/ 
Math-PlanePath-25
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::KnightSpiral

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 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 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

See math-image to draw the path lines too.

FUNCTIONS

$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

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 generating A068608. (As of March 2011 there's a typo in the A068609 sample numbers, the second "37" should be "39".)

SEE ALSO

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

HOME PAGE

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

LICENSE

Math-PlanePath is Copyright 2010, 2011 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/>.