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.

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 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/>.