/usr/share/perl5/Math/PlanePath/AztecDiamondRings.pm is in libmath-planepath-perl 113-1.
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 | # Copyright 2011, 2012, 2013 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/>.
# cute groupings
# AztecDiamondRings, FibonacciWord fibonacci_word_type plain, 10x10 scale 15
package Math::PlanePath::AztecDiamondRings;
use 5.004;
use strict;
#use List::Util 'min', 'max';
*min = \&Math::PlanePath::_min;
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 113;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
use Math::PlanePath::Base::Generic
'round_nearest';
# uncomment this to run the ### lines
# use Smart::Comments;
use constant n_frac_discontinuity => 0;
use constant xy_is_visited => 1;
use constant dx_minimum => -1;
use constant dx_maximum => 1;
use constant dy_minimum => -1;
use constant dy_maximum => 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 parameter_info_array =>
[
Math::PlanePath::Base::Generic::parameter_info_nstart1(),
];
#------------------------------------------------------------------------------
sub new {
my $self = shift->SUPER::new(@_);
if (! defined $self->{'n_start'}) {
$self->{'n_start'} = $self->default_n_start;
}
return $self;
}
# starting from X axis and n_start=0
# d = [ 1, 2, 3, 4, 5 ]
# n = [ 0,4,12,24,40 ]
# N = (2 d^2 - 2 d)
# = (2*$d**2 - 2*$d)
# = ((2*$d - 2)*$d)
# d = 1/2 + sqrt(1/2 * $n + 1/4)
# = (sqrt(2*$n+1) + 1)/2
#
# X negative axis
# d = [ 1, 2, 3, 4,5 ]
# n = [ 2, 8, 18, 32, 50 ]
# N = (2 d^2)
sub n_to_xy {
my ($self, $n) = @_;
#### n_to_xy: $n
# adjust to N=0 at origin X=0,Y=0
$n = $n - $self->{'n_start'};
if ($n < 0) { return; }
my $d = int( (sqrt(2*int($n)+1) + 1)/2 );
$n -= 2*$d*$d; # to $n=0 half way around at horiz Y=-1 X<-1
if ($n < 0) {
my $x = -$d-$n-1;
if ($n < -$d) {
# top-right
return ($x,
min($n+2*$d, $d-1));
} else {
# top-left
return (max($x, -$d),
-1-$n);
}
} else {
my $x = $n-$d;
if ($n < $d) {
# bottom-left
my $y = -1-$n;
return ($x,
max($y, -$d));
} else {
# bottom-right
return (min($x, $d-1),
$n-2*$d);
}
}
}
sub xy_to_n {
my ($self, $x, $y) = @_;
### AztecDiamondRings xy_to_n(): "$x, $y"
$x = round_nearest ($x);
$y = round_nearest ($y);
if ($x >= 0) {
my $d = $x + abs($y);
return (2*$d + 2)*$d + $y + $self->{'n_start'};
}
if ($y >= 0) {
my $d = $y - $x;
return 2*$d*$d - 1 - $y + $self->{'n_start'};
} else {
my $d = $y + $x;
return (2*$d + 4)*$d + 1 - $y + $self->{'n_start'};
}
}
# | | x2>=-x1 |
# M---+ | M-------M | +---M
# | | | | | | | | |
# +---m | +----m--+ | m---+
# | | |
# -----+------ -------+------- -----+--------
# | | |
#
# | | |
# M---+ | M-------M y2>=-y1 | +---M
# | | | | | | | | |
# | m | | | | | m |
# -------+------ -------m------- -----+--------
# | | | | | | | | |
# M---+ | M-------M | +---M
# | | |
#
# | | |
# -----+------ -------+------- -----+--------
# | | |
# +---m | +--m----+ | m---+
# | | | | | | | | |
# M---+ | M-------M | +---M
# | | |
# exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
### AztecDiamondRings rect_to_n_range(): "$x1,$y1, $x2,$y2"
$x1 = round_nearest ($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;
my $min_x = 0;
my $min_y = ($y2 < 0 ? ($min_x = -1, $y2)
: $y1 > 0 ? $y1
: 0);
if ($x2 < $min_x) { $min_x = $x2 } # right edge if 0/-1 not covered
elsif ($x1 > $min_x) { $min_x = $x1 } # left edge if 0/-1 not covered
my $max_y = ($y2 >= -$y1 ? $y2 : $y1);
my $max_x = ($x2 >= -$x1-($max_y<0) ? $x2 : $x1);
### min at: "$min_x, $min_y"
### max at: "$max_x, $max_y"
return ($self->xy_to_n($min_x,$min_y),
$self->xy_to_n($max_x,$max_y));
}
1;
__END__
=for stopwords eg Ryde Math-PlanePath ie xbase OEIS
=head1 NAME
Math::PlanePath::AztecDiamondRings -- rings around an Aztec diamond shape
=head1 SYNOPSIS
use Math::PlanePath::AztecDiamondRings;
my $path = Math::PlanePath::AztecDiamondRings->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This path makes rings around an Aztec diamond shape,
=cut
# math-image --path=AztecDiamondRings --all --output=numbers --size=60x14
=pod
46-45 4
/ \
47 29-28 44 3
/ / \ \
48 30 16-15 27 43 ... 2
/ / / \ \ \ \
49 31 17 7--6 14 26 42 62 1
/ / / / \ \ \ \ \
50 32 18 8 2--1 5 13 25 41 61 <- Y=0
| | | | | | | | | |
51 33 19 9 3--4 12 24 40 60 -1
\ \ \ \ / / / /
52 34 20 10-11 23 39 59 -2
\ \ \ / / /
53 35 21-22 38 58 -3
\ \ / /
54 36-37 57 -4
\ /
55-56 -5
^
-5 -4 -3 -2 -1 X=0 1 2 3 4 5
This is similar to the C<DiamondSpiral>, but has all four corners flattened
to 2 vertical or horizontal, instead of just one in the C<DiamondSpiral>.
This is only a small change to the alignment of numbers in the sides, but is
more symmetric.
X<Hexagonal numbers>Y axis N=1,6,15,28,45,66,etc are the hexagonal numbers
k*(2k-1). The hexagonal numbers of the "second kind" 3,10,21,36,55,78, etc
k*(2k+1), are the vertical at X=-1 going downwards. Combining those two is
the triangular numbers 3,6,10,15,21,etc, k*(k+1)/2, alternately on one line
and the other. Those are the positions of all the horizontal steps,
ie. where dY=0.
X<Centred square numbers>X axis N=1,5,13,25,etc is the "centred square
numbers". Those numbers are made by drawing concentric squares with an
extra point on each side each time. The path here grows the same way,
adding one extra point to each of the four sides.
*---*---*---*
| |
| *---*---* | count total "*"s for
| | | | centred square numbers
* | *---* | *
| | | | | |
| * | * | * |
| | | | | |
| | *---* | |
* | | *
| *---*---* |
| |
*---*---*---*
=head2 N Start
The default is to number points starting N=1 as shown above. An optional
C<n_start> can give a different start, in the same pattern. For example to
start at 0,
=cut
# math-image --path=AztecDiamondRings,n_start=0 --expression='i<=59?i:0' --output=numbers --size=50x10
=pod
n_start => 0
45 44
46 28 27 43
47 29 15 14 26 42
48 30 16 6 5 13 25 41
49 31 17 7 1 0 4 12 24 40
50 32 18 8 2 3 11 23 39 59
51 33 19 9 10 22 38 58
52 34 20 21 37 57
53 35 36 56
54 55
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::AztecDiamondRings-E<gt>new ()>
=item C<$path = Math::PlanePath::AztecDiamondRings-E<gt>new (n_start =E<gt> $n)>
Create and return a new Aztec diamond spiral object.
=item C<($x,$y) = $path-E<gt>n_to_xy ($n)>
Return the X,Y coordinates of point number C<$n> on the path.
For C<$n < 1> the return is an empty list, it being considered the path
starts at 1.
=item C<$n = $path-E<gt>xy_to_n ($x,$y)>
Return the point number for coordinates C<$x,$y>. C<$x> and C<$y> are
each rounded to the nearest integer, which has the effect of treating each
point in the path as a square of side 1, so the entire plane is covered.
=item C<($n_lo, $n_hi) = $path-E<gt>rect_to_n_range ($x1,$y1, $x2,$y2)>
The returned range is exact, meaning C<$n_lo> and C<$n_hi> are the smallest
and biggest in the rectangle.
=back
=head1 FORMULAS
=head2 X,Y to N
The path makes lines in each quadrant. The quadrant is determined by the
signs of X and Y, then the line in that quadrant is either d=X+Y or d=X-Y.
A quadratic in d gives a starting N for the line and Y (or X if desired) is
an offset from there,
Y>=0 X>=0 d=X+Y N=(2d+2)*d+1 + Y
Y>=0 X<0 d=Y-X N=2d^2 - Y
Y<0 X>=0 d=X-Y N=(2d+2)*d+1 + Y
Y<0 X<0 d=X+Y N=(2d+4)*d+2 - Y
For example
Y=2 X=3 d=2+3=5 N=(2*5+2)*5+1 + 2 = 63
Y=2 X=-1 d=2-(-1)=3 N=2*3*3 - 2 = 16
Y=-1 X=4 d=4-(-1)=5 N=(2*5+2)*5+1 + -1 = 60
Y=-2 X=-3 d=-3+(-2)=-5 N=(2*-5+4)*-5+2 - (-2) = 34
The two XE<gt>=0 cases are the same N formula and can be combined with an
abs,
X>=0 d=X+abs(Y) N=(2d+2)*d+1 + Y
This works because at Y=0 the last line of one ring joins up to the start of
the next. For example N=11 to N=15,
15 2
\
14 1
\
13 <- Y=0
12 -1
/
11 -2
^
X=0 1 2
=head2 Rectangle to N Range
Within each row N increases as X increases away from the Y axis, and within
each column similarly N increases as Y increases away from the X axis. So
in a rectangle the maximum N is at one of the four corners of the rectangle.
|
x1,y2 M---|----M x2,y2
| | |
-------O---------
| | |
| | |
x1,y1 M---|----M x1,y1
|
For any two rows y1 and y2, the values in row y2 are all bigger than in y1
if y2E<gt>=-y1. This is so even when y1 and y2 are on the same side of the
origin, ie. both positive or both negative.
For any two columns x1 and x2, the values in the part with YE<gt>=0 are all
bigger if x2E<gt>=-x1, or in the part of the columns with YE<lt>0 it's
x2E<gt>=-x1-1. So the biggest corner is at
max_y = (y2 >= -y1 ? y2 ? y1)
max_x = (x2 >= -x1 - (max_y<0) ? x2 : x1)
The difference in the X handling for Y positive or negative is due to the
quadrant ordering. When YE<gt>=0, at X and -X the bigger N is the X
negative side, but when YE<lt>0 it's the X positive side.
A similar approach gives the minimum N in a rectangle.
min_y = / y2 if y2 < 0, and set xbase=-1
| y1 if y1 > 0, and set xbase=0
\ 0 otherwise, and set xbase=0
min_x = / x2 if x2 < xbase
| x1 if x1 > xbase
\ xbase otherwise
The minimum row is Y=0, but if that's not in the rectangle then the y2 or y1
top or bottom edge is the minimum. Then within any row the minimum N is at
xbase=0 if YE<lt>0 or xbase=-1 if YE<gt>=0. If that xbase is not in range
then the x2 or x1 left or right edge is the minimum.
=head1 OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to
this path include
=over
L<http://oeis.org/A001844> (etc)
=back
n_start=1 (the default)
A001844 N on X axis, the centred squares 2k(k+1)+1
n_start=0
A023532 abs(dY), being 0 if N=k*(k+3)/2
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::DiamondSpiral>
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-planepath/index.html>
=head1 LICENSE
Copyright 2011, 2012, 2013 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
|