NAME

       Math::PlanePath::KnightSpiral -- integer points around a square, by chess knight moves

SYNOPSIS

        use Math::PlanePath::KnightSpiral;
        my $path = Math::PlanePath::KnightSpiral->new;
        my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

       This path traverses the plane by an infinite "knight's tour" in the form of a square
       spiral.

                                   ...
               21   4   9  14  19                 2

               10  15  20   3   8      28         1

                5  22   1  18  13            <- y=0

               16  11  24   7   2  27             1

               23   6  17  12  25                 2

                                       26

                        ^
               -2  -1  x=0  1   2   3

       Each step is a chess knight's move 1 across and 2 along, or vice versa.  The pattern makes
       4 cycles on a 2-wide path around a square before stepping outwards to do the same again to
       a now bigger square.  The above sample shows the first 4-cycle around the central 1, then
       stepping out at 26 and beginning to go around the outside of the 5x5 square.

       An attractive traced out picture of the path can be seen at the following page (quarter
       way down under "Open Knight's Tour"),

           <http://www.borderschess.org/KTart.htm> <http://www.borderschess.org/KTinfinity.gif>
           <http://www.borderschess.org/Infinite.gif>

       See math-image to draw the path lines too.

FUNCTIONS

       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

       "$path = Math::PlanePath::KnightSpiral->new ()"
           Create and return a new knight spiral 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, it being considered the path starts at 1.

       "$n = $path->xy_to_n ($x,$y)"
           Return the point number for coordinates "$x,$y".  $x and $y are each rounded to the
           nearest integer, which has the effect of treating each N in the path as centred in a
           square of side 1, so the entire plane is covered.

OEIS

       This Knight's tour is in Sloane's OEIS following the Knight spiral and giving the
       resulting X,Y location by the "SquareSpiral" numbering.  There's eight forms for 4
       rotations and spiralling the same or opposite directions.

           <http://oeis.org/A068608> (etc)

           permutations
             A068608   same knight and square spiral directions
             A068609   rotate 90 degrees
             A068610   rotate 180 degrees
             A068611   rotate 270 degrees
             A068612   rotate 180 degrees, spiral opp dir (X negate)
             A068613   rotate 270 degrees, spiral opp dir
             A068614   spiral opposite direction (Y negate)
             A068615   rotate 90 degrees, spiral opp dir (X,Y transpose)

       See examples/knights-oeis.pl for a sample program printing the values of A068608.

SEE ALSO

       Math::PlanePath, Math::PlanePath::SquareSpiral

HOME PAGE

       <http://user42.tuxfamily.org/math-planepath/index.html>

LICENSE

       Copyright 2010, 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/>.