NAME

Math::PlanePath::KnightSpiral -- integer points drawn around a square

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 with 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 then 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 now 5x5 square.

A 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

The math-image program can draw the path lines too. And see examples/knights-sloane.pl expressing the knight's tour by the numbering of the SquareSpiral (sequence A068608 of Sloane's On-Line Encyclopedia of Integer Sequences).

FUNCTIONS

$path = Math::PlanePath::KnightSpiral->new (key=>value, ...)

Create and return a new square 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.

SEE ALSO

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

HOME PAGE

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

LICENSE

Math-PlanePath is Copyright 2010 Kevin Ryde

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/>.