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.

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/>.