Provided by: libmath-planepath-perl_125-1_all bug

NAME

       Math::PlanePath::HexSpiral -- integer points around a hexagonal spiral

SYNOPSIS

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

DESCRIPTION

       This path makes a hexagonal spiral, with points spread out horizontally to fit on a square
       grid.

                    28 -- 27 -- 26 -- 25                  3
                   /                    \
                 29    13 -- 12 -- 11    24               2
                /     /              \     \
              30    14     4 --- 3    10    23            1
             /     /     /         \     \    \
           31    15     5     1 --- 2     9    22    <- Y=0
             \     \     \              /     /
              32    16     6 --- 7 --- 8    21           -1
                \     \                    /
                 33    17 -- 18 -- 19 -- 20              -2
                   \
                    34 -- 35 ...                         -3

            ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
           -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6

       Each horizontal gap is 2, so for instance n=1 is at X=0,Y=0 then n=2 is at X=2,Y=0.  The
       diagonals are just 1 across, so n=3 is at X=1,Y=1.  Each alternate row is offset from the
       one above or below.  The result is a triangular lattice per "Triangular Lattice" in
       Math::PlanePath.

       The octagonal numbers 8,21,40,65, etc 3*k^2-2*k fall on a horizontal straight line at
       Y=-1.  In general straight lines are 3*k^2 + b*k + c.  A plain 3*k^2 goes diagonally up to
       the left, then b is a 1/6 turn anti-clockwise, or clockwise if negative.  So b=1 goes
       horizontally to the left, b=2 diagonally down to the left, b=3 diagonally down to the
       right, etc.

   Wider
       An optional "wider" parameter makes the path wider, stretched along the top and bottom
       horizontals.  For example

           $path = Math::PlanePath::HexSpiral->new (wider => 2);

       gives

                                       ... 36----35                   3
                                                   \
                       21----20----19----18----17    34               2
                      /                          \     \
                    22     8---- 7---- 6---- 5    16    33            1
                   /     /                    \     \    \
                 23     9     1---- 2---- 3---- 4    15    32    <- Y=0
                   \     \                          /     /
                    24    10----11----12----13----14    31           -1
                      \                               /
                       25----26----27----28---29----30               -2

                  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
                 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7

       The centre horizontal from N=1 is extended by "wider" many extra places, then the path
       loops around that shape.  The starting point N=1 is shifted to the left by wider many
       places to keep the spiral centred on the origin X=0,Y=0.  Each horizontal gap is still 2.

       Each loop is still 6 longer than the previous, since the widening is basically a constant
       amount added into each loop.

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

           n_start => 0

                    27    26    25    24                    3
                 28    12    11    10    23                 2
              29    13     3     2     9    22              1
           30    14     4     0     1     8    21      <- Y=0
              31    15     5     6     7    20   ...       -1
                 32    16    17    18    19    38          -2
                    33    34    35    36    37             -3
                              ^
           -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6

       In this numbering the X axis N=0,1,8,21,etc is the octagonal numbers 3*X*(X+1).

FUNCTIONS

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

       "$path = Math::PlanePath::HexSpiral->new ()"
       "$path = Math::PlanePath::HexSpiral->new (wider => $w)"
           Create and return a new hex spiral object.  An optional "wider" parameter widens the
           path, it defaults to 0 which is no widening.

       "($x,$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 $n in the path as a square of
           side 1.

           Only every second square in the plane has an N, being those where X,Y both odd or both
           even.  If "$x,$y" is a position without an N, ie. one of X,Y odd the other even, then
           the return is "undef".

OEIS

       Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

           <http://oeis.org/A056105> (etc)

           A056105    N on X axis
           A056106    N on X=Y diagonal
           A056107    N on North-West diagonal
           A056108    N on negative X axis
           A056109    N on South-West diagonal
           A003215    N on South-East diagonal

           A063178    total sum N previous row or diagonal
           A135711    boundary length of N hexagons
           A135708    grid sticks of N hexagons

           n_start=0
             A000567    N on X axis, octagonal numbers
             A049451    N on X negative axis
             A049450    N on X=Y diagonal north-east
             A033428    N on north-west diagonal, 3*k^2
             A045944    N on south-west diagonal, octagonal numbers second kind
             A063436    N on WSW slope dX=-3,dY=-1
             A028896    N on south-east diagonal

SEE ALSO

       Math::PlanePath, Math::PlanePath::HexSpiralSkewed, Math::PlanePath::HexArms,
       Math::PlanePath::TriangleSpiral, Math::PlanePath::TriangularHypot

HOME PAGE

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

LICENSE

       Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017 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/>.