/ 
Math-PlanePath-127
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::DiamondArms

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 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/>.