A073097 number of 4s - 6s - 2s - 1 is -1,0,1
# A081769 positions of 2s
# A073088 cumulative total multiples of 4 roughly, hence (4n-3-cum)/2
#
# A088435 (contfrac+1)/2 of sum(k>=1,1/3^(2^k)).
# A007404 in decimal
#
package Math::PlanePath::DragonMidpoint;
use 5.004;
use strict;
use List::Util 'min'; # 'max'
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 115;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
*_divrem_mutate = \&Math::PlanePath::_divrem_mutate;
use Math::PlanePath::Base::Generic
'is_infinite',
'round_nearest';
use Math::PlanePath::Base::Digits
'bit_split_lowtohigh',
'digit_join_lowtohigh';
# uncomment this to run the ### lines
# use Smart::Comments;
# whole plane when arms==4
use Math::PlanePath::DragonCurve;
use constant n_start => 0;
use constant parameter_info_array => [ { name => 'arms',
share_key => 'arms_4',
display => 'Arms',
type => 'integer',
minimum => 1,
maximum => 4,
default => 1,
width => 1,
description => 'Arms',
} ];
{
my @_UNDOCUMENTED__x_negative_at_n = (undef, 6,5,2,2);
sub _UNDOCUMENTED__x_negative_at_n {
my ($self) = @_;
return $_UNDOCUMENTED__x_negative_at_n[$self->{'arms'}];
}
}
{
my @_UNDOCUMENTED__y_negative_at_n = (undef, 27,19,11,7);
sub _UNDOCUMENTED__y_negative_at_n {
my ($self) = @_;
return $_UNDOCUMENTED__y_negative_at_n[$self->{'arms'}];
}
}
use constant dx_minimum => -1;
use constant dx_maximum => 1;
use constant dy_minimum => -1;
use constant dy_maximum => 1;
*_UNDOCUMENTED__dxdy_list = \&Math::PlanePath::_UNDOCUMENTED__dxdy_list_four;
{
my @_UNDOCUMENTED__dxdy_list_at_n = (undef, 9, 9, 5, 3);
sub _UNDOCUMENTED__dxdy_list_at_n {
my ($self) = @_;
return $_UNDOCUMENTED__dxdy_list_at_n[$self->{'arms'}];
}
}
use constant dsumxy_minimum => -1; # straight only
use constant dsumxy_maximum => 1;
use constant ddiffxy_minimum => -1;
use constant ddiffxy_maximum => 1;
use constant dir_maximum_dxdy => (0,-1); # South
#------------------------------------------------------------------------------
sub new {
my $self = shift->SUPER::new(@_);
$self->{'arms'} = max(1, min(4, $self->{'arms'} || 1));
return $self;
}
# sub n_to_xy {
# my ($self, $n) = @_;
# ### DragonMidpoint n_to_xy(): $n
#
# if ($n < 0) { return; }
# if (is_infinite($n)) { return ($n, $n); }
#
# {
# my $int = int($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 = $int; # BigFloat int() gives BigInt, use that
# }
#
# my ($x1,$y1) = Math::PlanePath::DragonCurve->n_to_xy($n);
# my ($x2,$y2) = Math::PlanePath::DragonCurve->n_to_xy($n+1);
#
# my $dx = $x2-$x1;
# my $dy = $y2-$y1;
# return ($x1+$y1 + ($dx+$dy-1)/2,
# $y1-$x1 + ($dy-$dx+1)/2);
# }
sub n_to_xy {
my ($self, $n) = @_;
### DragonMidpoint n_to_xy(): $n
if ($n < 0) { return; }
if (is_infinite($n)) { return ($n, $n); }
my $frac;
{
my $int = int($n);
$frac = $n - $int; # inherit possible BigFloat
$n = $int; # BigFloat int() gives BigInt, use that
}
my $zero = ($n * 0); # inherit bignum 0
# arm as initial rotation
my $rot = _divrem_mutate ($n, $self->{'arms'});
### $arms
### rot from arm: $rot
### $n
# ENHANCE-ME: sx,sy just from len,len
my @digits = bit_split_lowtohigh($n);
my @sx;
my @sy;
{
my $sx = $zero + 1;
my $sy = -$sx;
foreach (@digits) {
push @sx, $sx;
push @sy, $sy;
# (sx,sy) + rot+90(sx,sy)
($sx,$sy) = ($sx - $sy,
$sy + $sx);
}
}
### @digits
my $rev = 0;
my $x = $zero;
my $y = $zero;
my $above_low_zero = 0;
for (my $i = $#digits; $i >= 0; $i--) { # high to low
my $digit = $digits[$i];
my $sx = $sx[$i];
my $sy = $sy[$i];
### at: "$x,$y $digit side $sx,$sy"
### $rot
if ($rot & 2) {
$sx = -$sx;
$sy = -$sy;
}
if ($rot & 1) {
($sx,$sy) = (-$sy,$sx);
}
### rotated side: "$sx,$sy"
if ($rev) {
if ($digit) {
$x -= $sy;
$y += $sx;
### rev add to: "$x,$y next is still rev"
} else {
$above_low_zero = $digits[$i+1];
$rot ++;
$rev = 0;
### rev rot, next is no rev ...
}
} else {
if ($digit) {
$rot ++;
$x += $sx;
$y += $sy;
$rev = 1;
### plain add to: "$x,$y next is rev"
} else {
$above_low_zero = $digits[$i+1];
}
}
}
# Digit above the low zero is the direction of the next turn, 0 for left,
# 1 for right.
#
### final: "$x,$y rot=$rot above_low_zero=".($above_low_zero||0)
if ($rot & 2) {
$frac = -$frac; # rotate 180
$x -= 1;
}
if (($rot+1) & 2) {
# rot 1 or 2
$y += 1;
}
if (!($rot & 1) && $above_low_zero) {
$frac = -$frac;
}
$above_low_zero ^= ($rot & 1);
if ($above_low_zero) {
$y = $frac + $y;
} else {
$x = $frac + $x;
}
### rotated return: "$x,$y"
return ($x,$y);
}
# or tables arithmetically,
#
# my $ax = ((($x+1) ^ ($y+1)) >> 1) & 1;
# my $ay = (($x^$y) >> 1) & 1;
# ### assert: $ax == - $yx_adj_x[$y%4]->[$x%4]
# ### assert: $ay == - $yx_adj_y[$y%4]->[$x%4]
#
my @yx_adj_x = ([0,1,1,0],
[1,0,0,1],
[1,0,0,1],
[0,1,1,0]);
my @yx_adj_y = ([0,0,1,1],
[0,0,1,1],
[1,1,0,0],
[1,1,0,0]);
# arm $x $y 2 | 1 Y=1
# 0 0 0 3 | 0 Y=0
# 1 0 1 ----+----
# 2 -1 1 X=-1 X=0
# 3 -1 0
my @xy_to_arm = ([0, # x=0,y=0
1], # x=0,y=1
[3, # x=-1,y=0
2]); # x=-1,y=1
sub xy_to_n {
my ($self, $x, $y) = @_;
### DragonMidpoint xy_to_n(): "$x, $y"
$x = round_nearest($x);
$y = round_nearest($y);
{ my $overflow = abs($x)+abs($y)+2;
if (is_infinite($overflow)) { return $overflow; }
}
my $zero = ($x * 0 * $y);
my @nbits; # low to high
while ($x < -1 || $x > 0 || $y < 0 || $y > 1) {
my $y4 = $y % 4;
my $x4 = $x % 4;
my $ax = $yx_adj_x[$y4]->[$x4];
my $ay = $yx_adj_y[$y4]->[$x4];
### at: "$x,$y n=$n axy=$ax,$ay bit=".($ax^$ay)
push @nbits, $ax^$ay;
$x -= $ax;
$y -= $ay;
### assert: ($x+$y)%2 == 0
($x,$y) = (($x+$y)/2, # rotate -45 and divide sqrt(2)
($y-$x)/2);
}
### final: "xy=$x,$y"
my $arm = $xy_to_arm[$x]->[$y];
### $arm
my $arms_count = $self->arms_count;
if ($arm >= $arms_count) {
return undef;
}
if ($arm & 1) {
### flip ...
@nbits = map {$_^1} @nbits;
}
return digit_join_lowtohigh(\@nbits, 2, $zero) * $arms_count + $arm;
}
#------------------------------------------------------------------------------
# xy_is_visited()
sub xy_is_visited {
my ($self, $x, $y) = @_;
return ($self->{'arms'} >= 4
|| _xy_to_arm($x,$y) < $self->{'arms'});
}
# return arm number 0,1,2,3
sub _xy_to_arm {
my ($x, $y) = @_;
### DragonMidpoint _xy_to_arm(): "$x, $y"
$x = round_nearest($x);
$y = round_nearest($y);
{ my $overflow = abs($x)+abs($y)+2;
if (is_infinite($overflow)) { return $overflow; }
}
while ($x < -1 || $x > 0 || $y < 0 || $y > 1) {
my $y4 = $y % 4;
my $x4 = $x % 4;
$x -= $yx_adj_x[$y4]->[$x4];
$y -= $yx_adj_y[$y4]->[$x4];
### assert: ($x+$y)%2 == 0
($x,$y) = (($x+$y)/2, # rotate -45 and divide sqrt(2)
($y-$x)/2);
}
return $xy_to_arm[$x]->[$y];
}
#------------------------------------------------------------------------------
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
### DragonMidpoint rect_to_n_range(): "$x1,$y1 $x2,$y2 arms=$self->{'arms'}"
$x1 = abs($x1);
$x2 = abs($x2);
$y1 = abs($y1);
$y2 = abs($y2);
my $xmax = int(max($x1,$x2));
my $ymax = int(max($y1,$y2));
return (0,
($xmax*$xmax + $ymax*$ymax + 1) * $self->{'arms'} * 5);
}
# sub rect_to_n_range {
# my ($self, $x1,$y1, $x2,$y2) = @_;
# ### DragonMidpoint rect_to_n_range(): "$x1,$y1 $x2,$y2"
#
# return Math::PlanePath::DragonCurve->rect_to_n_range
# (sqrt(2)*$x1, sqrt(2)*$y1, sqrt(2)*$x2, sqrt(2)*$y2);
# }
1;
__END__
# wider drawn arms ...
#
#
# ... 36---32 59---63-... 5
# | | | |
# 60 40 28 55 4
# | | | |
# 56---52---48---44 24---20---16 51 3
# | |
# 17---13----9----5 12 47---43---39 2
# | | | |
# 21 6--- 2 1 8 27---31---35 1
# | | | |
# 33---29---25 10 3 0--- 4 23 <- Y=0
# | | | |
# 37---41---45 14 7---11---15---19 -1
# | |
# 49 18---22---26 46---50---54---58 -2
# | | | |
# 53 30 42 62 -3
# | | | |
# ...--61---57 34---38 ... -4
#
#
#
# ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
# -5 -4 -3 -2 -1 X=0 1 2 3 4
# DragonMidpoint abs(dY) is A073089, but that seq has an extra leading 0
#
# --*--+ dy=+/-1 vert and left
# | horiz and right
# *
# |
# |
# *
# |
# +--*-- dy=+/-1
#
# +--*-- dx=+/-1 vert and right
# | horiz and left
# *
# |
# | dx=+/-1
# *
# |
# --*--+
#
# left turn ...01000
# right turn ...11000
# vert ...1
# horiz ...0
# Offset=1 0,0,1,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0,1,1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,1,
# mod16
# 0 1
# 1 8n+1=4n+1
# 2 0
# 3 1
# 4 1
# 5 1
# 6 0
# 7 0
# 8 1
# 9 8n+1=4n+1
# 10 0
# 11 1
# 12 1
# 13 0
# 14 0
# 15 0
#
# a(1) = a(4n+2) = a(8n+7) = a(16n+13) = 0,
# a(4n) = a(8n+3) = a(16n+5) = 1
# a(8n+1) = a(4n+1)
# N=0 0,1,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0,1,1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0,1,1,
=for stopwords eg Ryde Dragon Math-PlanePath Nlevel Heighway Harter et al bignum Xadj,Yadj lookup OEIS 0b.zz111 0b..zz11 ie tilingsearch
=head1 NAME
Math::PlanePath::DragonMidpoint -- dragon curve midpoints
=head1 SYNOPSIS
use Math::PlanePath::DragonMidpoint;
my $path = Math::PlanePath::DragonMidpoint->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This is the midpoint of each segment of the dragon curve by Heighway,
Harter, et al, per L<Math::PlanePath::DragonCurve>.
17--16 9---8 5
| | | |
18 15 10 7 4
| | | |
19 14--13--12--11 6---5---4 3
| |
20--21--22 3 2
| |
33--32 25--24--23 2 1
| | | |
34 31 26 0---1 <- Y=0
| | |
35 30--29--28--27 -1
|
36--37--38 43--44--45--46 -2
| | |
39 42 49--48--47 -3
| | |
40--41 50 -4
|
51 -5
|
52--53--54 -6
|
..--64 57--56--55 -7
| |
63 58 -8
| |
62--61--60--59 -9
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1
The dragon curve begins as follows. The midpoints of each segment are
numbered starting from 0,
+--8--+ +--4--+
| | | |
9 7 5 3
| | | | |
+-10--+--6--+ +--2--+ rotate 45 degrees |
| | v
11 1
| |
+-12--+ *--0--+ * = Origin
|
...
These midpoints are on fractions X=0.5,Y=0, X=1,Y=0.5, etc. For this
C<DragonMidpoint> path they're turned clockwise 45 degrees and shrunk by
sqrt(2) to be integer X,Y values a unit apart and initial direction to the
right.
The midpoints are distinct X,Y positions because the dragon curve traverses
each edge only once.
The dragon curve is self-similar in 2^level sections due to its unfolding.
This can be seen in the midpoints too as for example above N=0 to N=16 is
the same shape as N=16 to N=32, with the latter rotated 90 degrees and in
reverse.
Since the dragon curve always turns left or right, never straight ahead or
reverse, its segments are alternately horizontal and vertical. Rotated -45
degrees for the midpoints here this means alternately "opposite diagonal"
and "leading diagonal". They fall on X,Y alternately even or odd. So the
original dragon curve can be recovered from the midpoints by choosing
leading/opposite diagonal segment according to either X,Y even/odd, and
which is the same as N even/odd.
DragonMidpoint dragon segment
-------------- -----------------
"even" N==0 mod 2 opposite diagonal
which is X+Y==0 mod 2 too
"odd" N==1 mod 2 leading diagonal
which is X+Y==1 mod 2 too
/
3 0 at X=0,Y=0 "even", opposite diagonal
/ 1 at X=1,Y=0 "odd", leading diagonal
\ etc
2
\
\ /
0 1
\ /
=head2 Arms
Like the C<DragonCurve> the midpoints fill a quarter of the plane and four
copies mesh together perfectly when rotated by 90, 180 and 270 degrees. The
C<arms> parameter can choose 1 to 4 curve arms, successively advancing.
For example C<arms =E<gt> 4> begins as follows, with N=0,4,8,12,etc being
the first arm (the same as the plain curve above), N=1,5,9,13 the second,
N=2,6,10,14 the third and N=3,7,11,15 the fourth.
...-107-103 83--79--75--71 6
| | |
68--64 36--32 99 87 59--63--67 5
| | | | | | |
72 60 40 28 95--91 55 4
| | | | |
76 56--52--48--44 24--20--16 51 3
| | |
80--84--88 17--13---9---5 12 47--43--39 ... 2
| | | | | |
100--96--92 21 6---2 1 8 27--31--35 106 1
| | | | | |
104 33--29--25 10 3 0---4 23 94--98-102 <- Y=0
| | | | | |
... 37--41--45 14 7--11--15--19 90--86--82 -1
| | |
49 18--22--26 46--50--54--58 78 -2
| | | | |
53 89--93 30 42 62 74 -3
| | | | | | |
65--61--57 85 97 34--38 66--70 -4
| | |
69--73--77--81 101-105-... -5
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5
With four arms like this every X,Y point is visited exactly once, because
four arms of the C<DragonCurve> traverse every edge exactly once.
=head2 Tiling
Taking pairs of adjacent points N=2k and N=2k+1 gives little rectangles with
the following tiling of the plane repeating in 4x4 blocks.
+---+---+---+-+-+---+-+-+---+
| | | | | | | | | | |
+---+ | +---+ | +---+ | +---+
| | | |9 8| | | | | | |
+-+-+---+-+-+-+-+-+-+-+-+-+-+
| | | | |7| | | | | | |
| | +---+ | +---+ | +---+ | |
| | | | |6|5 4| | | | | |
+---+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | |3| | | | |
+---+ | +---+ | +---+ | +---+
| | | | | |2| | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | |0 1| | | | | | <- Y=0
| | +---+ | +---+ | +---+ | |
| | | | | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | | | | | | |
+---+ | +---+ | +---+ | +---+
| | | | | | | | | | |
+---+-+-+---+-+-+---+-+-+---+
^
X=0
The pairs follow this pattern both for the main curve N=0 etc shown, and
also for the rotated copies per L</Arms> above. This tiling is in the
tilingsearch database as
=over
L<http://tilingsearch.org/HTML/data24/K02A.html>
=back
Taking pairs N=2k+1 and N=2k+2, being odd N and its successor, gives a
regular pattern too, but this time repeating in blocks of 16x16.
|||--||||||--||--||--||||||--||||||--||||||--||||||--||||||--|||
|||--||||||--||--||--||||||--||||||--||||||--||||||--||||||--|||
-||------||------||------||------||------||------||------||-----
-||------||------||------||------||------||------||------||-----
|||--||||||||||||||--||||||||||||||--||||||||||||||--|||||||||||
|||--||||||||||||||--||||||||||||||--||||||||||||||--|||||||||||
-----||------||------||------||------||------||------||------||-
-----||------||------||------||------||------||------||------||-
-||--||--||--||--||--||||||--||--||--||--||--||--||--||||||--||-
-||--||--||--||--||--||||||--||--||--||--||--||--||--||||||--||-
-||------||------||------||------||------||------||------||-----
-||------||------||------||------||------||------||------||-----
|||||||||||--||||||||||||||--||||||||||||||--||||||||||||||--|||
|||||||||||--||||||||||||||--||||||||||||||--||||||||||||||--|||
-----||------||------||------||------||------||------||------||-
-----||------||------||------||------||------||------||------||-
|||--||||||--||--||--||||||--|| ||--||||||--||--||--||||||--|||
|||--||||||--||--||--||||||--|| ||--||||||--||--||--||||||--|||
-||------||------||------||------||------||------||------||-----
-||------||------||------||------||------||------||------||-----
|||--||||||||||||||--||||||||||||||--||||||||||||||--|||||||||||
|||--||||||||||||||--||||||||||||||--||||||||||||||--|||||||||||
-----||------||------||------||------||------||------||------||-
-----||------||------||------||------||------||------||------||-
-||--||||||--||--||--||--||--||--||--||||||--||--||--||--||--||-
-||--||||||--||--||--||--||--||--||--||||||--||--||--||--||--||-
-||------||------||------||------||------||------||------||-----
-||------||------||------||------||------||------||------||-----
|||||||||||--||||||||||||||--||||||||||||||--||||||||||||||--|||
|||||||||||--||||||||||||||--||||||||||||||--||||||||||||||--|||
-----||------||------||------||------||------||------||------||-
-----||------||------||------||------||------||------||------||-
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::DragonMidpoint-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.
Fractional positions give an X,Y position along a straight line between the
integer positions.
=item C<$n = $path-E<gt>n_start()>
Return 0, the first N in the path.
=back
=head1 FORMULAS
=head2 X,Y to N
An X,Y point is turned into N by dividing out digits of a complex base i+1.
This base is per the doubling of the C<DragonCurve> at each level. In
midpoint coordinates an adjustment subtracting 0 or 1 must be applied to
move an X,Y which is either N=2k or N=2k+1 to the position where dividing
out i+1 gives the N=k X,Y.
The adjustment is in a repeating pattern of 4x4 blocks. Points N=2k and
N=2k+1 both move to the same place corresponding to N=k multiplied by i+1.
The adjustment pattern is a little like the pair tiling shown above, but for
some pairs both the N=2k and N=2k+1 positions must move, it's not enough
just to shift the N=2k+1 to the N=2k.
Xadj Yadj
Ymod4 Ymod4
3 | 0 1 1 0 3 | 1 1 0 0
2 | 1 0 0 1 2 | 1 1 0 0
1 | 1 0 0 1 1 | 0 0 1 1
0 | 0 1 1 0 0 | 0 0 1 1
+-------- +--------
0 1 2 3 0 1 2 3
Xmod4 Xmod4
The same tables work for both the main curve and for the rotated copies per
L</Arms> above.
until -1<=X<=0 and 0<=Y<=1
Xm = X - Xadj(X mod 4, Y mod 4)
Ym = Y - Yadj(X mod 4, Y mod 4)
new X,Y = (Xm+i*Ym) / (i+1)
= (Xm+i*Ym) * (1-i)/2
= (Xm+Ym)/2, (Ym-Xm)/2 # Xm+Ym and Ym-Xm are both even
Nbit = Xadj xor Yadj # bits of N low to high
The X,Y reduction stops at one of the start points for the four arms
X,Y endpoint Arm +---+---+
------------ --- | 2 | 1 | Y=1
0, 0 0 +---+---+
0, 1 1 | 3 | 0 | Y=0
-1, 1 2 +---+---+
-1, 0 3 X=-1 X=0
For arms 1 and 3 the N bits must be flipped 0E<lt>-E<gt>1. The arm number
and hence whether this flip is needed is not known until reaching the
endpoint.
For bignum calculations there's no need to apply the "/2" shift in
newX=(Xm+Ym)/2 and newY=(Ym-Xm)/2. Instead keep a bit position which is the
logical low end and pick out two bits from there for the Xadj,Yadj lookup.
A whole word can be dropped when the bit position becomes a multiple of 32
or 64 or whatever.
=head1 OEIS
The C<DragonMidpoint> is in Sloane's Online Encyclopedia of Integer
Sequences as
=over
L<http://oeis.org/A073089> (etc)
=back
A073089 abs(dY) of n-1 to n, so 0=horizontal,1=vertical
(extra initial 0)
A077860 Y at N=2^k, being Re(-(i+1)^k + i-1)
For A073089, the midpoint curve is vertical when the C<DragonCurve> has a
vertical followed by a left turn, or horizontal followed by a right turn.
C<DragonCurve> verticals are whenever N is odd, and the turn is the bit
above the lowest 0 in N, as described in
L<Math::PlanePath::DragonCurve/Turn>. So
abs(dY) = lowbit(N) XOR bit-above-lowest-zero(N)
The n in A073089 is offset by 2 from the N numbering of the path here, being
n=N+2. The initial value at n=1 in A073089 has no corresponding N (it would
be N=-1).
The mod-16 definitions in A073089 express combinations of N odd/even and
bit-above-low-0 which are the vertical midpoint segments. The recurrence
a(8n+1)=a(4n+1) acts to strip strip of zeros above a low 1 bit,
ie. n=0b...00001 -E<gt> 0b...01. In terms of N=n-2 it reduces N=0b.zz111 to
0b..zz11 in order to seek a lowest 0 in range of the mod-16 conditions.
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::DragonCurve>,
L<Math::PlanePath::DragonRounded>
L<Math::PlanePath::AlternatePaperMidpoint>,
L<Math::PlanePath::R5DragonMidpoint>,
L<Math::PlanePath::TerdragonMidpoint>
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-planepath/index.html>
=head1 LICENSE
Copyright 2011, 2012, 2013, 2014 Kevin Ryde
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