NAME

       Math::PlanePath::DiamondArms -- four spiral arms

SYNOPSIS

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

DESCRIPTION

       This path follows four spiral arms, each advancing successively in a diamond pattern,

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

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

       Each arm makes a spiral widening out by 4 each time around, thus leaving room for four
       such arms.  Each arm loop is 64 longer than the preceding loop.  For example N=13 to N=85
       below is 84-13=72 points, and the next loop N=85 to N=221 is 221-85=136 which is an extra
       64, ie. 72+64=136.

                        25          ...
                       /  \           \
                     29  . 21  .  .  . 93
                    /        \           \
                  33  .  .  . 17  .  .  . 89
                 /              \           \
               37  .  .  .  .  . 13  .  .  . 85
              /                 /           /
            41  .  .  .  1  .  9  .  .  . 81
              \           \  /           /
               45  .  .  .  5  .  .  . 77
                 \                    /
                  49  .  .  .  .  . 73
                    \              /
                     53  .  .  . 69
                       \        /
                        57  . 65
                          \  /
                           61

       Each arm is N=4*k+rem for a remainder rem=0,1,2,3, so sequences related to multiples of 4
       or with a modulo 4 pattern may fall on particular arms.

       The starts of each arm N=1,2,3,4 are at X=0 or 1 and Y=0 or 1,

                      ..
                        \
                    4    3  ..          Y=1
                  /        /
                ..  1    2           <- Y=0
                     \
                      ..
                    ^    ^
                   X=0  X=1

       They could be centred around the origin by taking X-1/2,Y-1/2 so for example N=1 would be
       at -1/2,-1/2.  But the it's done as N=1 at 0,0 to stay in integers.

FUNCTIONS

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

       "$path = Math::PlanePath::DiamondArms->new ()"
           Create and return a new path 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, as the path starts at 1.

           Fractional $n gives a point on the line between $n and "$n+4", that "$n+4" being the
           next point on the same spiralling arm.  This is probably of limited use, but arises
           fairly naturally from the calculation.

   Descriptive Methods
       "$arms = $path->arms_count()"
           Return 4.

SEE ALSO

       Math::PlanePath, Math::PlanePath::SquareArms, 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/>.