Provided by: libmath-planepath-perl_129-1_all bug

NAME

       Math::PlanePath::AztecDiamondRings -- rings around an Aztec diamond shape

SYNOPSIS

        use Math::PlanePath::AztecDiamondRings;
        my $path = Math::PlanePath::AztecDiamondRings->new;
        my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

       This path makes rings around an Aztec diamond shape,

                        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
           * | *---* | *
           | | |   | | |
           | * | * | * |
           | | |   | | |
           | | *---* | |
           * |       | *
           | *---*---* |
           |           |
           *---*---*---*

   N Start
       The default is to number points starting N=1 as shown above.  An optional "n_start" can
       give a different start, in the same pattern.  For example to start at 0,

           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

FUNCTIONS

       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

       "$path = Math::PlanePath::AztecDiamondRings->new ()"
       "$path = Math::PlanePath::AztecDiamondRings->new (n_start => $n)"
           Create and return a new Aztec diamond spiral object.

       "($x,$y) = $path->n_to_xy ($n)"
           Return the X,Y coordinates of point number $n on the path.

           For "$n < 1" the return is an empty list, it being considered the path starts at 1.

       "$n = $path->xy_to_n ($x,$y)"
           Return the point number for coordinates "$x,$y".  $x and $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.

       "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
           The returned range is exact, meaning $n_lo and $n_hi are the smallest and biggest in
           the rectangle.

FORMULAS

   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 X>=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

   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 y2>=-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 Y>=0 are all bigger if x2>=-x1,
       or in the part of the columns with Y<0 it's x2>=-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 Y>=0, at X and -X the bigger N is the X negative side, but when Y<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 Y<0 or xbase=-1
       if Y>=0.  If that xbase is not in range then the x2 or x1 left or right edge is the
       minimum.

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 2k(k+1)+1

           n_start=0
             A046092    N on X axis, 4*triangular
             A139277    N on diagonal X=Y
             A023532    abs(dY), being 0 if N=k*(k+3)/2

SEE ALSO

       Math::PlanePath, Math::PlanePath::DiamondSpiral

HOME PAGE

       <http://user42.tuxfamily.org/math-planepath/index.html>

LICENSE

       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.