Math::PlanePath::DiamondArms -- four spiral arms
use Math::PlanePath::DiamondArms; my $path = Math::PlanePath::DiamondArms->new; my ($x, $y) = $path->n_to_xy (123);
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.
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.
$arms = $path->arms_count()
Return 4.
Math::PlanePath, Math::PlanePath::SquareArms, Math::PlanePath::DiamondSpiral
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 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.
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