/usr/share/perl5/Math/PlanePath/DragonRounded.pm is in libmath-planepath-perl 117-1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 | # 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=DragonRounded --lines --scale=10
# math-image --path=DragonRounded,arms=4 --all --output=numbers_dash --size=132x60
#
package Math::PlanePath::DragonRounded;
use 5.004;
use strict;
#use List::Util 'min','max';
*min = \&Math::PlanePath::_min;
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 117;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
*_divrem_mutate = \&Math::PlanePath::_divrem_mutate;
use Math::PlanePath::Base::Generic
'is_infinite',
'round_nearest',
'floor';
use Math::PlanePath::Base::Digits
'round_down_pow';
use Math::PlanePath::DragonMidpoint;
# uncomment this to run the ### lines
#use Smart::Comments;
use constant n_start => 0;
use constant parameter_info_array => [ { name => 'arms',
share_key => 'arms_4',
display => 'Arms',
type => 'integer',
minimum => 1,
maximum => 4,
default => 1,
width => 1,
description => 'Arms',
} ];
{
my @x_negative_at_n = (undef, 8,5,2,2);
sub x_negative_at_n {
my ($self) = @_;
return $x_negative_at_n[$self->{'arms'}];
}
}
{
my @y_negative_at_n = (undef, 26,17,8,3);
sub y_negative_at_n {
my ($self) = @_;
return $y_negative_at_n[$self->{'arms'}];
}
}
use constant sumabsxy_minimum => 1;
use constant absdiffxy_minimum => 1; # X=Y doesn't occur
use constant rsquared_minimum => 1; # minimum X=1,Y=0
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_eight;
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
#------------------------------------------------------------------------------
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) = @_;
### DragonRounded n_to_xy(): $n
if ($n < 0) { return; }
if (is_infinite($n)) { return ($n, $n); }
my $frac;
{
my $int = int($n);
$frac = $n - $int;
$n = $int; # BigFloat int() gives BigInt, use that
}
### $frac
my $zero = ($n * 0); # inherit bignum 0
# arm as initial rotation
my $rot = _divrem_mutate ($n, $self->{'arms'});
# two points per edge
my $x_offset = _divrem_mutate ($n, 2);
# ENHANCE-ME: sx,sy just from len=3*2**level
my @digits;
my @sx;
my @sy;
{
my $sx = $zero + 3;
my $sy = $zero;
while ($n) {
push @digits, ($n % 2);
push @sx, $sx;
push @sy, $sy;
$n = int($n/2);
# (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) {
($sx,$sy) = (-$sx,-$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, and that determines the y_offset to apply to go across
# towards the next edge. When original input $n is odd, which means
# $x_offset 0 at this point, there's no y_offset as going along the edge
# not across the vertex.
#
my $y_offset = ($x_offset ? ($above_low_zero ? -$frac : $frac)
: 0);
$x_offset = $frac + 1 + $x_offset;
### final: "$x,$y rot=$rot above_low_zero=$above_low_zero offset=$x_offset,$y_offset"
if ($rot & 2) {
($x_offset,$y_offset) = (-$x_offset,-$y_offset); # rotate 180
}
if ($rot & 1) {
($x_offset,$y_offset) = (-$y_offset,$x_offset); # rotate +90
}
$x = $x_offset + $x;
$y = $y_offset + $y;
### rotated offset: "$x_offset,$y_offset return $x,$y"
return ($x,$y);
}
my @yx_rtom_dx = ([undef, 1, 1, undef, 1, 1],
[ 0, undef, undef, 1, undef, undef],
[ 0, undef, undef, 1, undef, undef],
[undef, 1, 1, undef, 1, 1],
[ 1, undef, undef, 0, undef, undef],
[ 1, undef, undef, 0, undef, undef]);
my @yx_rtom_dy = ([undef, 0, 0, undef, -1, -1],
[ 0, undef, undef, 0, undef, undef],
[ 0, undef, undef, 0, undef, undef],
[undef, -1, -1, undef, 0, 0],
[ 0, undef, undef, 0, undef, undef],
[ 0, undef, undef, 0, undef, undef]);
sub xy_to_n {
my ($self, $x, $y) = @_;
### DragonRounded xy_to_n(): "$x, $y"
$x = round_nearest($x);
$y = round_nearest($y);
my $x6 = $x % 6;
my $y6 = $y % 6;
my $dx = $yx_rtom_dx[$y6][$x6]; defined $dx or return undef;
my $dy = $yx_rtom_dy[$y6][$x6]; defined $dy or return undef;
# my $n = $self->Math::PlanePath::DragonMidpoint::xy_to_n
# ($x - floor($x/3) - $dx,
# $y - floor($y/3) - $dy);
# ### dxy: "$dx, $dy"
# ### to: ($x - floor($x/3) - $dx).", ".($y - floor($y/3) - $dy)
# ### $n
return $self->Math::PlanePath::DragonMidpoint::xy_to_n
($x - floor($x/3) - $dx,
$y - floor($y/3) - $dy);
}
# level 21 n=1048576 .. 2097152
# min 1052677 0b100000001000000000101 at -1026,1 factor 1.99610706057474
# n=2^20 min r^2=2^20 plus a bit
# maybe ...
#
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
### DragonRounded 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) / 3);
my $ymax = int(max($y1,$y2) / 3);
return (0,
($xmax*$xmax + $ymax*$ymax + 1) * $self->{'arms'} * 16);
}
#------------------------------------------------------------------------------
# each 2 points is a line segment, so 2*DragonMidpoint
# level 0 0--1
# level 1 0--1 2--3
# level 2 0--1 2--3 4--5 6--7
#
# arms=4
# level 0 0--3 / 1--4 / 2--5 / 3--7
# level 1
#
# 2^level segments
# 2*2^level rounded points
# arms*2^level when multi-arm
# numbered starting 0
#
sub level_to_n_range {
my ($self, $level) = @_;
return (0, 2**($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 ($n, 2*$self->{'arms'});
my ($pow, $exp) = round_down_pow ($n, 2);
return $exp + 1;
}
#------------------------------------------------------------------------------
1;
__END__
=for stopwords eg Ryde Dragon Math-PlanePath Nlevel Heighway Harter et al vertices multi-arm Xadj,Yadj OEIS Xadj
=head1 NAME
Math::PlanePath::DragonRounded -- dragon curve, with rounded corners
=head1 SYNOPSIS
use Math::PlanePath::DragonRounded;
my $path = Math::PlanePath::DragonRounded->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This is a version of the dragon curve by Harter, Heighway, et al, done with
two points per edge and skipping vertices so as to make rounded-off corners,
17-16 9--8 6
/ \ / \
18 15 10 7 5
| | | |
19 14 11 6 4
\ \ / \
20-21 13-12 5--4 3
\ \
22 3 2
| |
23 2 1
/ /
33-32 25-24 . 0--1 Y=0
/ \ /
34 31 26 -1
| | |
35 30 27 -2
\ \ /
36-37 29-28 44-45 -3
\ / \
38 43 46 -4
| | |
39 42 47 -5
\ / /
40-41 49-48 -6
/
50 -7
|
...
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
-15-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 ...
The two points on an edge have one of X or Y a multiple of 3 and the other Y
or X at 1 mod 3 or 2 mod 3. For example N=19 and N=20 are on the X=-9 edge
(a multiple of 3), and at Y=4 and Y=5 (1 and 2 mod 3).
The "rounding" of the corners ensures that for example N=13 and N=21 don't
touch as they approach X=-6,Y=3. The curve always approaches vertices like
this and never crosses itself.
=head2 Arms
The dragon curve fills a quarter of the plane and four copies mesh together
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> gives
36-32 59-... 6
/ \ /
... 40 28 55 5
| | | |
56 44 24 51 4
\ / \ \
52-48 13--9 20-16 47-43 3
/ \ \ \
17 5 12 39 2
| | | |
21 1 8 35 1
/ / /
29-25 6--2 0--4 27-31 <- Y=0
/ / /
33 10 3 23 -1
| | | |
37 14 7 19 -2
\ \ \ /
41-45 18-22 11-15 50-54 -3
\ \ / \
49 26 46 58 -4
| | | |
53 30 42 ... -5
/ \ /
...-57 34-38 -6
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
With 4 arms like this all 3x3 blocks are visited, using 4 out of 9 points in
each.
=head2 Midpoint
The points of this rounded curve correspond to the C<DragonMidpoint> with a
little squish to turn each 6x6 block into a 4x4 block. For instance in the
following N=2,3 are pushed to the left, and N=6 through N=11 shift down and
squashes up horizontally.
DragonRounded DragonMidpoint
9--8
/ \
10 7 9---8
| | | |
11 6 10 7
/ \ | |
5--4 <=> -11 6---5---4
\ |
3 3
| |
2 2
/ |
. 0--1 0---1
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::DragonRounded-E<gt>new ()>
=item C<$path = Math::PlanePath::DragonRounded-E<gt>new (arms =E<gt> $aa)>
Create and return a new path object.
The optional C<arms> parameter makes a multi-arm curve. The default is 1
for just one arm.
=item C<($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.
=item C<$n = $path-E<gt>n_start()>
Return 0, the first N in the path.
=back
=head2 Level Methods
=over
=item C<($n_lo, $n_hi) = $path-E<gt>level_to_n_range($level)>
Return C<(0, 2 * 2**$level - 1)>, or for multiple arms return C<(0, $arms *
2 * 2**$level - 1)>.
There are 2^level segments comprising the dragon, or arms*2^level when
multiple arms. Each has 2 points in this rounded curve, numbered starting
from 0.
=back
=head1 FORMULAS
=head2 X,Y to N
The correspondence with the C<DragonMidpoint> noted above allows the method
from that module to be used for the rounded C<xy_to_n()>.
The correspondence essentially reckons each point on the rounded curve as
the midpoint of a dragon curve of one greater level of detail, and segments
on 45-degree angles.
The coordinate conversion turns each 6x6 block of C<DragonRounded> to a 4x4
block of C<DragonMidpoint>. There's no rotations or anything.
Xmid = X - floor(X/3) - Xadj[X%6][Y%6]
Ymid = Y - floor(Y/3) - Yadj[X%6][Y%6]
N = DragonMidpoint n_to_xy of Xmid,Ymid
Xadj[][] is a 6x6 table of 0 or 1 or undef
Yadj[][] is a 6x6 table of -1 or 0 or undef
The Xadj,Yadj tables are a handy place to notice X,Y points not on the
C<DragonRounded> style 4 of 9 points. Or 16 of 36 points since the tables
are 6x6.
=head1 OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this
path include the various C<DragonCurve> sequences at even N, and in addition
=over
L<http://oeis.org/A152822> (etc)
=back
A152822 abs(dX), so 0=vertical,1=not, being 1,1,0,1 repeating
A166486 abs(dY), so 0=horizontal,1=not, being 0,1,1,1 repeating
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::DragonCurve>,
L<Math::PlanePath::DragonMidpoint>,
L<Math::PlanePath::TerdragonRounded>
=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
|