# 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 . # cute groupings # AztecDiamondRings, FibonacciWord fibonacci_word_type plain, 10x10 scale 15 package Math::PlanePath::AztecDiamondRings; use 5.004; use strict; use vars '$VERSION', '@ISA'; $VERSION = 100; use Math::PlanePath; @ISA = ('Math::PlanePath'); use Math::PlanePath::Base::Generic 'round_nearest'; # uncomment this to run the ### lines #use Devel::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 dir4_maximum => 3.5; # South-East # use constant dir_maximum_360 => 315; # South-East 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 $self; } # d = [ 1, 2, 3, 4 ] # n = [ 1,5,13,25 ] # N = (2 d^2 - 2 d + 1) # = (2*$d**2 - 2*$d + 1) # = ((2*$d - 2)*$d + 1) # d = 1/2 + sqrt(1/2 * $n + -1/4) # sub n_to_xy { my ($self, $n) = @_; #### n_to_xy: $n # adjust to N=1 at origin X=0,Y=0 $n = $n - $self->{'n_start'} + 1; if ($n < 1) { return; } my $frac; { my $int = int($n); $frac = $n - $int; $n = $int; # BigFloat int() gives BigInt, use that ### assert: $frac >= 0 ### assert: $frac < 1 } my $d = int( (1 + sqrt(2*$n-1))/2 ); #### $d #### d frac: (1 + sqrt(2*$n-1))/2 #### base: ((2*$d - 2)*$d + 1) #### base with offset: (2*$d*$d + 1) # and base+2d = (2 d^2 - 2d + 1) + 2d # = 2*d*d + 1 $n -= (2*$d*$d + 1); ### rem from left: $n if ($n < 0) { my $x = -$d-$n-1; if ($n != -1) { $x = -$frac + $x; } if ($n < -$d) { # top-right my $y = $n+2*$d; if ($n != -$d-1) { $y = $frac + $y; } return ($x, $y); } else { # top-left return ($x, -$frac-1-$n); } } else { my $x = $n-$d; if ($n != 2*$d-1) { $x = $frac + $x; } if ($n < $d) { # bottom-left my $y = -1-$n; if ($n != $d-1) { $y = -$frac + $y; } return ($x, $y); } else { # bottom-right return ($x, $frac-2*$d+$n); } } my $y = $d - abs($n); # y=+$d at the top, down to y=-$d my $x = abs($y) - $d; # 0 to $d on the right #### uncapped y: $y #### abs x: $x return (($n >= 0 ? $x : -$x), # negate if on the right max ($y, -$d)); # cap for horiz at 5 to 6, 13 to 14 etc } 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 + $self->{'n_start'}; } else { if ($y >= 0) { my $d = $y - $x; return 2*$d*$d - $y - 1 + $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__ # 67 66 5 # 68 46 45 65 4 # 69 47 29 28 44 64 3 # 70 48 30 16 15 27 43 63 2 # 71 49 31 17 7 6 14 26 42 62 1 # 72 50 32 18 8 2 1 5 13 25 41 61 <- Y=0 # 73 51 33 19 9 3 4 12 24 40 60 84 -1 # 74 52 34 20 10 11 23 39 59 83 -2 # 75 53 35 21 22 38 58 82 -3 # 76 54 36 37 57 81 -4 # 77 55 56 80 -5 # 78 79 -6 =for stopwords SquareSpiral eg AztecDiamondRings Ryde Math-PlanePath DiamondSpiral 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 DiamondSpiral, but has all four corners flattened to 2 vertical or horizontal, instead of just one in the DiamondSpiral. This is only a small change to the alignment of numbers in the sides, but is more symmetric. 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 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 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 for behaviour common to all path classes. =over 4 =item C<$path = Math::PlanePath::AztecDiamondRings-Enew ()> =item C<$path = Math::PlanePath::AztecDiamondRings-Enew (n_start =E $n)> Create and return a new Aztec diamond spiral object. =item C<($x,$y) = $path-En_to_xy ($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 = $path-Exy_to_n ($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) = $path-Erect_to_n_range ($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=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=-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=0 are all bigger if x2E=-x1, or in the part of the columns with YE0 it's x2E=-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=0, at X and -X the bigger N is the X negative side, but when YE0 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 YE0 or xbase=-1 if YE=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 http://oeis.org/A001844 (etc) n_start=1 (the default) A001844 N on X axis, the centred squares 2n(n+1)+1 n_start=0 A023532 abs(dY), being 0 if N=k*(k+3)/2 =head1 SEE ALSO L, L =head1 HOME PAGE 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 . =cut