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

NAME

       Math::PlanePath::HexSpiralSkewed -- integer points around a skewed hexagonal spiral

SYNOPSIS

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

DESCRIPTION

       This path makes a hexagonal spiral with points skewed so as to fit a square grid and fully
       cover the plane.

           13--12--11   ...              2
            |         \   \
           14   4---3  10  23            1
            |   |     \   \   \
           15   5   1---2   9  22    <- Y=0
             \   \          |   |
               16   6---7---8  21       -1
                 \              |
                   17--18--19--20       -2

            ^   ^   ^   ^   ^   ^
           -2  -1  X=0  1   2   3  ...

       The kinds of N=3*k^2 numbers which fall on straight lines in the plain "HexSpiral" also
       fall on straight lines when skewed.  See Math::PlanePath::HexSpiral for notes on this.

   Skew
       The skewed path is the same shape as the plain "HexSpiral", but fits more points on a
       square grid.  The skew pushes the top horizontal to the left, as shown by the following
       parts, and the bottom horizontal is similarly skewed but to the right.

           HexSpiralSkewed               HexSpiral

           13--12--11                   13--12--11
            |         \                /          \
           14          10            14            10
            |             \         /                \
           15               9     15                   9

           -2  -1  X=0  1   2     -4 -3 -2  X=0  2  3  4

       In general the coordinates can be converted each way by

           plain X,Y -> skewed (X-Y)/2, Y

           skewed X,Y -> plain 2*X+Y, Y

Corners

       "HexSpiralSkewed" is similar to the "SquareSpiral" but cuts off the top-right and bottom-
       left corners so that each loop is 6 steps longer than the previous, whereas for the
       "SquareSpiral" it's 8.  See "Corners" in Math::PlanePath::SquareSpiral for other corner
       cutting.

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

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

       gives

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

            ^   ^   ^   ^   ^   ^   ^   ^
           -4  -3  -2  -1  X=0  1   2   3  ...

       The centre horizontal from N=1 is extended by "wider" many further places, then the path
       loops around that shape.  The starting point 1 is shifted to the left by wider/2 places
       (rounded up to an integer) to keep the spiral centred on the origin X=0,Y=0.

       Each loop is still 6 longer than the previous, since the widening is basically a constant
       amount added into each loop.  The result is the same as the plain "HexSpiral" of the same
       widening too.  The effect looks better in the plain "HexSpiral".

   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  39           -1
                   32  16  17  18  19  38           -2
                       33  34  35  36  37           -3

           -3  -2  -1  X=0  1   2   3   4

       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.

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

       "$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.

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, 3n^2-2n+1
           A056106    N on Y axis, 3n^2-n+1
           A056107    N on North-West diagonal, 3n^2+1
           A056108    N on X negative axis, 3n^2+n+1
           A056109    N on Y negative axis, 3n^2+2n+1
           A003215    N on South-East diagonal, centred hexagonals

           n_start=0
             A000567    N on X axis, octagonal numbers
             A049450    N on Y axis
             A049451    N on X negative axis
             A045944    N on Y negative axis, octagonal numbers second kind
             A062783    N on X=Y diagonal north-east
             A033428    N on north-west diagonal, 3*k^2
             A063436    N on south-west diagonal
             A028896    N on south-east diagonal

SEE ALSO

       Math::PlanePath, Math::PlanePath::HexSpiral, Math::PlanePath::HeptSpiralSkewed,
       Math::PlanePath::PentSpiralSkewed, Math::PlanePath::DiamondSpiral

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/>.