SYNOPSIS
use Math::PlanePath::SquareArms;
my $path = Math::PlanePath::SquareArms>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This path follows four spiral arms, each advancing successively,
...3329 3  2622181410 25 2    30 11 7 3 6 21 1     ... 15 4 1 2 17 ... < Y=0      19 8 5 913 32 1    23 1216202428 2  2731... 3 ^ ^ ^ ^ ^ ^ ^ 3 2 1 X=0 1 2 3 ...
Each arm is quadratic, with each loop 128 longer than the preceding. The perfect squares fall in eight straight lines 4, with the even squares on the X and Y axes and the odd squares on the diagonals X=Y and X=Y.
Some novel straight lines arise from numbers which are a repdigit in one or more bases (Sloane's A167782). ``111'' in various bases falls on straight lines. Numbers ``[16][16][16]'' in bases 17,19,21,etc are a horizontal at Y=3 because they're perfect squares, and ``[64][64][64]'' in base 65,66,etc go a vertically downwards from X=12,Y=266 similarly because they're squares.
Each arm is N=4*k+rem for a remainder rem=0,1,2,3, so sequences related to multiples of 4 or with a modulo 4 pattern may fall on particular arms.
FUNCTIONS
See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes. "$path = Math::PlanePath::SquareArms>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.
FORMULAS
Rectangle N Range
Within a square X=d...+d, and Y=d...+d the biggest N is the end of the N=5 arm in that square, which is N=9, 25, 49, 81, etc, (2d+1)^2, in successive corners of the square. So for a rectangle find a surrounding d square,
d = max(abs(x1),abs(y1),abs(x2),abs(y2))
from which
Nmax = (2*d+1)^2 = (4*d + 4)*d + 1
This can be used for a minimum too by finding the smallest d covered by the rectangle.
dlo = max (0, min(abs(y1),abs(y2)) if x=0 not covered min(abs(x1),abs(x2)) if y=0 not covered )
from which the maximum of the preceding dlo1 square,
Nlo = / 1 if dlo=0 \ (2*(dlo1)+1)^2 +1 if dlo!=0 = (2*dlo  1)^2 = (4*dlo  4)*dlo + 1
For a tighter maximum, horizontally N increases to the left or right of the diagonal X=Y line (or X=Y+/1 line), which means one end or the other is the maximum. Similar vertically N increases above or below the offdiagonal X=Y so the top or bottom is the maximum. This means for a rectangle the biggest N is at one of the four corners,
Nhi = max (xy_to_n (x1,y1), xy_to_n (x1,y2), xy_to_n (x2,y1), xy_to_n (x2,y2))
The current code uses a dlo for Nlo and the corners for Nhi, which means the high is exact but the low is not.
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016 Kevin RydeThis 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/>.