Provided by: libmath-planepath-perl_122-1_all 
NAME
Math::PlanePath::SquareArms -- four spiral arms
SYNOPSIS
use Math::PlanePath::SquareArms;
my $path = Math::PlanePath::SquareArms->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This path follows four spiral arms, each advancing successively,
...--33--29 3
|
26--22--18--14--10 25 2
| | |
30 11-- 7-- 3 6 21 1
| | | |
... 15 4 1 2 17 ... <- Y=0
| | | | |
19 8 5-- 9--13 32 -1
| | |
23 12--16--20--24--28 -2
|
27--31--... -3
^ ^ ^ ^ ^ ^ ^
-3 -2 -1 X=0 1 2 3 ...
Each arm is quadratic, with each loop 128 longer than the preceding. The perfect squares
fall in eight straight lines 4, with the even squares on the X and Y axes and the odd
squares on the diagonals X=Y and X=-Y.
Some novel straight lines arise from numbers which are a repdigit in one or more bases
(Sloane's A167782). "111" in various bases falls on straight lines. Numbers
"[16][16][16]" in bases 17,19,21,etc are a horizontal at Y=3 because they're perfect
squares, and "[64][64][64]" in base 65,66,etc go a vertically downwards from X=12,Y=-266
similarly because they're squares.
Each arm is N=4*k+rem for a remainder rem=0,1,2,3, so sequences related to multiples of 4
or with a modulo 4 pattern may fall on particular arms.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$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 dlo-1 square,
Nlo = / 1 if dlo=0
\ (2*(dlo-1)+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 off-diagonal 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.
SEE ALSO
Math::PlanePath, Math::PlanePath::DiamondArms, Math::PlanePath::HexArms,
Math::PlanePath::SquareSpiral
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015 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/>.