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 8413=72 points, and the next loop N=85 to N=221 is 22185=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 X1/2,Y1/2 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
SEE ALSO
Math::PlanePath, Math::PlanePath::SquareArms, Math::PlanePath::DiamondSpiral
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.